Similar triangles · Ages 14–15
Areas of similar figures
Area ratio = side ratio2; perimeter ratio = side ratio.
How to do it
Method
Area ratio = side ratio^2; perimeter ratio = side ratio
Sides 2 : 5, area 8 × 254 = 50
Another way
By a proportion
4 : 25 = 8 : A, A = 8 × 25 ÷ 4 = 50
In full
When similar figures' sides are in the ratio a : b, their areas are in the ratio a2 : b2. Sides 3 times as long make the area 32 = 9 times as big, not 3: every length is multiplied by 3, the length and the width (the base and the height) both, so the area by 3 × 3. One way: by the factor. A small triangle of 8 cm² with sides in the ratio 2 : 5 to a big one: big area = 8 × (52)2 = 8 × 254 = 50 cm². Another: by a proportion. The areas are 4 : 25, so 8 : A = 4 : 25 and A = 50. The other way round, from areas to sides, take the square root: areas 16 : 49 mean sides 4 : 7. In the A shape with DE // BC, AD : AB = 2 : 5 makes △ADE and △ABC 4 : 25, so the trapezium DBCE is 25 − 4 = 21 of those parts. Perimeters go as the sides do, not squared.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
△ABC ~ △PQR, with AB = 4 cm and PQ = 7 cm. The area of △ABC is 80 cm². The area of △PQR =
Show the answer and working
Answer: 245 cm²
Areas are 42 : 72: 80 × 72 ÷ 42 = 245 cm².
Example 2
In the picture, DE // BC, and the area of △ADE is 48 cm². The area of △ABC =
Show the answer and working
Answer: 243 cm²
△ADE ~ △ABC with sides AD : AB = 4 : 9, so areas 42 : 92: 48 × 92 ÷ 42 = 243 cm².
Example 3
Two similar polygons have sides in the ratio 4 : 7. The small one's area is 32 cm². What is the big one's?
- A. 112 cm²
- B. 51249 cm²
- C. 56 cm²
- D. 98 cm²
Choose the right answer
Show the answer and working
Answer: D. 98 cm²
Areas are 42 : 72, so the big one = 32 × 72 ÷ 42 = 98 cm².
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Two similar triangles have corresponding sides in the ratio 1 : 7. The big triangle's area is times the small one's.
Answer:
2.Two similar figures have sides in the ratio 3 : 7, so their areas are in the ratio 9 :
Answer:
3.△ABC ~ △PQR, with AB = 5 cm and PQ = 8 cm. The area of △ABC is 50 cm². The area of △PQR =
Answer:
4.In the picture, DE // BC, and the area of △ADE is 75 cm². The area of △ABC =
Answer:
5.In the picture, DE // BC, and the area of △ADE is 16 cm². The area of △ABC = cm², and of DBCE = cm².
Fill in each blank
Answer:
All the answers
- 1. 49
- 2. 49
- 3. 128 cm²
- 4. 243 cm²
- 5. 49, 33
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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