[{"data":1,"prerenderedAt":1093},["ShallowReactive",2],{"questions:shape-and-space":3},{"bank":4,"contentHash":1081,"dependencyHashes":1082,"releaseId":1092},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":46,"sourceIds":1068,"reviewStatus":1077,"authoring":1078},1,"shape-and-space","Shape and space: question bank","Practise every part of shape and space, from naming polygons to Euler’s formula. Questions are grouped by skill and graded **foundation**, **core**, **stretch** and **challenge**. Every question has a full worked solution: try first, then read it even when you are right, because the method matters more than the answer.",[10,14,18,22,26,30,34,38,42],{"id":11,"title":12,"description":13},"polygons","2D shapes and polygons","2D versus 3D, open and closed curves, polygons named by sides, regular polygons, diagonals and perimeter.",{"id":15,"title":16,"description":17},"triangles","Triangles","Classifying triangles by sides and angles, the 180° angle sum, exterior angles and the triangle inequality.",{"id":19,"title":20,"description":21},"quadrilaterals","Quadrilaterals","Square, rectangle, rhombus, parallelogram, trapezium and kite: definitions, properties, diagonals and the family tree.",{"id":23,"title":24,"description":25},"circles","Circles","Centre, radius, diameter, chord, arc, sector and circumference, with wheels and the Ashoka Chakra.",{"id":27,"title":28,"description":29},"solids","3D solids: faces, edges, vertices","Counting F, E and V for prisms, pyramids and curved solids, and using Euler’s formula.",{"id":31,"title":32,"description":33},"nets","Nets","Which flat patterns fold into cubes, cylinders and pyramids; opposite faces; surface area from a net.",{"id":35,"title":36,"description":37},"views","Views","Top, front and side views, plans and elevations, and counting cubes from views.",{"id":39,"title":40,"description":41},"symmetry","Symmetry","Lines of symmetry of shapes and letters, and rotational symmetry.",{"id":43,"title":44,"description":45},"reasoning","Reasoning and puzzles","Tangrams, tilings, angle sums, painted cubes, chessboards, footballs and Platonic solids.",[47,72,84,96,112,132,143,153,163,174,184,195,212,223,241,260,270,279,289,301,320,330,343,354,371,387,397,406,427,446,465,475,486,503,514,532,543,552,562,571,581,599,607,617,626,635,651,661,680,687,697,714,729,739,748,756,767,783,795,806,824,834,843,853,871,879,896,913,921,936,945,953,973,982,991,1001,1012,1023,1032,1049],{"id":48,"section":11,"level":49,"prompt":50,"check":51,"hints":67,"solution":69,"skills":70},"shape-and-space.q001","foundation","Which of these is a **3D** shape?",{"kind":52,"options":53,"correct":66},"choice",[54,57,60,63],{"id":55,"label":56},"a","A drawing of a square",{"id":58,"label":59},"b","A rangoli pattern",{"id":61,"label":62},"c","The shadow of a ball",{"id":64,"label":65},"d","A brick",[64],[68],"Which one could you pick up and turn round?","A brick has length, breadth **and** height, so it takes up space: it is a 3D shape (a cuboid). A drawing, a rangoli and a shadow all lie flat, so they are 2D.",[71],"2D vs 3D",{"id":73,"section":11,"level":49,"prompt":74,"check":75,"hints":79,"solution":81,"skills":82},"shape-and-space.q002","What is the name of a polygon with **8** sides?",{"kind":76,"accept":77},"text",[78],"octagon",[80],"Think of an octopus, which has 8 arms.","A polygon with 8 sides is an **octagon** (*octa* means eight, as in octopus). A STOP sign is an octagon.",[83],"naming polygons",{"id":85,"section":11,"level":49,"prompt":86,"check":87,"hints":91,"solution":93,"skills":94},"shape-and-space.q003","How many vertices does a pentagon have?",{"kind":88,"answer":89,"tolerance":90},"number",5,0,[92],"Count sides and corners together: they always match.","A polygon always has as many vertices as sides. A pentagon has 5 sides, so it has **5** vertices.",[95],"vertices",{"id":97,"section":11,"level":49,"prompt":98,"check":99,"hints":108,"solution":109,"skills":110},"shape-and-space.q004","Is a circle a polygon?",{"kind":52,"options":100,"correct":107},[101,104],{"id":102,"label":103},"t","True",{"id":105,"label":106},"f","False",[105],[],"**False.** A polygon must be made only of straight line segments. A circle is one smooth curve, with no straight sides and no vertices.",[111],"polygon definition",{"id":113,"section":11,"level":114,"prompt":115,"check":116,"hints":127,"solution":129,"skills":130},"shape-and-space.q005","core","Which figure is a **simple closed curve**?",{"kind":52,"options":117,"correct":126},[118,120,122,124],{"id":55,"label":119},"The letter C",{"id":58,"label":121},"The digit 8",{"id":61,"label":123},"The letter S",{"id":64,"label":125},"The letter O",[64],[128],"Closed means it ends where it began; simple means it does not cross itself.","O starts and ends at the same point and never crosses itself: a **simple closed curve**. C and S are open; 8 is closed but crosses itself, so it is not simple.",[131],"curves",{"id":133,"section":11,"level":114,"prompt":134,"check":135,"hints":137,"solution":140,"skills":141},"shape-and-space.q006","How many diagonals does a heptagon (7 sides) have?",{"kind":88,"answer":136,"tolerance":90},14,[138,139],"From one vertex you cannot draw a diagonal to itself or its 2 neighbours.","Multiply, then halve.","From each vertex, 7 − 3 = 4 diagonals can be drawn. 