Geometry & measures

Volume and Surface Area of Prisms and Cylinders

Year 7 · Year 8 · Year 9

✓Calculate the volume of various prisms, including cuboids and triangular prisms.
✓Calculate the volume of cylinders.
✓Calculate the surface area of cuboids and other simple prisms.
✓Calculate the surface area of cylinders.
✓Understand and use appropriate units for volume and surface area.

Key concepts

Prism

A prism is a three-dimensional shape that has the same cross-section throughout its length. The two end faces are parallel and identical. Examples include cuboids, triangular prisms, and hexagonal prisms.

Cylinder

A cylinder is a three-dimensional shape with a circular cross-section that is uniform along its length. It can be thought of as a circular prism.

Volume

Volume is the amount of three-dimensional space an object occupies. It is measured in cubic units, such as cubic centimetres (cm³) or cubic metres (m³).

Volume of a Prism

The volume of any prism is found by multiplying the area of its cross-section by its length (or height).

Volume = Area of cross-section × length

Volume of a Cylinder

Since a cylinder has a circular cross-section, its area is πr². Therefore, the volume of a cylinder is the area of its circular base multiplied by its height.

Volume = πr²h

Surface Area

Surface area is the total area of all the faces (surfaces) of a three-dimensional object. It is measured in square units, such as square centimetres (cm²) or square metres (m²).

Surface Area of a Cuboid

A cuboid has 6 rectangular faces. The surface area is the sum of the areas of these 6 faces. Since opposite faces are identical, we can calculate the area of three unique faces and multiply by two.

Surface Area = 2(lw + lh + wh)

Surface Area of a Cylinder

The surface area of a cylinder consists of two circular ends and one curved rectangular surface. The curved surface unrolls into a rectangle with length equal to the circumference of the base (2πr) and width equal to the height (h).

Surface Area = 2πr² + 2πrh

Key facts to remember

1Volume is the amount of space a 3D object occupies and is measured in cubic units (e.g., cm³, m³).
2Surface area is the total area of all the faces of a 3D object and is measured in square units (e.g., cm², m²).
3A prism is a 3D shape with a uniform cross-section along its length.
4The volume of any prism is calculated by: Volume = Area of cross-section × length.
5The volume of a cylinder is calculated by: Volume = πr²h, where r is the radius and h is the height.
6The surface area of a cuboid is calculated by: Surface Area = 2(lw + lh + wh).
7The surface area of a cylinder is calculated by: Surface Area = 2πr² + 2πrh.

Worked examples

Example 1

Calculate the volume and surface area of a cuboid with length 8 cm, width 3 cm, and height 5 cm.

I**Volume:**

IIFormula for volume of a cuboid: V = lwh

IIISubstitute values: V = 8 cm × 3 cm × 5 cm

IVCalculate: V = 120 cm³

V**Surface Area:**

VIFormula for surface area of a cuboid: SA = 2(lw + lh + wh)

VIISubstitute values: SA = 2((8 × 3) + (8 × 5) + (3 × 5))

VIIICalculate areas of faces: SA = 2(24 + 40 + 15)

9Sum areas: SA = 2(79)

10Calculate total surface area: SA = 158 cm²

Answer

Volume = 120 cm³, Surface Area = 158 cm²

Example 2

A triangular prism has a cross-section that is a right-angled triangle with base 6 cm and height 8 cm. The length of the prism is 10 cm. Calculate its volume and surface area.

I**Volume:**

IIArea of triangular cross-section: A = ½ × base × height = ½ × 6 cm × 8 cm = 24 cm²

IIIVolume of prism: V = Area of cross-section × length = 24 cm² × 10 cm = 240 cm³

IV**Surface Area:**

VFirst, find the hypotenuse of the triangular cross-section using Pythagoras' theorem: c² = a² + b² = 6² + 8² = 36 + 64 = 100. So, c = √100 = 10 cm.

VIArea of two triangular ends: 2 × 24 cm² = 48 cm²

VIIArea of rectangular base: 6 cm × 10 cm = 60 cm²

VIIIArea of rectangular side 1: 8 cm × 10 cm = 80 cm²

9Area of rectangular side 2 (hypotenuse side): 10 cm × 10 cm = 100 cm²

10Total Surface Area = 48 cm² + 60 cm² + 80 cm² + 100 cm² = 288 cm²

Answer

Volume = 240 cm³, Surface Area = 288 cm²

Remember to find the area of *all* faces for surface area, including the two identical end faces and all rectangular side faces.

Example 3

Calculate the volume and surface area of a cylinder with radius 4 cm and height 12 cm. Give your answers to 1 decimal place.

I**Volume:**

IIFormula for volume of a cylinder: V = πr²h

IIISubstitute values: V = π × (4 cm)² × 12 cm

IVCalculate: V = π × 16 cm² × 12 cm = 192π cm³

VV ≈ 603.185... cm³

VIRound to 1 decimal place: V = 603.2 cm³

VII**Surface Area:**

VIIIFormula for surface area of a cylinder: SA = 2πr² + 2πrh

9Substitute values: SA = (2 × π × (4 cm)²) + (2 × π × 4 cm × 12 cm)

10Calculate areas: SA = (2 × π × 16) + (2 × π × 48)

11SA = 32π + 96π

12SA = 128π cm²

13SA ≈ 402.123... cm³

14Round to 1 decimal place: SA = 402.1 cm²

Answer

Volume = 603.2 cm³, Surface Area = 402.1 cm² (to 1 d.p.)

Use the π button on your calculator for accuracy and only round at the final step.

Common mistakes

✗Confusing volume and surface area, leading to incorrect calculations or units.
✗Using incorrect units for the final answer (e.g., cm² for volume, cm³ for surface area).
✗Forgetting to include all faces when calculating the surface area of a prism or cylinder.
✗Incorrectly calculating the area of the cross-section for a prism (e.g., forgetting the ½ for a triangle).
✗Using the diameter instead of the radius (or vice versa) in formulas involving circles for cylinders.

Exam tips

★Always write down the correct formula before substituting any values.
★Draw a clear diagram of the shape and label all given dimensions to help visualise the problem.
★Pay close attention to the units given in the question and ensure your final answer has the correct units.
★Show all your working steps clearly, as marks are often awarded for method even if the final answer is incorrect.
★For calculations involving π, use the π button on your calculator and only round your final answer to the specified degree of accuracy.

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