The same shape at a different size
A photo printed small and the same photo printed large show the same picture. Everything in the large print is bigger, but nothing is stretched or squashed: the shape is the same, and only the size has changed.
Triangles can be like this too. Here are two triangles with the same shape. The smaller one has a base of 4 and a height of 3, with its top corner above the middle of its base. The larger one shares its bottom left corner.
Two triangles sharing their bottom left corner. The small one has a base of 4 and a height of 3. The large one has a base of 8 and a height of 6, and each of its sides is parallel to a side of the small one.
Every length times the same number
The larger triangle is the smaller one with every length doubled. Its base is 4 × 2 = 8 and its height is 3 × 2 = 6. Count the dots along a sloping side: on the small triangle it goes 2 across and 3 up, and on the large one 4 across and 6 up, so it is twice as long as well.
No angle changes. Each side of the larger triangle points in exactly the same direction as the matching side of the smaller one, so the sides meet at the same angles. Only the lengths have grown.
Similar shapes
Multiply every length by 3 instead, and the base becomes 4 × 3 = 12 and the height 3 × 3 = 9. Again no angle changes.
Two shapes are called similar when one is an enlargement of the other. That means two things together: every angle in one shape is equal to the matching angle in the other, and every length is multiplied by the same number. That number is the scale factor.
Scale factor 3: the base 4 becomes 12, the height 3 becomes 9, and each sloping side goes 6 across and 9 up instead of 2 across and 3 up.
Finding a missing side
A triangle has sides of 3 cm, 4 cm and 5 cm, and a similar triangle has 9 cm as its shortest side. The shortest side of one matches the shortest side of the other, so the scale factor is 9 ÷ 3 = 3. The other sides are 4 × 3 = 12 cm and 5 × 3 = 15 cm.
Check by dividing each side of the larger triangle by the matching side of the smaller one: 9 ÷ 3 = 12 ÷ 4 = 15 ÷ 5 = 3. For similar shapes every pair gives the same answer.
A triangle with sides 6 cm, 8 cm and 11 cm is not similar to the 3, 4, 5 triangle: 6 ÷ 3 = 2 and 8 ÷ 4 = 2, but 11 ÷ 5 = 2.2.
Equal angles alone
For triangles, equal angles are enough. If the three angles of one triangle are equal to the three angles of another, the two triangles are similar, and their sides are in the same ratio.
For other shapes they are not enough. Every rectangle has four right angles, but a 2 by 3 rectangle and a 2 by 4 rectangle are not similar: the widths match with factor 2 ÷ 2 = 1, but the lengths need 4 ÷ 3, which is not 1.
Measuring what you cannot reach
At the same moment, the sun shines on a stick and a tree at the same angle, because its rays are parallel. The stick and its shadow make a right triangle, and so do the tree and its shadow, and the two triangles have the same angles. So they are similar.
A stick 1 m tall casts a shadow 0.4 m long, and the tree casts a shadow 12 m long. The scale factor from the stick’s triangle to the tree’s triangle is 12 ÷ 0.4 = 30, so the tree is 1 × 30 = 30 m tall.
sun at 40°: the stick's shadow is 1.2 m and the tree's is 10.7 m; the rays are parallel, so the two triangles are similar and tree ÷ shadow = 1 ÷ 1.2, tree = 10.7 ÷ 1.2 = 9 m
Move the sun until the stick's shadow is 0.4 m, then read the tree from its shadow
A stick 1 m tall and a tree, in the same sunlight. Drag the end of the tree’s shadow to move the sun, and drag the top of the tree to change its height. Height ÷ shadow is the same number for the stick and the tree. Move the sun until the stick’s shadow is 0.4 m.
The usual mistakes
Adding instead of multiplying. With scale factor 3, a side of 4 becomes 4 × 3 = 12, not 4 + 3 = 7. Adding the same amount to every side changes the shape: 3, 4, 5 with 3 added to each side becomes 6, 7, 8, and 6 ÷ 3 = 2 but 8 ÷ 5 = 1.6.
Matching the wrong sides. A side is matched with the side in the same place in the other shape: base with base, height with height, the shortest side with the shortest side.
Worked example: The Height of a Tree from the Length of Its Shadow
Question At the same moment on a sunny day, a vertical pole 1.5 m tall casts a shadow 2 m long on level ground, and a tree casts a shadow 16 m long. (a) How tall is the tree? (b) A bird flies in a straight line from the top of the tree to the tip of the tree's shadow. How far does the bird fly?