7 × 4 = 28 counts every diagonal twice (once from each end), so there are 28 ÷ 2 = **14** diagonals.",[142],"diagonals",{"id":144,"section":11,"level":114,"prompt":145,"check":146,"hints":148,"solution":150,"skills":151},"shape-and-space.q007","The outline of a five-pointed star (cut out of paper) is a polygon. How many sides does it have?",{"kind":88,"answer":147,"tolerance":90},10,[149],"Count both the points and the inner corners.","Going round the outline you pass 5 outer points and 5 inner corners: **10** vertices, so **10** sides. It is a concave decagon.",[11,152],"concave",{"id":154,"section":11,"level":155,"prompt":156,"check":157,"hints":159,"solution":161,"skills":162},"shape-and-space.q008","stretch","A polygon has **90** diagonals. How many sides does it have?",{"kind":88,"answer":158,"tolerance":90},15,[160],"Double 90 and look for two numbers that differ by 3 with that product.","We need n(n − 3) ÷ 2 = 90, so n(n − 3) = 180. Look for two numbers 3 apart that multiply to 180: 15 × 12 = 180. So n = **15**. Check: 15 × 12 ÷ 2 = 90. ✓",[142,43],{"id":164,"section":11,"level":114,"prompt":165,"check":166,"hints":169,"solution":171,"skills":172},"shape-and-space.q009","A rectangular school ground is 35 m long and 22 m wide. How far do you run in **one lap** round its edge?",{"kind":88,"answer":167,"tolerance":90,"unit":168},114,"m",[170],"Add length and breadth, then double.","Perimeter of a rectangle = 2 × (length + breadth) = 2 × (35 + 22) = 2 × 57 = **114 m**.",[173],"perimeter",{"id":175,"section":11,"level":49,"prompt":176,"check":177,"hints":180,"solution":182,"skills":183},"shape-and-space.q010","A regular hexagonal table top has sides of 14 cm. How long is the lace needed to go once round its edge?",{"kind":88,"answer":178,"tolerance":90,"unit":179},84,"cm",[181],"How many equal sides does a hexagon have?","A regular hexagon has 6 equal sides: 6 × 14 = **84 cm**.",[173],{"id":185,"section":15,"level":49,"prompt":186,"check":187,"hints":191,"solution":192,"skills":193},"shape-and-space.q011","A triangle has sides 6 cm, 6 cm and 6 cm. What kind of triangle is it (by its sides)?",{"kind":76,"accept":188},[189,190],"equilateral","equilateral triangle",[],"All three sides are equal, so it is **equilateral**. Its angles are all 60° too.",[194],"classifying triangles",{"id":196,"section":15,"level":49,"prompt":197,"check":198,"hints":209,"solution":210,"skills":211},"shape-and-space.q012","A triangle has one angle of exactly 90°. What is it called?",{"kind":52,"options":199,"correct":208},[200,202,204,206],{"id":55,"label":201},"Acute-angled",{"id":58,"label":203},"Right-angled",{"id":61,"label":205},"Obtuse-angled",{"id":64,"label":207},"Equilateral",[58],[],"A triangle with a 90° angle is **right-angled**. Set squares are right-angled triangles.",[194],{"id":213,"section":15,"level":114,"prompt":214,"check":215,"hints":218,"solution":220,"skills":221},"shape-and-space.q013","Two angles of a triangle are 72° and 45°. Find the third angle.",{"kind":88,"answer":216,"tolerance":90,"unit":217},63,"°",[219],"The three angles add to 180°.","Angles of a triangle add to 180°. 72° + 45° = 117°, so the third angle is 180° − 117° = **63°**.",[222],"angle sum",{"id":224,"section":15,"level":114,"prompt":225,"check":226,"hints":237,"solution":239,"skills":240},"shape-and-space.q014","Can a triangle have angles 95°, 50° and 35°? If so, classify it by angles.",{"kind":52,"options":227,"correct":236},[228,230,232,234],{"id":55,"label":229},"No, the angles do not add to 180°",{"id":58,"label":231},"Yes, it is obtuse-angled",{"id":61,"label":233},"Yes, it is acute-angled",{"id":64,"label":235},"Yes, it is right-angled",[58],[238],"Check the total first, then look at the largest angle.","95 + 50 + 35 = 180, so the triangle exists. Its largest angle, 95°, is more than 90°, so it is **obtuse-angled**.",[222,194],{"id":242,"section":15,"level":114,"prompt":243,"check":244,"hints":255,"solution":257,"skills":258},"shape-and-space.q015","Which set of lengths **cannot** form a triangle?",{"kind":52,"options":245,"correct":254},[246,248,250,252],{"id":55,"label":247},"5 cm, 7 cm, 10 cm",{"id":58,"label":249},"4 cm, 4 cm, 7 cm",{"id":61,"label":251},"3 cm, 6 cm, 10 cm",{"id":64,"label":253},"8 cm, 8 cm, 8 cm",[61],[256],"Add the two shorter sides and compare with the longest.","The two shorter sides must add to more than the longest. 3 + 6 = 9, which is less than 10, so **3, 6, 10 cannot** form a triangle. The others pass: 5 + 7 = 12 > 10, 4 + 4 = 8 > 7, 8 + 8 = 16 > 8.",[259],"triangle inequality",{"id":261,"section":15,"level":155,"prompt":262,"check":263,"hints":265,"solution":267,"skills":268},"shape-and-space.q016","An isosceles triangle has one angle of 40°, and this is one of the two **equal** angles. Find the largest angle of the triangle.",{"kind":88,"answer":264,"tolerance":90,"unit":217},100,[266],"The two equal angles are both 40°.","The two equal angles are 40° each: 40° + 40° = 80°. The third angle is 180° − 80° = **100°**, which is the largest. (The triangle is obtuse-angled.)",[269,222],"isosceles",{"id":271,"section":15,"level":155,"prompt":272,"check":273,"hints":275,"solution":277,"skills":278},"shape-and-space.q017","Two sides of a triangle are 6 cm and 10 cm. The third side is a whole number of centimetres. How many different lengths are possible?",{"kind":88,"answer":274,"tolerance":90},11,[276],"The third side lies strictly between the difference and the sum of the other two.","The third side must be more than 10 − 6 = 4 cm and less than 10 + 6 = 16 cm. Whole numbers from 5 to 15: that is 15 − 5 + 1 = **11** lengths.",[259],{"id":280,"section":15,"level":155,"prompt":281,"check":282,"hints":284,"solution":286,"skills":287},"shape-and-space.q018","In triangle PQR, angle P = 58° and angle Q = 67°. Side QR is extended beyond R. What is the exterior angle at R?",{"kind":88,"answer":283,"tolerance":90,"unit":217},125,[285],"Exterior angle = sum of the two opposite interior angles.","The exterior angle equals the sum of the two interior opposite angles: 58° + 67° = **125°**. (Check: interior angle R = 180° − 125° = 55°, and 55° + 125° = 180° on the straight line.)",[288],"exterior angle",{"id":290,"section":15,"level":291,"prompt":292,"check":293,"hints":295,"solution":298,"skills":299},"shape-and-space.q019","challenge","How many different triangles with **whole-number** sides have a perimeter of **15 cm**? (Triangles with the same three side lengths in a different order count as the same.)",{"kind":88,"answer":294,"tolerance":90},7,[296,297],"Let the longest side be c. It must be less than half the perimeter.","Try c = 7, 6 and 5 in turn.","List sides a ≤ b ≤ c with a + b + c = 15 and a + b > c. Since a + b > c, c is less than 7.5, so c ≤ 7; and c ≥ 5 because it is the largest. c = 7: (1,7,7), (2,6,7), (3,5,7), (4,4,7). c = 6: (3,6,6), (4,5,6). c = 5: (5,5,5). Total **7** triangles.",[259,300],"systematic listing",{"id":302,"section":15,"level":114,"prompt":303,"check":304,"hints":315,"solution":317,"skills":318},"shape-and-space.q020","**Spot the mistake.** Riya says: “My triangle has angles 90°, 60° and 40°, so it is right-angled.” What is wrong?",{"kind":52,"options":305,"correct":314},[306,308,310,312],{"id":55,"label":307},"Nothing: it is right-angled",{"id":58,"label":309},"It is obtuse-angled, not right-angled",{"id":61,"label":311},"The angles add to 190°, so no such triangle exists",{"id":64,"label":313},"A right angle must be the smallest angle",[61],[316],"Add the angles.","90 + 60 + 40 = **190°**, but a triangle’s angles always add to 180°. No triangle has these angles, so it cannot be classified at all.",[222,319],"spot the mistake",{"id":321,"section":19,"level":49,"prompt":322,"check":323,"hints":327,"solution":328,"skills":329},"shape-and-space.q021","Which quadrilateral has four equal sides and four right angles?",{"kind":76,"accept":324},[325,326],"square","a square",[],"Four equal sides and four right angles make a **square**.",[19],{"id":331,"section":19,"level":49,"prompt":332,"check":333,"hints":338,"solution":340,"skills":341},"shape-and-space.q022","True or false: every square is a rectangle.",{"kind":52,"options":334,"correct":337},[335,336],{"id":102,"label":103},{"id":105,"label":106},[102],[339],"What is the definition of a rectangle?","**True.** A rectangle is a quadrilateral with four right angles. A square has four right angles, so it passes the test. It is a special rectangle with equal sides.",[342],"quadrilateral family",{"id":344,"section":19,"level":114,"prompt":345,"check":346,"hints":351,"solution":352,"skills":353},"shape-and-space.q023","True or false: every rectangle is a square.",{"kind":52,"options":347,"correct":350},[348,349],{"id":102,"label":103},{"id":105,"label":106},[105],[],"**False.** A 6 cm by 4 cm rectangle has four right angles but unequal sides, so it is not a square. Squares are only the rectangles whose sides are all equal.",[342],{"id":355,"section":19,"level":114,"prompt":356,"check":357,"hints":368,"solution":369,"skills":370},"shape-and-space.q024","Which quadrilateral has exactly **one** line of symmetry (in general)?",{"kind":52,"options":358,"correct":367},[359,361,363,365],{"id":55,"label":360},"Rectangle",{"id":58,"label":362},"Rhombus",{"id":61,"label":364},"Kite",{"id":64,"label":366},"Parallelogram",[61],[],"A **kite** folds onto itself along the diagonal between its two pairs of equal sides: 1 line. A rectangle and a rhombus have 2 each; a general parallelogram has none.",[39,19],{"id":372,"section":19,"level":114,"prompt":373,"check":374,"hints":382,"solution":384,"skills":385},"shape-and-space.q025","The diagonals of a quadrilateral bisect each other and are equal in length, but they are **not** perpendicular. What is its most special name?",{"kind":52,"options":375,"correct":381},[376,378,379,380],{"id":55,"label":377},"Square",{"id":58,"label":362},{"id":61,"label":360},{"id":64,"label":364},[61],[383],"Bisecting diagonals = parallelogram. What does “equal” add?","Diagonals that bisect each other make it a parallelogram. Equal diagonals make that parallelogram a **rectangle**. If they were also perpendicular it would be a square.",[386],"diagonal properties",{"id":388,"section":19,"level":114,"prompt":389,"check":390,"hints":392,"solution":394,"skills":395},"shape-and-space.q026","In a parallelogram, one angle is 65°. What is the angle next to it?",{"kind":88,"answer":391,"tolerance":90,"unit":217},115,[393],"Neighbouring angles of a parallelogram add to 180°.","Neighbouring angles of a parallelogram lie between a pair of parallel sides, so they add to 180°: 180° − 65° = **115°**.",[396],"parallelogram",{"id":398,"section":19,"level":114,"prompt":399,"check":400,"hints":402,"solution":404,"skills":405},"shape-and-space.q027","Three angles of a quadrilateral are 85°, 95° and 110°. Find the fourth angle.",{"kind":88,"answer":401,"tolerance":90,"unit":217},70,[403],"The angles of a quadrilateral add to 360°.","A diagonal splits a quadrilateral into two triangles, so its angles add to 2 × 180° = 360°. 