Solids: volume and surface area · Grade 12 · 3 units
In Solids: volume and surface area (Grade 12 · 3 units) you practise 9 skills: What volume and surface area mean · Boxes and cubes · Triangular prisms and cylinders · Pyramids and cones · Spheres · A missing dimension · Changing the dimensions · Comparing and deciding · Units of volume and area. 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
What volume and surface area mean
For example: To find how much sand goes into a crate, which measure do you work out?
- The surface area
- The lateral area
- The base area
- The volume
Answer and solution
The answer: The volume
The volume measures the space inside the solid, that is, how much it holds. So you work out the volume.
Boxes and cubes
For example:
A box has length 12 cm, width 3 cm and height 7 cm. What is the volume of the box, in cm³?Answer and solution
The answer: 252 cm³
The volume is the length times the width times the height: 12 × 3 × 7 = 252 .
Triangular prisms and cylinders
For example: A cylinder has a radius of 6 cm and a height of 2 cm. What is the volume of the cylinder? Give your answer in terms of π .
Answer and solution
The answer: 72π
The area of the base is π × 6² = 36π , and times the height: 36π × 2 = 72π . The volume of the cylinder is 72π cm³.
Pyramids and cones
For example: 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 and solution
The answer: 288 cm³
The base area: 12² = 144 . The volume: 144 × 6 ÷ 3 = 288 .
Spheres
For example:
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 and solution
The answer: 36π
Substitute in the formula for the volume: 4π × 3³ ÷ 3 = 36π .
A missing dimension
For 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 and solution
The answer: 4 cm
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
For 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 and solution
The answer: 2
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
For 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 .
- Container B
- Container A
Answer and solution
The answer: Container A
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.
Units of volume and area
For example: How many cm² are there in 39 m²?
Answer and solution
The answer: 390000
A metre is 100 cm, so one square metre holds 100 × 100 = 10000 cm². So multiply: 39 × 10000 = 390000 .
Not on the site yet
Parts of this topic that are not built yet: Angles between lines and planes · Truncated pyramids and composite solids · Proofs in three dimensions.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 3 units (grades 10 and 11).