1.The rays of the sun are parallel, so they meet the ground at the same angle at the pole and at the tree. The pole and the tree are vertical, so each makes a right angle with the ground. The two triangles have equal angles, so they are similar.
The rays of the sun are parallel and both objects are vertical, so the two triangles have equal angles and are similar. 2.In similar triangles, corresponding sides are in the same ratio. The two shadows correspond, so the scale factor from the pole's triangle to the tree's triangle is 16 ÷ 2 = 8.
The shadows correspond, so the scale factor is 16 ÷ 2 = 8. 3.The two heights correspond as well. (a) The tree is 1.5 × 8 = 12 m tall.
(a) The heights correspond too: the tree is 1.5 × 8 = 12 m tall. 4.The bird flies along the hypotenuse of the tree's triangle. Let this distance be c m. By Pythagoras' theorem, c2 = 122 + 162 = 144 + 256 = 400, so c = √400 = 20.
The bird flies along the hypotenuse: c2 = 122 + 162 = 400, so c = 20. 5.(b) The bird flies 20 m. Check with the pole's triangle: its hypotenuse is √1.52 + 22 = √6.25 = 2.5 m, and 2.5 × 8 = 20 m.
(b) The bird flies 20 m. The pole's hypotenuse is 2.5 m, and 2.5 × 8 = 20 m.
Answer: (a) 12 m; (b) 20 m
Common mistakes
- Adding the same amount instead of multiplying: the tree's shadow is 14 m longer than the pole's shadow, so the tree is said to be 1.5 + 14 = 15.5 m tall. Similar shapes keep the ratio of their sides, not the difference between them.
- Matching sides that do not correspond, such as 1.516 = 2h. A height must be matched with a height and a shadow with a shadow: h1.5 = 162.
More pythagoras and similar shapes problems, worked step by step →
What happens to an area
An enlargement with scale factor 3 multiplies every length by 3, but it does not multiply an area by 3. A rectangle 2 cm by 1 cm has an area of 2 cm². Enlarged by 3, it is 6 cm by 3 cm, with an area of 18 cm², which is 9 times as much. The larger rectangle holds 3 rows of 3 copies of the smaller one.
An area is a length times a length, and each length is multiplied by the scale factor k, so the area is multiplied by . The same is true for a triangle: its base and its height are both multiplied by k, so ½ × base × height is multiplied by . The triangle scaled by 2 above has times the area of the original: ½ × 8 × 6 = 24 against ½ × 4 × 3 = 6.
Worked example: A Logo Enlarged for a Banner: the New Height and the Ink It Uses
Question A rectangular school logo is 12 cm wide and 8 cm high. It is enlarged to make a banner that is 36 cm wide, and the banner is similar to the logo. (a) How high is the banner? (b) Printing the logo uses 20 ml of ink, and the amount of ink is proportional to the area that is printed. How much ink does the banner use?
1.The banner is similar to the logo, so every length is multiplied by the same scale factor. The two widths correspond, so the scale factor is k = 36 ÷ 12 = 3.
The widths correspond, so the scale factor is k = 36 ÷ 12 = 3. 2.The height is multiplied by 3 as well. (a) The banner is 8 × 3 = 24 cm high.
(a) The height is multiplied by 3 as well: 8 × 3 = 24 cm. 3.An area is a length multiplied by a length. Both lengths are multiplied by 3, so the area is multiplied by 3 × 3 = 32 = 9. The banner holds 9 copies of the logo, in 3 rows of 3.
Both lengths are multiplied by 3, so the area is multiplied by 32 = 9: the banner holds 9 copies of the logo. 4.(b) The ink is proportional to the area, so the banner uses 20 × 9 = 180 ml of ink. Check with the areas: the logo is 12 × 8 = 96 cm2 and the banner is 36 × 24 = 864 cm2, and 864 ÷ 96 = 9.
(b) The ink is proportional to the area, so the banner uses 20 × 9 = 180 ml.
Answer: (a) 24 cm; (b) 180 ml
Common mistakes
- Multiplying the ink by the scale factor for lengths, 20 × 3 = 60 ml. The ink covers an area, and the area is multiplied by 32 = 9, not by 3.
- Adding 24 cm to the height because the width grew by 24 cm, which gives a banner 36 cm by 32 cm. That rectangle is not similar to the logo, because 36 ÷ 12 = 3 but 32 ÷ 8 = 4. An enlargement multiplies every length by the same number.
More pythagoras and similar shapes problems, worked step by step →