Geometry and measures, Year 10: pyramids, cones, spheres and sectors

120 cm³

HintWork out the area of the square first, then take the fraction named in the formula.

The whyBase area = 6 × 6 = 36 cm². Volume = 1/3 × 36 × 10 = 120 cm³. A pyramid fills exactly one third of the prism with the same base and height.

12π cm³

HintThe formula's h is measured straight up the middle, not along the sloping edge.

The whyVolume = 1/3 × π × 3² × 4 = 1/3 × 36π = 12π cm³. The slant height is not used for volume; it belongs to the curved surface area. AQA gives the cone formula in the question.

36π cm³

HintCube the radius first, then deal with the fraction.

The whyr³ = 27. Volume = 4/3 × π × 27 = 4 × 9 × π = 36π cm³. AQA gives the sphere formula in the question.

one third

HintCompare πr²h with the formula for the pointed solid.

The whyCylinder: πr²h. Cone: 1/3 × πr²h. The same fraction links a pyramid to the prism with the same base and height.

12

HintPut the known values into the formula, cancel the symbol on both sides, and solve for the missing letter.

The why100π = 1/3 × π × 25 × h. Dividing by π gives 100 = 25h ÷ 3, so 25h = 300 and h = 12 cm.

100π cm²

HintSquare the radius before multiplying by anything else.

The why4 × π × 5² = 4 × 25 × π = 100π cm². That is exactly four times the area of a circle with the same radius. AQA gives this formula in the question.

13

HintThe radius, the perpendicular height and the sloping edge make a right-angled triangle.

The whyThe slant height l is the hypotenuse: l² = 5² + 12² = 25 + 144 = 169, so l = 13 cm. The curved surface area formula, π × r × l, needs the slant height, so this step often comes first.

60; 96

HintFind the sloping part first, then remember the flat face the cone stands on.

The whyCurved surface = π × 6 × 10 = 60π cm². The base is a circle of area π × 6² = 36π cm². Total = 60π + 36π = 96π cm². AQA gives the curved surface formula in the question.

340 cm²

HintCount the faces: one four-sided, the rest three-sided and all alike.

The whyBase = 10 × 10 = 100 cm². Each triangular face = 1/2 × 10 × 12 = 60 cm², and there are four of them: 240 cm². Total = 100 + 240 = 340 cm².

48π cm²

HintHalf of the ball's skin is only part of it: cutting a ball in two exposes a new flat face.

The whyCurved part = half of 4 × π × 4² = 32π cm². The flat face is a circle of area π × 4² = 16π cm². Total = 32π + 16π = 48π cm².

1/8

HintCompare the angle with a complete turn.

The whyA full turn is 360°, so the sector is 45/360 = 1/8 of the circle. The same fraction gives the sector's share of both the area and the circumference.

4π cm

HintFind the whole distance round the circle, then take the share that the angle gives.

The whyCircumference = 2 × π × 9 = 18π cm. The sector is 80/360 = 2/9 of the circle, so the arc is 2/9 × 18π = 4π cm.

6π cm²

HintSixty degrees is a simple fraction of a full turn: apply it to the area of the whole circle.

The whyArea of the circle = π × 6² = 36π cm². The sector is 60/360 = 1/6 of it: 36π ÷ 6 = 6π cm².

90

HintCompare the sector's area with the area of the complete circle to see what share it is.

The whyThe full circle has area π × 10² = 100π cm². The sector is 25π ÷ 100π = 1/4 of it, and 1/4 of 360° is 90°.

15.7; 35.7

HintWalk all the way round the edge of the slice: one curved part and two straight parts.

The whyArc = 90/360 × 2 × 3.14 × 10 = 15.7 cm. The perimeter also includes the two radii: 15.7 + 10 + 10 = 35.7 cm.