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This page sets out the method for similar shapes: lengths in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Similar shapes: lengths topic page.
When two similar shapes are shown and one side is known on both.
When a small triangle sits inside a larger one, with a line parallel to the base.
With length scale factor 3, a side of 4 becomes 4 × 3 = 12. Multiply by the scale factor.
A scale factor below 1 shrinks the shape. With length scale factor 1/2, a side of 10 becomes 5.
If matching sides of two similar shapes are 4 and 12, the scale factor from the smaller shape to the larger is 12 ÷ 4 = 3. Divide the larger side by the smaller side.
A map scale of 1:100 means 1 cm on the map is 100 cm in real life. A length of 3 cm on the map is 300 cm in real life. Multiply by the second number.
From the smaller to the larger, the factor is greater than 1. From the larger to the smaller, it is a fraction.
A scale factor multiplies. Adding the factor to the side is a different operation.
Before you divide, decide which sides are a pair. On similar triangles, a side corresponds to the side opposite the equal angle. If both triangles have a right angle, the hypotenuses correspond, the sides opposite the shared acute angle correspond, and the remaining legs correspond. On rectangles, a length corresponds to a length, not to a width, unless the enlargement has turned the shape and the diagram shows that.
A common diagram gives the two triangles pointing the same way, one inside the other, or one beside the other. Mark the equal angles, then tick a pair of sides you already know. The scale factor comes only from that pair. Mixing a side of 6 cm with a non-matching side of 20 cm produces a scale factor that will not fit the other sides.
Method: 1. Match sides that sit opposite equal angles before you divide. 2. On right-angled triangles, match the hypotenuses first. 3. Do not pair a 6cm side with a non-matching 20cm side.
Divide the image side by the object side. If a side of 8 cm corresponds to a side of 12 cm, the scale factor is 12/8 = 1.5. Every length on the small triangle is multiplied by 1.5 to give the large triangle. A side of 10 cm on the small triangle becomes 10 × 1.5 = 15 cm on the large one.
Write the scale factor as a fraction if it is not a tidy decimal. 20/8 = 5/2 = 2.5 is comfortable either way. If the image is smaller than the object, the scale factor is less than 1. A reduction from 12 cm to 3 cm has scale factor 3/12 = 1/4. You can also say the shapes are similar with scale factor 4 from the small one to the large one. Say which direction you are using, and stay with it.
Method: 1. Divide a side on the image by the matching side on the object. 2. Sides 12cm and 8cm give scale factor 12/8=1.5. 3. Multiply every other object length by 1.5, so 10cm becomes 15cm.
Multiplying by the scale factor moves you from the object to the image. Dividing by it moves you back. Suppose the scale factor from a small photo to a large photo is 2.5. A detail that is 5 cm wide on the small photo is 5 × 2.5 = 12.5 cm on the large photo. A frame that is 30 cm on the large photo matches 30 ÷ 2.5 = 12 cm on the small photo.
Check the size. After an enlargement, the new length must be longer than the old one. After a reduction, it must be shorter. If your answer goes the wrong way, you have multiplied when you should have divided, or you have divided the sides in the wrong order. Angles are copied, not multiplied. A 40^° angle stays 40^° on both shapes.
Method: 1. Multiply to go from the object to the image, and divide to come back. 2. Scale factor 2.5 turns 5cm into 5× 2.5=12.5cm. 3. A large length of 30cm matches 30÷ 2.5=12cm on the small shape.
Once the scale factor is known, apply it to each remaining side. A small triangle with sides 6 cm, 8 cm and 10 cm is similar to a larger triangle. The side corresponding to 8 cm is 20 cm, so the scale factor is 20/8 = 2.5. The side corresponding to 6 cm is 6 × 2.5 = 15 cm. The side corresponding to 10 cm is 10 × 2.5 = 25 cm. The large sides are 15 cm, 20 cm and 25 cm.