85 + 95 + 110 = 290, so the fourth angle is 360° − 290° = **70°**.",[222],{"id":407,"section":19,"level":155,"prompt":408,"check":409,"hints":423,"solution":425,"skills":426},"shape-and-space.q028","Which statements are **always** true? Choose all that apply.",{"kind":52,"options":410,"correct":422},[411,413,415,417,419],{"id":55,"label":412},"A rhombus is a parallelogram",{"id":58,"label":414},"A square is a rhombus",{"id":61,"label":416},"A parallelogram is a rectangle",{"id":64,"label":418},"A rectangle has equal diagonals",{"id":420,"label":421},"e","A kite has two pairs of parallel sides",[55,58,64],[424],"Test each statement against the definitions.","A rhombus has both pairs of opposite sides parallel (**a** ✓). A square has four equal sides (**b** ✓). A rectangle’s diagonals are always equal (**d** ✓). A parallelogram is a rectangle only if it has right angles (c ✗). A kite usually has no parallel sides at all (e ✗).",[342],{"id":428,"section":19,"level":155,"prompt":429,"check":430,"hints":441,"solution":443,"skills":444},"shape-and-space.q029","A rhombus has one angle of 50°. What are its other three angles, going round in order?",{"kind":52,"options":431,"correct":440},[432,434,436,438],{"id":55,"label":433},"50°, 50°, 50°",{"id":58,"label":435},"130°, 50°, 130°",{"id":61,"label":437},"130°, 130°, 50°",{"id":64,"label":439},"90°, 90°, 130°",[58],[442],"A rhombus is a special parallelogram.","A rhombus is a parallelogram, so neighbouring angles add to 180° and opposite angles are equal. Going round: 50°, **130°, 50°, 130°**. Check: 50 + 130 + 50 + 130 = 360.",[445],"rhombus",{"id":447,"section":19,"level":291,"prompt":448,"check":449,"hints":460,"solution":462,"skills":463},"shape-and-space.q030","A quadrilateral has all four sides equal **and** equal diagonals. Prove which shape it must be, and choose the answer.",{"kind":52,"options":450,"correct":459},[451,453,455,457],{"id":55,"label":452},"Any rhombus",{"id":58,"label":454},"A kite that is not a rhombus",{"id":61,"label":456},"A rectangle that is not a square",{"id":64,"label":458},"A square",[64],[461],"First name it from its sides, then use the diagonal fact.","Four equal sides make it a rhombus, so it is a parallelogram whose diagonals bisect each other. A parallelogram with **equal** diagonals is a rectangle (it has right angles). A shape that is both a rhombus and a rectangle is a **square**.",[464,342],"proof",{"id":466,"section":23,"level":49,"prompt":467,"check":468,"hints":470,"solution":472,"skills":473},"shape-and-space.q031","A circular clock has a radius of 12 cm. What is its diameter?",{"kind":88,"answer":469,"tolerance":90,"unit":179},24,[471],"Diameter = 2 × radius.","The diameter is twice the radius: 2 × 12 = **24 cm**.",[474],"radius and diameter",{"id":476,"section":23,"level":49,"prompt":477,"check":478,"hints":482,"solution":483,"skills":484},"shape-and-space.q032","What is a line segment from the centre of a circle to any point on the circle called?",{"kind":76,"accept":479},[480,481],"radius","a radius",[],"It is a **radius**. All radii of a circle are the same length.",[485],"parts of a circle",{"id":487,"section":23,"level":114,"prompt":488,"check":489,"hints":500,"solution":501,"skills":502},"shape-and-space.q033","Which is the longest chord of a circle?",{"kind":52,"options":490,"correct":499},[491,493,495,497],{"id":55,"label":492},"A radius",{"id":58,"label":494},"An arc",{"id":61,"label":496},"A diameter",{"id":64,"label":498},"All chords are equal",[61],[],"A chord joins two points on the circle. The one through the centre, the **diameter**, is the longest. A radius is not a chord (one end is at the centre), and an arc is curved.",[485],{"id":504,"section":23,"level":114,"prompt":505,"check":506,"hints":509,"solution":511,"skills":512},"shape-and-space.q034","A **cycle-rickshaw** wheel has a diameter of 50 cm. Using π ≈ 3.14, how far does the rickshaw move in one turn of the wheel?",{"kind":88,"answer":507,"tolerance":508,"unit":179},157,0.5,[510],"Distance per turn = circumference = π × diameter.","In one turn the wheel rolls out its whole circumference: C = π × d ≈ 3.14 × 50 = **157.0 cm** (about 1.57 m).",[513],"circumference",{"id":515,"section":23,"level":114,"prompt":516,"check":517,"hints":528,"solution":530,"skills":531},"shape-and-space.q035","A slice of a round cake cut from the centre, with two straight cuts and a curved crust, is which part of a circle?",{"kind":52,"options":518,"correct":527},[519,521,523,525],{"id":55,"label":520},"Segment",{"id":58,"label":522},"Chord",{"id":61,"label":524},"Semicircle",{"id":64,"label":526},"Sector",[64],[529],"Two straight cuts from the centre…","The region between two radii and the arc joining them is a **sector**. A segment is the region between a chord and its arc.",[485],{"id":533,"section":23,"level":155,"prompt":534,"check":535,"hints":537,"solution":540,"skills":541},"shape-and-space.q036","A cycle wheel has a diameter of 70 cm. About how many complete turns does it make in 1 km? (Use π ≈ 22\u002F7.)",{"kind":88,"answer":536,"tolerance":5},455,[538,539],"Find the circumference in metres first.","1 km = 1,000 m.","Circumference = 22\u002F7 × 70 = 220 cm = 2.2 m. Number of turns = 1,000 m ÷ 2.2 m ≈ 454.5, so about **455** turns (454 complete turns plus a bit).",[513,542],"word problem",{"id":544,"section":23,"level":155,"prompt":545,"check":546,"hints":547,"solution":549,"skills":550},"shape-and-space.q037","The Ashoka Chakra has 24 equally spaced spokes. What angle, in degrees, is between two neighbouring spokes?",{"kind":88,"answer":158,"tolerance":90,"unit":217},[548],"A full turn is 360°.","The spokes share the full turn of 360° equally: 360° ÷ 24 = **15°**.",[551],"angles at the centre",{"id":553,"section":23,"level":291,"prompt":554,"check":555,"hints":557,"solution":559,"skills":560},"shape-and-space.q038","Circle A has a diameter of 14 cm. Circle B has a circumference **three times** as long as circle A’s. What is the radius of circle B?",{"kind":88,"answer":556,"tolerance":90,"unit":179},21,[558],"If C = π × d, what happens to d when C is tripled?","Circumference is proportional to diameter (C = π × d). Three times the circumference means three times the diameter: 3 × 14 = 42 cm. The radius is half of that: **21 cm**.",[513,561],"proportion",{"id":563,"section":27,"level":49,"prompt":564,"check":565,"hints":567,"solution":568,"skills":569},"shape-and-space.q039","How many faces does