# Solids: volume and surface area · Year 10

> Solids: volume and surface area, Year 10: Pyramids and cones and more. Explanation, worked examples and free practice for the national curriculum in England.

In Solids: volume and surface area (Year 10) you practise 5 skills: Pyramids and cones · Spheres · A missing dimension · Changing the dimensions · Comparing and deciding.

The volume of a solid is how many unit cubes fit inside it, and its surface area is the sum of the areas of all its faces. The volume of a box is its length times its width times its height. The volume of a prism or a cylinder is the area of its base times its height, and the volume of a pyramid or a cone is a third of that. The lateral area is the surface area without the bases.

The volume of a sphere is four thirds of π times the radius cubed, and its surface area is four π times the radius squared. When a length is not given, find it from a right triangle with Pythagoras’ theorem, or substitute the given values in a formula and solve the equation. When every dimension grows by the same factor, the area grows by the square of the factor and the volume by its cube.

The units of volume are cm³ and m³, and the units of area are cm² and m². A metre is 100 cm, so a square metre is 10,000 cm² and a cubic metre is 1,000,000 cm³. A litre is 1,000 cm³, and a cubic metre holds 1,000 litres. To decide which container holds more, or which box needs less material, work out the volume or the surface area of each and compare them.

**Tip:** Before any calculation, ask whether the solid is to be filled or wrapped: filling is volume, and wrapping is surface area.

## What you learn

### Pyramids and cones

- Example: (A pyramid on a square, its side 12 and its height 6) The base of a right pyramid is a square with side 12 cm, and the pyramid is 6 cm high. What is the volume of the pyramid, in cm³?
- Answer: 288 cm³
- Solution: The base area: 12² = 144 . The volume: 144 × 6 ÷ 3 = 288 .

### Spheres

- Example: (A drawing of a whole sphere) A sphere has a radius of 3 cm. What is the volume of the sphere, in cm³? Give your answer in terms of π .
- Answer: 36π
- Solution: Substitute in the formula for the volume: 4π × 3³ ÷ 3 = 36π .

### A missing dimension

- Example: A box has a volume of 180 cm³. Its base is a rectangle with sides 3 cm and 15 cm. What is the height of the box?
- Answer: 4 cm
- Solution: The base area: 3 × 15 = 45 . The volume is the base area times the height, so the height is 180 ÷ 45 = 4 .

### Changing the dimensions

- Example: The height of a box is doubled, and its other two dimensions stay the same. By what factor does the volume of the box grow?
- Answer: 2
- Solution: The volume is the product of the three dimensions. When one of them is multiplied by 2, the product is multiplied by 2 too.

### Comparing and deciding

- Example: Container A is a box 2 cm by 2 cm by 2 cm. Container B is a cylinder whose base is 2 cm across and which is 2 cm high. Which container holds more? Use π ≈ 3.14 .
- Options: Container B · Container A
- Answer: Container A
- Solution: The volume of the box: 2 × 2 × 2 = 8 . The radius of the cylinder is 1 cm, and its volume 3.14 × 1² × 2 = 6.28 . So container A holds more.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

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