Those three large sides are still in the ratio 6 : 8 : 10, which is 3 : 4 : 5. That is a useful check. You can also check the longest side corresponds to the longest side. Here 10 × 2.5 = 25, and 25 is the longest side of the image, opposite the same angle as the side of 10 cm. This lesson stops at lengths. Do not turn 2.5 into an area by squaring it unless a later question about area asks you to.
Method: 1. Find one scale factor, then multiply each remaining side by it. 2. From 8cm to 20cm, the scale factor is 20/8=2.5. 3. The other sides become 6× 2.5=15cm and 10× 2.5=25cm.
A "show that" question gives you two shapes and asks you to prove they are similar. Shapes are similar when all the corresponding angles are equal and all the corresponding sides are in the same ratio. For rectangles the angles are all 90^°, so you only need to check that the length ratio and the width ratio are equal.
Divide each larger side by the matching smaller side. If both results are 1.5, the rectangles are similar with scale factor 1.5. Do not compare the differences. Rectangles of 8 × 6 and 12 × 10 both grow by 4 cm, but 12 ÷ 8 = 1.5 while 10 ÷ 6 ≈ 1.67, so they are not similar. Write the two ratios side by side so the comparison is clear to the person marking your work.
Method: 1. Match each side of one shape with its corresponding side on the other. 2. Divide each larger side by its matching smaller side. 3. Check that every ratio gives the same scale factor. 4. Write a conclusion that uses the word similar and quotes the scale factor.
Tip: Never use subtraction to compare shapes. Similar means multiply by the same number, not add the same number.
Many exam diagrams hide two similar triangles. If a line DE is parallel to the side BC of triangle ABC, then triangle ADE and triangle ABC are similar, because the corresponding angles are equal. The small triangle sits inside the large one and they share the angle at A. Redrawing the two triangles side by side is the most reliable way to avoid matching the wrong sides.
The key skill is matching whole sides with whole sides. The side AB is the whole side, made of AD and DB. So the scale factor from the small triangle to the large one is AB ÷ AD, not DB ÷ AD. Find the whole length first by adding, and subtract at the end if you need the part EC or DB.
Method: 1. Redraw the small and large triangle separately and label the sides you know. 2. Add parts to find the whole sides, for example AB = AD + DB. 3. Use a pair of whole corresponding sides to find the scale factor. 4. Multiply or divide to find the unknown side, then subtract if a part is needed.
Similar shapes appear in real contexts. At the same time of day, a pole and a tree each make a right-angled triangle with their shadows, and these triangles are similar. A photograph and its enlargement are similar rectangles, and a model of a building is a similar solid. Photographs are a favourite exam context, because the width and height must be enlarged by the same scale factor.
Set up the scale factor from the pair of lengths you know in both shapes, such as the two shadow lengths. Then apply it to the length you want. Keep the units the same on both sides. If a model length is in centimetres and the real length is in metres, convert one of them first.
Method: 1. Identify the two similar shapes and the pair of lengths you know for both. 2. Write the scale factor as larger divided by smaller, in the same units. 3. Apply the scale factor to the length you want, multiplying to enlarge or dividing to reduce. 4. Check the answer is bigger or smaller than the starting length as the context suggests.
In harder questions one or more sides are written with x. Write a ratio of corresponding sides as an equation, then solve it. For two similar rectangles with widths 6 and 9, the scale factor is 9/6 = 1.5. If the lengths are (x + 6) and (3x - 3), then 3x - 3 = 1.5(x + 6). A diagram that gives only the expressions for the sides is typical of the Higher paper.
Expand, collect the x terms on one side, and solve. Substitute the value of x back into the expressions to find actual lengths and check that the ratios agree. Reject any answer that gives a negative length. These are typically worth four marks, with one method mark for the equation.
Method: 1. Work out the scale factor from the pair of sides with no x. 2. Write an equation for the pair of sides with x using the same scale factor. 3. Expand brackets, collect terms and solve for x. 4. Substitute back to find lengths and check the ratios match.
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