Two similar solids have matching lengths in the ratio 1 : 4. Their surface areas are in the ratio □ : □, and their volumes □ : □. Fill in each blank

1, 16, 1, 64. Areas go as the lengths squared, 1 : 16; volumes as cubed, 1 : 64.

Two spheres have radii in the ratio 2 : 7, so their volumes are in the ratio 4 : 49. True or false?

False. Volumes go as the radii cubed, 8 : 343; 4 : 49 is the surface areas'.

A model of a building is made to a scale of 1 : 100, and its volume is 27 cm³. What is the real building's volume, in m³?

27 m³. 100 times the lengths is 100^3 = 1,000,000 times the volume: 27 × 1,000,000 cm³; 1 m³ = 1 000 000 cm³, so 27 m³.

Cones & spheres · Ages 14–15

Similar figures and solids

Lengths a : b → areas a² : b² → volumes a³ : b³.

How to do it

Method

Lengths a : b → areas a² : b² → volumes a³ : b³

Heights 2 : 3: volume 16 × 278 = 54 cm³

Another way

The scale factor k

16 × (32)³ = 54

In full

Similar figures have one shape at different sizes: their matching lengths are all in one ratio a : b. Their areas are in the ratio a² : b², the lengths' ratio squared, and their volumes a³ : b³, cubed. Two similar cones 2 cm and 3 cm high: their surface areas are in the ratio 4 : 9 and their volumes 8 : 27. The small one holds 16 cm³, so the big one 16 × 278 = 54 cm³. The other way round, from an area or volume ratio to the lengths': the square root or the cube root. Volumes 8 : 125 make lengths 2 : 5. Or think of the scale factor: k times the lengths makes k² times the areas and k³ times the volumes: k = 32, so 16 × (32)³ = 54.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
Two similar solids have matching lengths in the ratio 1 : 4. Their surface areas are in the ratio : , and their volumes : .
Fill in each blank
Show the answer and working
Answer: 1, 16, 1, 64
Areas go as the lengths squared, 1 : 16; volumes as cubed, 1 : 64.
Example 2
Two spheres have radii in the ratio 2 : 7, so their volumes are in the ratio 4 : 49.
True or false?
Show the answer and working
Answer: False
Volumes go as the radii cubed, 8 : 343; 4 : 49 is the surface areas'.
Example 3
A model of a building is made to a scale of 1 : 100, and its volume is 27 cm³. What is the real building's volume, in m³?
Show the answer and working
Answer: 27 m³
100 times the lengths is 100³ = 1,000,000 times the volume: 27 × 1,000,000 cm³; 1 m³ = 1 000 000 cm³, so 27 m³.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.Every length of a solid is made 5 times as long (its shape the same). How many times its old surface area is its new one?
- A. 125 times
- B. 5 times
- C. 25 times
- D. 10 times
Choose the right answer
Answer:
2.Two spheres have radii in the ratio 4 : 5, so their volumes are in the ratio 16 : 25.
True or false?
Answer:
3.Two similar cylinders have volumes in the ratio 1 : 64. The small one is 1 cm high. How high is the big one, in cm?
Answer:
4.A model of a building is made to a scale of 1 : 10, and its volume is 21 cm³. What is the real building's volume, in m³?
Answer:
5.Two similar triangles have matching sides 3 cm and 7 cm long. The small triangle's area is 81 cm². What is the big one's, in cm²?
Answer:

All the answers

1. C. 25 times
2. False
3. 4 cm
4. 0.021 m³
5. 441 cm²

Free worksheet: Similar figures and solidsWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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