a cube have?",{"kind":88,"answer":566,"tolerance":90},6,[],"A cube has **6** square faces: top, bottom, front, back, left and right. A dice shows this: numbers 1 to 6, one on each face.",[570],"faces",{"id":572,"section":27,"level":49,"prompt":573,"check":574,"hints":576,"solution":578,"skills":579},"shape-and-space.q040","How many edges does a cuboid (like a matchbox) have?",{"kind":88,"answer":575,"tolerance":90},12,[577],"Top ring, bottom ring, then the upright edges.","Count by layers: 4 edges round the top, 4 round the bottom and 4 upright ones joining them: 4 + 4 + 4 = **12**.",[580],"edges",{"id":582,"section":27,"level":49,"prompt":583,"check":584,"hints":595,"solution":596,"skills":597},"shape-and-space.q041","Which solid has one circular flat face, one curved surface and one vertex?",{"kind":52,"options":585,"correct":594},[586,588,590,592],{"id":55,"label":587},"Cone",{"id":58,"label":589},"Cylinder",{"id":61,"label":591},"Sphere",{"id":64,"label":593},"Square pyramid",[55],[],"A **cone**: a circular base, a curved surface narrowing to the tip, and one vertex (the apex). A cylinder has 2 flat faces and no vertex; a sphere has no flat face.",[598],"curved solids",{"id":600,"section":27,"level":114,"prompt":601,"check":602,"hints":603,"solution":605,"skills":606},"shape-and-space.q042","How many vertices does a square pyramid have?",{"kind":88,"answer":89,"tolerance":90},[604],"Base corners plus the top.","4 vertices round the square base and 1 at the apex: **5**.",[95],{"id":608,"section":27,"level":114,"prompt":609,"check":610,"hints":612,"solution":614,"skills":615},"shape-and-space.q043","A tent is shaped like a triangular prism. How many edges does it have?",{"kind":88,"answer":611,"tolerance":90},9,[613],"Count each triangle end, then the long edges.","3 edges round each triangular end (6) and 3 edges running along the length: 6 + 3 = **9**. In general a prism has 3n edges; 3 × 3 = 9.",[580,616],"prisms",{"id":618,"section":27,"level":155,"prompt":619,"check":620,"hints":621,"solution":623,"skills":624},"shape-and-space.q044","An octagonal prism has how many faces, edges and vertices? Enter the number of **edges**.",{"kind":88,"answer":469,"tolerance":90},[622],"Edges of a prism = 3 × number of sides of the base.","For a prism with an n-sided base: faces n + 2, edges 3n, vertices 2n. With n = 8: 10 faces, **24 edges**, 16 vertices. Check Euler: 10 + 16 − 24 = 2 ✓.",[616,625],"Euler",{"id":627,"section":27,"level":155,"prompt":628,"check":629,"hints":631,"solution":633,"skills":634},"shape-and-space.q045","A polyhedron has 14 faces and 24 vertices. Use Euler’s formula to find its number of edges.",{"kind":88,"answer":630,"tolerance":90},36,[632],"F + V − E = 2.","F + V − E = 2, so 14 + 24 − E = 2 and E = 38 − 2 = **36**.",[625],{"id":636,"section":27,"level":114,"prompt":637,"check":638,"hints":647,"solution":648,"skills":649},"shape-and-space.q046","Which of these is **not** a polyhedron?",{"kind":52,"options":639,"correct":646},[640,641,643,645],{"id":55,"label":589},{"id":58,"label":642},"Cube",{"id":61,"label":644},"Triangular prism",{"id":64,"label":593},[55],[],"A polyhedron has only flat polygon faces. A **cylinder** has a curved surface, so it is not a polyhedron.",[650],"polyhedra",{"id":652,"section":27,"level":291,"prompt":653,"check":654,"hints":656,"solution":658,"skills":659},"shape-and-space.q047","A pyramid has **30 edges**. How many faces and vertices does it have? Enter the number of **faces**.",{"kind":88,"answer":655,"tolerance":90},16,[657],"Edges of a pyramid = 2 × base sides.","A pyramid with an n-sided base has 2n edges, so 2n = 30 and n = 15. Faces = n + 1 = **16** (15 triangles and the base); vertices = n + 1 = 16. Check: 16 + 16 − 30 = 2 ✓.",[660,625],"pyramids",{"id":662,"section":27,"level":291,"prompt":663,"check":664,"hints":675,"solution":678,"skills":679},"shape-and-space.q048","Can a polyhedron have exactly **7** edges?",{"kind":52,"options":665,"correct":674},[666,668,670,672],{"id":55,"label":667},"Yes, a triangular pyramid with one extra edge",{"id":58,"label":669},"No: this contradicts Euler’s formula together with 2E ≥ 3F and 2E ≥ 3V",{"id":61,"label":671},"Yes, but only if it has curved faces",{"id":64,"label":673},"We cannot know",[58],[676,677],"How many edges must each face have at least?","Combine with F + V = E + 2.","Each face has at least 3 edges and each edge is shared by 2 faces, so 2E ≥ 3F; similarly 2E ≥ 3V. With E = 7: F ≤ 4 and V ≤ 4, so F + V ≤ 8. But Euler requires F + V = E + 2 = 9. Contradiction, so **no polyhedron has 7 edges**.",[625,464],{"id":681,"section":31,"level":49,"prompt":682,"check":683,"hints":684,"solution":685,"skills":686},"shape-and-space.q049","How many squares are there in the net of a cube?",{"kind":88,"answer":566,"tolerance":90},[],"A cube has 6 square faces, so its net is made of **6** squares.",[31],{"id":688,"section":31,"level":114,"prompt":689,"check":690,"hints":694,"solution":695,"skills":696},"shape-and-space.q050","Which solid does a net of **2 circles and 1 rectangle** fold into?",{"kind":76,"accept":691},[692,693],"cylinder","a cylinder",[],"The two circles are the ends and the rectangle rolls into the curved surface: a **cylinder**.",[31],{"id":698,"section":31,"level":114,"prompt":699,"check":700,"hints":711,"solution":712,"skills":713},"shape-and-space.q051","The net of a square pyramid contains:",{"kind":52,"options":701,"correct":710},[702,704,706,708],{"id":55,"label":703},"1 square and 4 triangles",{"id":58,"label":705},"2 squares and 4 triangles",{"id":61,"label":707},"4 squares and 1 triangle",{"id":64,"label":709},"1 square and 3 triangles",[55],[],"A square pyramid has 5 faces: **1 square** base and **4 triangles**, one on each side of the base.",[31],{"id":715,"section":31,"level":114,"prompt":716,"check":717,"hints":724,"solution":726,"skills":727},"shape-and-space.q052","Six squares are joined in a single straight row. Does this fold into a cube?",{"kind":52,"options":718,"correct":723},[719,721],{"id":55,"label":720},"Yes",{"id":58,"label":722},"No",[58],[725],"Try it with paper.","**No.