GCSE · Maths · AQA · Spec 8300 · Foundation
Compare lengths, areas, volumes using ratio
Double every side of a photo. Do you need twice as much paper to print it? Hold on to your guess, because this lesson settles it.
Similar shapes
Your target: similar triangles
When angle R = angle C, the triangles are similar: PQ : AB = QR : BC = PR : AC = 2 : 1. One scale factor, 2, for every pair of sides.
Both triangles have a right angle, at B and at Q, and QR is exactly twice BC. Drag P up or down. Almost every position gives you a different shape. Only one gives you the same shape, just bigger. Watch the angles to find it.
Predict, then check
Stay with the two triangles you just matched up.
Triangle ABC has an area of 6 cm². Every side of PQR is twice as long as the matching side of ABC. What is the area of PQR?
Watch the method
Problem
Two bottles are mathematically similar. The small one is 12 cm tall and the large one is 18 cm tall. The small bottle has a surface area of 80 cm² and a volume of 160 cm³. Find the surface area and the volume of the large bottle.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Same shape, different size. What happens to the lengths, the areas and the volumes?
What you need to know
- Similar shapes have equal matching angles, and every pair of matching lengths is in the same ratio. They're the same shape, but not necessarily the same size.
- Write a length ratio in simplest form by dividing both parts by their highest common factor.
- Length ratio a : b gives area ratio a² : b² and volume ratio a³ : b³.
- To work backwards, square-root an area ratio or cube-root a volume ratio to get back to the length ratio.
- In similar right-angled triangles, sin, cos and tan of matching angles are equal.
The big picture
Similar shapes have the same angles, and every pair of matching lengths is in the same ratio: the scale factor. Enlarge by scale factor k and every length is multiplied by k, every area by k² and every volume by k³. So a length ratio a : b becomes a² : b² for areas and a³ : b³ for volumes. Because the angles don't change, sin, cos and tan of matching angles are equal too.
Key points
Worked example
Problem
Two cylinders are mathematically similar. Their surface areas are 45 cm² and 80 cm². The larger cylinder has a volume of 192 cm³. Find the volume of the smaller cylinder.
⚠ Watch out
Using the length scale factor for an area or a volume. If every side is 3 times as long, the area is 9 times as big and the volume is 27 times as big, not 3 times.
Memory hook
Count the dimensions. Length is 1-D, so use the ratio as it is. Area is 2-D, so square it. Volume is 3-D, so cube it.
Check yourself
Two similar cones have heights 5 cm and 15 cm. Write their length ratio in simplest form, then work out the ratio of their volumes.
Flashcards
(7)What makes two shapes similar?
Lengths are in the ratio a : b. What is the ratio of the areas?
A solid is enlarged by scale factor k. What happens to its volume?
You're given an area ratio or a volume ratio. How do you get back to the length ratio?
How do you write a ratio in its simplest form?
Lengths are in the ratio 4 : 5. What is the scale factor from the smaller shape to the larger?
Why is tan of an angle the same in two similar right-angled triangles?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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Keep me postedMore AQA GCSE Maths topics
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- Alternate and corresponding angles on parallel lines
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- Angle sum in a triangle and polygons
- Approximate solutions of equations from a graph
- Approximate solutions of quadratics from a graph
- Arc lengths, angles and sector areas
- Area and volume formulae — triangles, parallelograms, trapezia, prisms, cylinders
- Box plots, quartiles, inter-quartile range (Higher)
- Calculate with roots and integer indices
- Circle definitions and properties
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