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

120 cm³

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

为什么Base 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

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

为什么Volume = 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

提示Cube the radius first, then deal with the fraction.

为什么r³ = 27. Volume = 4/3 × π × 27 = 4 × 9 × π = 36π cm³. AQA gives the sphere formula in the question.

one third

提示Compare πr²h with the formula for the pointed solid.

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

12

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

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

100π cm²

提示Square the radius before multiplying by anything else.

为什么4 × 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

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

为什么The 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

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

为什么Curved 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

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

为什么Base = 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

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

为什么Curved 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

提示Compare the angle with a complete turn.

为什么A 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

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

为什么Circumference = 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

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

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

90

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

为什么The 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

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

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