** Four squares wrap into a ring; the 5th and 6th land on top of the 1st and 2nd, and the two ends of the ring stay open. A row of 5 or more squares never folds into a cube.",[728],"cube nets",{"id":730,"section":31,"level":155,"prompt":731,"check":732,"hints":735,"solution":737,"skills":738},"shape-and-space.q053","A tin of diameter 8 cm needs a paper label that wraps exactly once round it. Using π ≈ 3.14, how long must the label be?",{"kind":88,"answer":733,"tolerance":734,"unit":179},25.12,0.1,[736],"The rectangle wraps round the circle.","The label is the rectangle in the cylinder’s net. Its length must equal the circumference: 3.14 × 8 = **25.12 cm**.",[31,513],{"id":740,"section":31,"level":155,"prompt":741,"check":742,"hints":743,"solution":745,"skills":746},"shape-and-space.q054","In a cross-shaped cube net, the row of four squares is numbered 1, 2, 6, 5 from left to right; the square above the “2” is 3 and the square below the “2” is 4. Which number is opposite 2 when it is folded?",{"kind":88,"answer":89,"tolerance":90},[744],"In the ring of four, which squares face each other?","In a row of four that wraps into a ring, the 1st and 3rd squares are opposite, and the 2nd and 4th are opposite. So 2 is opposite **5** (and 1 is opposite 6, 3 is opposite 4: every pair adds to 7, a proper dice).",[728,747],"dice",{"id":749,"section":31,"level":291,"prompt":750,"check":751,"hints":752,"solution":754,"skills":755},"shape-and-space.q055","How many different nets does a cube have (turned or flipped copies count as the same)?",{"kind":88,"answer":274,"tolerance":90},[753],"Group them by the longest row.","There are 35 hexominoes (shapes of 6 squares joined edge to edge), and exactly **11** fold into a cube: 6 of the 1-4-1 type, 3 of the 1-3-2 type, one 2-2-2 staircase and one 3-3 shape.",[728],{"id":757,"section":31,"level":155,"prompt":758,"check":759,"hints":762,"solution":764,"skills":765},"shape-and-space.q056","A cube has edges of 7 cm. What is the total area of its net (its surface area)?",{"kind":88,"answer":760,"tolerance":90,"unit":761},294,"cm²",[763],"Area of one face × number of faces.","The net has 6 squares, each 7 × 7 = 49 cm². Total: 6 × 49 = **294 cm²**.",[31,766],"surface area",{"id":768,"section":35,"level":49,"prompt":769,"check":770,"hints":780,"solution":781,"skills":782},"shape-and-space.q057","A cylinder stands upright on a table. What do you see from directly above?",{"kind":52,"options":771,"correct":779},[772,774,776,778],{"id":55,"label":773},"A circle",{"id":58,"label":775},"A rectangle",{"id":61,"label":777},"A triangle",{"id":64,"label":458},[55],[],"From the top you look down on the circular end: a **circle**. From the front it looks like a rectangle.",[35],{"id":784,"section":35,"level":114,"prompt":785,"check":786,"hints":791,"solution":793,"skills":794},"shape-and-space.q058","A solid looks like a **square** from the top and a **triangle** from the front and the side. What is it?",{"kind":76,"accept":787},[788,789,790],"square pyramid","a square pyramid","pyramid",[792],"Which solid has a square base and comes to a point?","A square base seen from above, sloping triangular faces seen from the front and side: a **square pyramid**.",[35],{"id":796,"section":35,"level":114,"prompt":797,"check":798,"hints":803,"solution":804,"skills":805},"shape-and-space.q059","Which solid looks like a **circle** from every direction?",{"kind":76,"accept":799},[800,801,802],"sphere","a sphere","ball",[],"Only a **sphere** looks the same, a circle, from every direction.",[35],{"id":807,"section":35,"level":114,"prompt":808,"check":809,"hints":820,"solution":821,"skills":822},"shape-and-space.q060","Architects call the top view of a building its…",{"kind":52,"options":810,"correct":819},[811,813,815,817],{"id":55,"label":812},"Elevation",{"id":58,"label":814},"Plan",{"id":61,"label":816},"Section",{"id":64,"label":818},"Perspective",[58],[],"The top view is the **plan**. Front and side views are elevations; a section shows an imaginary slice through the building.",[35,823],"architecture",{"id":825,"section":35,"level":155,"prompt":826,"check":827,"hints":829,"solution":831,"skills":832},"shape-and-space.q061","Cubes are stacked in one straight row of 4 columns. The front view shows columns of heights 3, 1, 2 and 2. How many cubes are there?",{"kind":88,"answer":828,"tolerance":90},8,[830],"Is anything hidden behind the columns?","With only one row, nothing is hidden: the front view shows every column’s full height. 