Start with the recap, study the fully worked examples, then use the practice problems to
check your understanding of Similarity.
This page combines explanation, solved examples, and follow-up practice so you can move
from recognition to confident problem-solving in Math.
Concept Recap
Two figures are similar if they have the same shape but possibly different sizes, meaning all corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor).
A photo and its enlargement are similar—same shape, different size.
Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Two figures are similar when all matching angles are equal and all matching sides share one scale factor.
Common stuck point:The procedure for similarity is the easy part; the trap is expecting similar figures to have equal sides. Asking "Are all corresponding angles equal and all corresponding sides in the same ratio?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Are all corresponding angles equal and all corresponding sides in the same ratio?
Worked Examples
Example 1
medium
Triangle ABC is similar to triangle DEF. If AB=6, BC=8, AC=10, and DE=9, find EF and DF.
Answer
EF=12,DF=15
First step
1
Find the scale factor: k=ABDE=69=1.5.
See the full worked solution + why-it-works coaching
Setup·Key insight·Why it works·Common pitfall·Connection
A tree casts a 15 m shadow at the same time a 2 m pole casts a 3 m shadow. How tall is the tree?
Example 3
medium
In △ABC, a line through D on AB parallel to BC meets AC at E. If AD=3,DB=6, AE=4, find EC.
Example 4
hard
In △ABC, the altitude from the right angle to the hypotenuse has length h. Legs are a=6,b=8, hypotenuse c=10. Find h.Right triangle ABC with legs 6 and 8 and hypotenuse 10; find the altitude h from the right angle to the hypotenuse
Example 5
challenge
Triangles ABC and XYZ are similar. The longest side of ABC is 14 and the longest side of XYZ is 21. The area of ABC is 40. Find the area of XYZ.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
medium
Two similar rectangles have widths of 4 cm and 10 cm. If the smaller rectangle has a length of 6 cm, find the length of the larger rectangle.
Example 2
medium
Two similar triangles have corresponding side lengths in the ratio 3:5. If the perimeter of the smaller triangle is 27 cm, find the perimeter of the larger triangle.
Example 3
easy
Two similar triangles have a scale factor of 3. A side of the small one is 4. Find the matching side of the large one.
Example 4
easy
Two similar figures have corresponding sides 5 and 15. What is the scale factor (large to small)?
Example 5
easy
Are all squares similar to each other?
Example 6
easy
In two similar triangles, corresponding angles are related how?
Example 7
easy
A 6-foot person casts a 4-foot shadow. A tree casts a 20-foot shadow at the same time. Set up the proportion for the tree's height h.
Example 8
easy
Two triangles each have angles 50∘ and 60∘. Are they similar?Both triangles share these three angles — AA similarity applies
Example 9
easy
Two similar rectangles have a scale factor of 2. The small one has area 10. Find the large one's area.
Example 10
easy
True or false: two figures can be congruent but not similar.
Example 11
medium
Triangle ABC∼ triangle DEF. AB=8, DE=12, and BC=10. Find EF.Triangle ABC — find the corresponding side EF in the similar triangle DEF
Example 12
medium
In triangle ABC, a line parallel to BC cuts AB at D and AC at E. If AD=4, DB=6, and AE=6, find EC.
Example 13
medium
Two similar solids have a scale factor of 2. How do their volumes compare?
Example 14
medium
A map has scale 1:50000. Two towns are 4 cm apart on the map. Find the real distance in km.
Example 15
medium
Two similar triangles have areas 9 and 25. Find the ratio of their corresponding sides.
Example 16
medium
A photo 4 in by 6 in is enlarged so its longer side becomes 15 in. If the enlargement is similar, find the new shorter side.
Example 17
medium
In a right triangle, the altitude to the hypotenuse creates two smaller triangles. How do they relate to the original?
Example 18
medium
Triangle A has sides 3,4,5. Triangle B has sides 9,12,16. Are they similar?Triangle A (sides 3-4-5) — check whether Triangle B (sides 9-12-16) is similar
Example 19
challenge
Two similar triangles have areas in the ratio 4:9. The perimeter of the smaller is 24. Find the perimeter of the larger.
Example 20
challenge
A cone is filled with water to half its height. What fraction of the cone's total volume is the water?
Example 21
challenge
In triangle ABC, point D on AB and E on AC make △ADE∼△ABC with DE∥BC. If AD=x, AB=x+6, and the area of △ADE is one-quarter the area of △ABC, find x.
Example 22
challenge
Explain why all circles are similar to each other, and why this means π is the same for every circle.
Example 23
easy
Two similar triangles have scale factor 4. A side on the small triangle is 7. Find the matching side on the large triangle.
Example 24
easy
Two similar rectangles have sides 3×5 and 9×k. Find k.
Example 25
easy
A 5-ft post casts a 3-ft shadow. At the same time a flagpole casts a 24-ft shadow. Find the flagpole's height.
Example 26
easy
A map uses scale 1:25000. Two parks are 6 cm apart on the map. Find the real distance in km.
Example 27
medium
Triangles ABC∼DEF with AB=9,AC=12,BC=15. If DE=6, find EF.Triangle ABC (scale factor = DE/AB
Example 28
medium
Two similar triangles have areas 36 and 100. Find the ratio of their corresponding sides.
Example 29
medium
In a right triangle with legs 3 and 4, the altitude to the hypotenuse divides the triangle into two smaller right triangles. Are all three triangles similar?Right triangle with legs 3 and 4; an altitude from C to the hypotenuse creates two smaller right triangles
Example 30
medium
Two similar solids have scale factor 3. The smaller has volume 40 cm3. Find the larger's volume.
Example 31
medium
In two similar triangles, one has sides 5,12,13 and the other has its shortest side 20. Find the longer leg of the second triangle.Triangle 1 (5-12-13 right triangle); the second similar triangle has its shortest side
Example 32
medium
Triangles ABC∼XYZ with ∠A=∠X, ∠B=∠Y. If AB=10,XY=15, XZ=12, find AC.
Example 33
hard
In △ABC, D lies on AB with AD:DB=2:3, and DE∥BC meets AC at E. Find the ratio of the area of △ADE to △ABC.
Example 34
hard
Two similar cones have volumes 125 and 343 cm3. Find the ratio of their slant heights.
Example 35
hard
A scale model of a building has scale 1:200. The real building has a roof area of 4000 m2. Find the model's roof area in cm2.
Example 36
hard
△ADE∼△ABC with DE∥BC. AD=x, AB=12. If area of △ADE equals 1/9 of △ABC, find x.
Example 37
challenge
A cone of height H is filled with water to height h from the apex. Find the fraction of the cone's volume that is filled, in terms of h and H.