Measurement, Space & Statistics

Measurement & Similarity: Surface Area, Volume & Similar Figures

Year 9

✓By the end of this lesson students will be able to calculate the surface area and volume of right prisms, cylinders, pyramids, cones, and spheres.
✓By the end of this lesson students will be able to identify similar two-dimensional and three-dimensional figures.
✓By the end of this lesson students will be able to apply the relationships between corresponding lengths, areas, and volumes of similar figures to solve problems.
✓By the end of this lesson students will be able to use appropriate units for surface area and volume calculations.

Key concepts

Surface Area of 3D Objects

The surface area (SA) of a three-dimensional object is the total area of all its faces or surfaces. It is measured in square units (e.g., cm², m²).

Right Prism: SA = 2 × A_base + P_base × h (where A_base is base area, P_base is base perimeter, h is height)\nCylinder: SA = 2πr² + 2πrh (where r is radius, h is height)\nRight Pyramid: SA = A_base + A_lateral (sum of areas of triangular faces)\nRight Cone: SA = πr² + πrs (where r is radius, s is slant height)\nSphere: SA = 4πr² (where r is radius)

Volume of 3D Objects

The volume (V) of a three-dimensional object is the amount of space it occupies. It is measured in cubic units (e.g., cm³, m³).

Right Prism: V = A_base × h (where A_base is base area, h is height)\nCylinder: V = πr²h (where r is radius, h is height)\nRight Pyramid: V = (1/3)A_base × h (where A_base is base area, h is perpendicular height)\nRight Cone: V = (1/3)πr²h (where r is radius, h is perpendicular height)\nSphere: V = (4/3)πr³ (where r is radius)

Similar Figures

Two figures are similar if they have the same shape but not necessarily the same size. For similar figures, corresponding angles are equal, and the ratio of corresponding lengths is constant. This applies to both 2D and 3D shapes.

Ratios of Lengths, Areas, and Volumes for Similar Figures

If two figures are similar and the ratio of their corresponding lengths (or any linear dimension like radius, height, perimeter) is k, then there are specific relationships for their areas and volumes:

Ratio of corresponding lengths (L₁ : L₂) = k\nRatio of corresponding areas (A₁ : A₂) = k²\nRatio of corresponding volumes (V₁ : V₂) = k³

Key facts to remember

1Surface area is measured in square units (e.g., cm², m²).
2Volume is measured in cubic units (e.g., cm³, m³).
3For prisms and cylinders, Volume = Area of Base × Height.
4For pyramids and cones, Volume = (1/3) × Area of Base × Perpendicular Height.
5For a sphere, SA = 4πr² and V = (4/3)πr³.
6Similar figures have the same shape; corresponding angles are equal, and corresponding lengths are in proportion.
7If the ratio of corresponding lengths of similar figures is k, then the ratio of their areas is k², and the ratio of their volumes is k³.

Worked examples

Example 1

Calculate the surface area and volume of a cylinder with a radius of 5 cm and a height of 12 cm. Give your answers correct to two decimal places.

I1. Write down the formulas for surface area and volume of a cylinder:

II SA = 2πr² + 2πrh

III V = πr²h

IV2. Substitute the given values (r = 5 cm, h = 12 cm) into the surface area formula:

V SA = 2 × π × (5)² + 2 × π × 5 × 12

VI SA = 2 × π × 25 + 2 × π × 60

VII SA = 50π + 120π

VIII SA = 170π

9 SA ≈ 534.0707... cm²

103. Substitute the given values into the volume formula:

11 V = π × (5)² × 12

12 V = π × 25 × 12

13 V = 300π

14 V ≈ 942.4777... cm³

154. Round the answers to two decimal places and include units.

Answer

Surface Area ≈ 534.07 cm²\nVolume ≈ 942.48 cm³

Remember to use the correct units: square units for area and cubic units for volume.

Example 2

A square-based pyramid has a base side length of 6 cm and a perpendicular height of 4 cm. The slant height of each triangular face is 5 cm. Calculate its total surface area and volume.

I1. Calculate the area of the square base (A_base):

II A_base = side × side = 6 cm × 6 cm = 36 cm²

III2. Calculate the area of one triangular face (A_face):

IV Each triangular face has a base of 6 cm and a slant height (perpendicular height of the triangle) of 5 cm.

V A_face = (1/2) × base × height = (1/2) × 6 cm × 5 cm = 15 cm²

VI3. Calculate the total lateral surface area (A_lateral):

VII There are 4 triangular faces, so A_lateral = 4 × A_face = 4 × 15 cm² = 60 cm²

VIII4. Calculate the total surface area (SA):

9 SA = A_base + A_lateral = 36 cm² + 60 cm² = 96 cm²

105. Calculate the volume (V) using the perpendicular height of the pyramid (h = 4 cm):

11 V = (1/3) × A_base × h

12 V = (1/3) × 36 cm² × 4 cm

13 V = 12 × 4 cm³

14 V = 48 cm³

Answer

Surface Area = 96 cm²\nVolume = 48 cm³

Distinguish between the perpendicular height of the pyramid (for volume) and the slant height of the triangular faces (for surface area).

Example 3

Two similar cones have radii in the ratio 2:3. If the volume of the smaller cone is 160 cm³, what is the volume of the larger cone?

I1. Identify the ratio of corresponding lengths (radii):

II k = 2/3 (smaller to larger)

III2. Recall the relationship between the ratio of lengths and the ratio of volumes for similar figures:

IV Ratio of volumes = k³

V3. Calculate the cube of the length ratio:

VI k³ = (2/3)³ = 2³/3³ = 8/27

VII4. Set up a proportion using the ratio of volumes and the given volume:

VIII V_smaller / V_larger = k³

9 160 / V_larger = 8/27

105. Solve for V_larger:

11 V_larger = 160 × (27/8)

12 V_larger = 20 × 27

13 V_larger = 540 cm³

Answer

The volume of the larger cone is 540 cm³.

Ensure you cube the ratio when dealing with volumes, and square it for areas. Pay attention to whether the ratio is 'smaller to larger' or 'larger to smaller'.

Common mistakes

✗Confusing surface area and volume formulas, or using the wrong formula for a specific shape.
✗Using slant height instead of perpendicular height for volume calculations of pyramids and cones.
✗Forgetting to square the length ratio for areas or cube it for volumes when dealing with similar figures.
✗Incorrectly calculating the area of the base for complex prisms or pyramids.
✗Not including correct units (e.g., cm instead of cm² or cm³) in the final answer.

Exam tips

★Always draw a diagram if one isn't provided, labelling all known dimensions.
★Write down the correct formula first before substituting values.
★Show all steps of your working clearly, especially when dealing with similar figures and ratios.
★Double-check your units and ensure they are consistent throughout the problem and in your final answer.

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