3 + 1 + 2 + 2 = **8** cubes.",[35,833],"counting cubes",{"id":835,"section":35,"level":291,"prompt":836,"check":837,"hints":838,"solution":841,"skills":842},"shape-and-space.q062","A stack of cubes has a top view of 6 squares in a 2 × 3 block (2 rows, 3 columns). The front view shows heights 3, 1, 2 from left to right. What is the **largest** number of cubes the stack could have?",{"kind":88,"answer":575,"tolerance":90},[839,840],"Each position has 2 columns, one behind the other.","The front view gives the maximum height in each position.","The front view shows the tallest column in each of the 3 positions. For the most cubes, both columns in each position reach that height: 2 × 3 + 2 × 1 + 2 × 2 = 6 + 2 + 4 = **12**. (The fewest would be 3 + 1 + 2 + 1 + 1 + 1 = 9.)",[35,43],{"id":844,"section":39,"level":49,"prompt":845,"check":846,"hints":848,"solution":850,"skills":851},"shape-and-space.q063","How many lines of symmetry does a rectangle (that is not a square) have?",{"kind":88,"answer":847,"tolerance":90},2,[849],"Try folding a rectangular sheet of paper.","A rectangle has **2** lines of symmetry: one across the middle and one down the middle. The diagonals are **not** lines of symmetry: folding along one, the corners do not match.",[852],"line symmetry",{"id":854,"section":39,"level":49,"prompt":855,"check":856,"hints":867,"solution":868,"skills":869},"shape-and-space.q064","Which letter, written as a plain block capital, has a vertical line of symmetry?",{"kind":52,"options":857,"correct":866},[858,860,862,864],{"id":55,"label":859},"F",{"id":58,"label":861},"N",{"id":61,"label":863},"R",{"id":64,"label":865},"M",[64],[],"**M** is the same on both sides of a vertical line through its middle. F, N and R have no vertical mirror line.",[870],"letters",{"id":872,"section":39,"level":114,"prompt":873,"check":874,"hints":875,"solution":876,"skills":877},"shape-and-space.q065","How many lines of symmetry does a regular hexagon have?",{"kind":88,"answer":566,"tolerance":90},[],"A regular polygon with n sides has n lines of symmetry. A regular hexagon has **6**: 3 through opposite vertices and 3 through midpoints of opposite sides.",[878],"regular polygons",{"id":880,"section":39,"level":114,"prompt":881,"check":882,"hints":893,"solution":894,"skills":895},"shape-and-space.q066","Which of these letters, written as plain block capitals, has **both** a vertical and a horizontal line of symmetry?",{"kind":52,"options":883,"correct":892},[884,886,888,890],{"id":55,"label":885},"A",{"id":58,"label":887},"E",{"id":61,"label":889},"H",{"id":64,"label":891},"K",[61],[],"**H** matches left-to-right and top-to-bottom. A has only a vertical line; E and K have only a horizontal line.",[870],{"id":897,"section":39,"level":114,"prompt":898,"check":899,"hints":910,"solution":911,"skills":912},"shape-and-space.q067","How many lines of symmetry does a circle have?",{"kind":52,"options":900,"correct":909},[901,903,905,907],{"id":55,"label":902},"1",{"id":58,"label":904},"2",{"id":61,"label":906},"4",{"id":64,"label":908},"Infinitely many",[64],[],"Every diameter is a line of symmetry, and there are **infinitely many** diameters.",[852],{"id":914,"section":39,"level":155,"prompt":915,"check":916,"hints":917,"solution":918,"skills":919},"shape-and-space.q068","What is the order of rotational symmetry of a regular pentagon?",{"kind":88,"answer":89,"tolerance":90},[],"A regular pentagon fits onto itself after every turn of 360° ÷ 5 = 72°: 5 positions in a full turn, so order **5**.",[920],"rotational symmetry",{"id":922,"section":39,"level":155,"prompt":923,"check":924,"hints":933,"solution":934,"skills":935},"shape-and-space.q069","Which shape has rotational symmetry of order 2 but **no** lines of symmetry?",{"kind":52,"options":925,"correct":932},[926,927,929,930],{"id":55,"label":360},{"id":58,"label":928},"Isosceles triangle",{"id":61,"label":364},{"id":64,"label":931},"Parallelogram with sides 8 cm and 5 cm and angles 60° and 120°",[64],[],"A **general parallelogram** looks the same after a half turn about its centre, but no fold makes its halves match. A rectangle has 2 lines; an isosceles triangle and a kite have 1 line and no rotational symmetry.",[920],{"id":937,"section":39,"level":291,"prompt":938,"check":939,"hints":941,"solution":943,"skills":944},"shape-and-space.q070","A rangoli design has rotational symmetry of order 8 and also 8 lines of symmetry. What is the smallest angle between two of its lines of symmetry?",{"kind":88,"answer":940,"tolerance":734,"unit":217},22.5,[942],"Lines through the centre split 180° (not 360°), because each line points in two directions.","With order 8, the design repeats every 360° ÷ 8 = 45°. Its 8 mirror lines all pass through the centre and are equally spaced, dividing the half-turn of 180° into 8 equal parts: 180° ÷ 8 = **22.5°**. (Two mirror lines 22.5° apart combine to give a turn of 2 × 22.5° = 45°, matching the order-8 rotation.)",[920,852],{"id":946,"section":43,"level":49,"prompt":947,"check":948,"hints":949,"solution":950,"skills":951},"shape-and-space.q071","A tangram is made of how many pieces?",{"kind":88,"answer":294,"tolerance":90},[],"A tangram has **7** pieces: 2 large triangles, 1 medium triangle, 2 small triangles, 1 square and 1 parallelogram.",[952],"tangram",{"id":954,"section":43,"level":114,"prompt":955,"check":956,"hints":968,"solution":970,"skills":971},"shape-and-space.q072","Which regular polygons can tile a floor with **no gaps** using copies of just one shape? Choose all that apply.",{"kind":52,"options":957,"correct":967},[958,960,961,963,965],{"id":55,"label":959},"Equilateral triangle",{"id":58,"label":377},{"id":61,"label":962},"Regular pentagon",{"id":64,"label":964},"Regular hexagon",{"id":420,"label":966},"Regular octagon",[55,58,64],[969],"Does the interior angle divide 360° exactly?","The angles at a meeting point must total 360°. Triangle 60° (6 fit), square 90° (4 fit), hexagon 120° (3 fit) all work. Pentagon 108° and octagon 135° do not divide 360 exactly, so they leave gaps.",[972],"tiling",{"id":974,"section":43,"level":114,"prompt":975,"check":976,"hints":978,"solution":980,"skills":981},"shape-and-space.q073","Find each interior angle of a regular nonagon (9 sides).",{"kind":88,"answer":977,"tolerance":90,"unit":217},140,[979],"Angle sum of an n-gon is (n − 2) × 180°.","Angle sum = (9 − 2) × 180° = 1,260°. Each of the 9 equal angles = 1,260° ÷ 9 = **140°**. (Or: exterior angle 360° ÷ 9 = 40°, so interior 180° − 40° = 140°.)",[222,878],{"id":983,"section":43,"level":155,"prompt":984,"check":985,"hints":986,"solution":988,"skills":989},"shape-and-space.q074","Each exterior angle of a regular polygon is 30°. How many sides does it have?",{"kind":88,"answer":575,"tolerance":90},[987],"Exterior angles add to 360°.","The exterior angles of a polygon total 360°. 