EDEXCEL GCSE MATHS · HIGHER
Similarity in 3D solids
Similarity and congruence · Lesson 5 of 5
Warm-up
Answer each one, then check.
1. Work out 2³.
8
2. Work out 1.5³.
3.375
3. What is the area scale factor if k = 3?
9
4. Work out ∛27.
3
5. What is the formula for the volume of a cone?
⅓πr² h
Learning Objectives
1. Use length, area and volume scale factors.
2. Find a volume, capacity or mass of a similar solid.
3. Work out the length ratio from a volume ratio.
4. Solve problems with similar cones, cylinders and frustums.
Similar Cubes
|
Doubling lengths multiplies areas by 4 and volumes by 8. |
Three cubes with edges 1, 2 and 3 showing that when lengths are doubled areas are multiplied by 4 and volumes by 8.
Scale Factors for Similar Solids
If the length scale factor is k.
|
Measure |
Scale factor |
If k = 2 |
|---|---|---|
|
Lengths |
k |
2 |
|
Areas (including surface area) |
k² |
4 |
|
Volumes (and capacity, mass) |
k³ |
8 |
Volume of a Similar Cone
|
Cone A has height 6 cm and volume 40 cm³. Cone B is similar to A and has height 9 cm. Find the volume of cone B. |
1. Length scale factor
9/6 = 1.5
2. Volume scale factor
1.5³ = 3.375
3. Volume of B
40 × 3.375
Answer: 135 cm³
A Model and the Real Thing
|
A model statue is 20 cm tall and has mass 2 kg. The real statue is 1.2 m tall and made of the same material. Find the mass of the real statue. |
1. Length scale factor
120 ÷ 20 = 6
2. Volume (and mass) scale factor
6³ = 216
3. Mass
2 × 216
Answer: 432 kg
A Frustum
|
A cone of height 15 cm has volume 500 cm³. A small cone of height 6 cm is cut off the top, leaving a frustum. Find the volume of the frustum. |
1. The small cone is similar to the large cone
Length scale factor 6/15 = 0.4
2. Volume scale factor
0.4³ = 0.064
3. Volume of the small cone
500 × 0.064 = 32
4. Frustum = large cone minus small cone
500 − 32
Answer: 468 cm³
From Volumes to Lengths
|
Two similar solids have volumes 8 cm³ and 125 cm³. Find the ratio of their lengths and of their surface areas. |
1. Take the cube root of the volumes
∛8 : ∛125 = 2 : 5
2. Lengths
2 : 5
3. Surface areas: square the ratio
4 : 25
Answer: Lengths 2 : 5; surface areas 4 : 25.
Which Scale Factor?
|
USE K (LENGTHS) |
USE K² OR K³ |
|
▸ Heights, radii, edges, perimeters. ▸ Anything measured in a straight line. ▸ Read directly from the two shapes. |
▸ k²: area, surface area, cross-section. ▸ k³: volume, capacity, mass (same material). ▸ Cube or square root to go backwards. |
Key Terms
|
Similar solids Solids with the same shape and all lengths in the same ratio. |
Volume scale factor k³. |
|
Surface area scale factor k². |
Frustum A cone or pyramid with the top cut off parallel to the base. |
|
Capacity The volume a container holds. |
Density Mass per unit volume; the same in similar solids of the same material. |
Your Task: Bottle Sizes
12 minutes
|
A small bottle is 10 cm tall and holds 250 ml. A similar large bottle is 15 cm tall. Find the capacity of the large bottle. A label on the small bottle has area 40 cm²; find the label area on the large bottle. 1. Find k. 2. Cube for volume. 3. Square for area. |
A good answer shows: Scale factor 1.5. Capacity 250 × 1.5³ = 843.75 ml, about 844 ml. Label 40 × 1.5² = 90 cm².
Can I...?
☐ State the scale factors k, k², k³.
☐ Find a volume of a similar solid.
☐ Find a mass using a volume factor.
☐ Find the surface area of a similar solid.
☐ Use a cube root to find a length ratio.
☐ Find the volume of a frustum.
☐ Choose the right scale factor.
☐ Show clear working.
Summary
✓ Lengths k, areas k², volumes k³.
✓ For the same material, mass scales like volume.
✓ From a volume ratio take the cube root for lengths.
✓ A frustum is a large cone minus a similar small cone.
|
EXAM FOCUS Cone A has height 6 cm and volume 40 cm³. Cone B is similar to A and has height 9 cm. Work out the volume of cone B. (3 marks) Find the length scale factor first. Cube it for volume. Do not square it. |