360° ÷ 30° = **12** sides: a regular dodecagon.",[990],"exterior angles",{"id":992,"section":43,"level":155,"prompt":993,"check":994,"hints":995,"solution":997,"skills":998},"shape-and-space.q075","A wooden 3 × 3 × 3 cube is painted on the outside and cut into 27 small cubes. How many small cubes have exactly **two** painted faces?",{"kind":88,"answer":575,"tolerance":90},[996],"Where on the big cube do two painted faces meet?","Cubes with exactly two painted faces sit along the edges but not at the corners. Each of the 12 edges has 3 − 2 = 1 such cube: **12**.",[999,1000],"painted cube","visualisation",{"id":1002,"section":43,"level":155,"prompt":1003,"check":1004,"hints":1007,"solution":1009,"skills":1010},"shape-and-space.q076","With 36 m of fencing, Arjun wants a rectangular vegetable patch with the largest possible area (whole-metre sides). What area can he enclose?",{"kind":88,"answer":1005,"tolerance":90,"unit":1006},81,"m²",[1008],"Length + breadth must be half the perimeter.","Length + breadth = 36 ÷ 2 = 18. Try: 17 × 1 = 17, 14 × 4 = 56, 10 × 8 = 80, **9 × 9 = 81** m². The square gives the largest area.",[173,1011],"optimisation",{"id":1013,"section":43,"level":291,"prompt":1014,"check":1015,"hints":1017,"solution":1019,"skills":1020},"shape-and-space.q077","How many squares of **all sizes** are there on a standard 8 × 8 chessboard?",{"kind":88,"answer":1016,"tolerance":90},204,[1018],"Count 1 × 1, 2 × 2, … 8 × 8 squares separately.","A k × k square fits in (9 − k)² positions. Total = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = **204**.",[1021,1022],"counting","puzzles",{"id":1024,"section":43,"level":291,"prompt":1025,"check":1026,"hints":1028,"solution":1030,"skills":1031},"shape-and-space.q078","A football is stitched from 12 pentagons and 20 hexagons. How many seams (edges) does it have?",{"kind":88,"answer":1027,"tolerance":90},90,[1029],"Count all the patch sides, then remember each seam is shared.","Pentagons give 12 × 5 = 60 sides and hexagons 20 × 6 = 120 sides: 180 in all. Each seam joins two patches, so there are 180 ÷ 2 = **90** seams. (With 32 faces and 60 vertices, 32 + 60 − 90 = 2 ✓.)",[625,1021],{"id":1033,"section":43,"level":114,"prompt":1034,"check":1035,"hints":1046,"solution":1047,"skills":1048},"shape-and-space.q079","Why are wheels circular?",{"kind":52,"options":1036,"correct":1045},[1037,1039,1041,1043],{"id":55,"label":1038},"Circles are the easiest shape to cut",{"id":58,"label":1040},"Circles have no edges",{"id":61,"label":1042},"Circles have the largest perimeter",{"id":64,"label":1044},"Every point of the rim is the same distance from the centre, so the axle stays at a steady height",[64],[],"The axle sits at the centre, exactly one radius above the road at every moment, so the cart rolls smoothly without bumping. A square wheel would lift and drop its axle on every corner.",[23,43],{"id":1050,"section":43,"level":291,"prompt":1051,"check":1052,"hints":1063,"solution":1065,"skills":1066},"shape-and-space.q080","Why are there only five Platonic solids? Choose the best reason.",{"kind":52,"options":1053,"correct":1062},[1054,1056,1058,1060],{"id":55,"label":1055},"Only five have been discovered so far",{"id":58,"label":1057},"Euler’s formula only has five solutions",{"id":61,"label":1059},"At each vertex at least 3 identical regular faces must meet with angles totalling less than 360°; only 5 combinations work",{"id":64,"label":1061},"Only triangles and squares can make solids",[61],[1064],"Try 3 regular hexagons at a vertex.","At least 3 faces meet at a vertex, and their angles must total less than 360° to fold into a corner. Triangles: 3, 4 or 5 (180°, 240°, 300°); squares: 3 (270°); pentagons: 3 (324°). Hexagons already give 360° (flat). So there are exactly **5**, proved, not just found.",[1067,464],"Platonic solids",[1069,1070,1071,1072,1073,1074,1075,1076],"shape-and-space-ncert-class6","shape-and-space-ncert-class7","shape-and-space-ncert-class8","shape-and-space-khan-geometry","shape-and-space-mathsisfun-circle","shape-and-space-mathsisfun-euler","shape-and-space-mathsisfun-platonic","shape-and-space-mathsisfun-quadrilaterals","needs_review",{"generatedBy":1079,"notes":1080},"claude-code","Draft generated locally; numeric answers computed and asserted in Python. Pending owner review.","28c08932e6304fbfb91bd9407f65d961db125077bf7c939dc5aa29b930cb5d88",{"logic:questions":1083,"source:shape-and-space-khan-geometry":1084,"source:shape-and-space-mathsisfun-circle":1085,"source:shape-and-space-mathsisfun-euler":1086,"source:shape-and-space-mathsisfun-platonic":1087,"source:shape-and-space-mathsisfun-quadrilaterals":1088,"source:shape-and-space-ncert-class6":1089,"source:shape-and-space-ncert-class7":1090,"source:shape-and-space-ncert-class8":1091},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","10f385d6be5a688f5df4a8b9ea08a6c101d7bf683d87c29e7d9d2827a5e11963","ff337c822df6bd0ba5ef54450ce49945d4ef414438058441e656ee73ca685947","512952430fd4d2cad9d5ad14eb51fe077bf3637c74e2c604e6458fe725b018b6","4a5b30c2c4240904f42b67c39dc2176e4a85ade7223fe6b2b7f47916eb25e6e5","292166035f591d3d313675b23dcd587f26add442ae362358b916e3e15caa24b4","e1821bd507663f06793be51247da07646b65ba5a734a31c173a63c03f426f078","20d54bcdc9a5cc5930102f965a7435de619c7bc9731d1de2ff67ae1d0307ca5c","9516472286b1156fa1275e7ea5b6574beff3acc596b8dfe8563f5c446cb7d6b6","preview-7e1cbbcc4f",1789899599942]