Mathematics Year 7: Ratios of perimeters and areas of similar figures
Mathematics, Year 7 (Portugal): Perimeter is multiplied by k and area by k²: find out why and use it with plans, enlargements and problems.
Perimeter times k, area times k²
If two figures are similar with ratio of similarity k, the perimeter of the larger is k times that of the smaller, and its area is k × k = k² times as big. A square of side 3 cm and a square of side 6 cm are similar with k = 2. Perimeters: 12 cm and 24 cm, double. Areas: 9 cm² and 36 cm², four times as big, because 2² = 4. The ratio of the areas is the square of the ratio of the perimeters.
Why area is multiplied by k squared
A perimeter is a sum of lengths, and every length becomes k times bigger, so the sum does too. Area counts little squares, and both the length and the width become k times bigger. With k = 2, a big square holds 2 × 2 = 4 small squares; with k = 3, it holds 3 × 3 = 9. In general, k × k. With k below 1 it works the same way: with k = 1/2, the area becomes 1/2 × 1/2 = 1/4.
Enlarging a triangle three times
A triangle has sides 5 cm, 12 cm and 13 cm, with a right angle between the 5 cm and 12 cm sides. Perimeter: 5 + 12 + 13 = 30 cm. Area: 5 × 12 ÷ 2 = 30 cm². Enlarge it with k = 3: the sides become 15, 36 and 39 cm. Perimeter: 15 + 36 + 39 = 90 cm = 3 × 30. Area: 15 × 36 ÷ 2 = 270 cm² = 9 × 30 = 3² × 30. The other way round: a similar figure with a quarter of the area needs k² = 1/4, so k = 1/2, and its perimeter is half.
The trap: doubling the sides does not double the area
It is natural to think that if the sides double, the area doubles too. A square of side 5 cm has area 25 cm². With the sides doubled, side 10 cm, its area is 100 cm², not 50 cm²: the area became four times as big, because 2² = 4. Only the perimeter doubles, from 20 cm to 40 cm. Do not mix up the two ratios: for perimeters it is k, for areas it is k². And to double the area, the ratio of similarity is not 2.
Where you see this outside school
On a plan at scale 1:50, a room of 4 m by 6 m is drawn 8 cm by 12 cm. On the drawing the area is 8 × 12 = 96 cm²; in reality it is 24 m² = 240 000 cm². The ratio of the areas is 240 000 ÷ 96 = 2500 = 50², the square of the scale. In the same way, painting a wall with twice the length and twice the height takes four times as much paint.
Test what you learned
Two similar figures have ratio of similarity 3. What are the ratio of their perimeters and the ratio of their areas?
- a) Perimeters 3, areas 9
- b) Perimeters 3, areas 3
- c) Perimeters 9, areas 9
- d) Perimeters 9, areas 3
Correct answer: a) Perimeters 3, areas 9 — Right: perimeters are multiplied by k = 3 and areas by k² = 3 × 3 = 9.
A 5 cm by 8 cm rectangle is enlarged with ratio of similarity 3. What is the area of the enlarged rectangle?
- a) 120 cm²
- b) 360 cm²
- c) 1080 cm²
Correct answer: b) 360 cm² — Right: 40 × 3² = 40 × 9 = 360 cm². Check: the new sides are 15 and 24, and 15 × 24 = 360.
Joana doubles the sides of a square of side 5 cm and expects double the area, 50 cm². What is the area of the new square?
- a) 50 cm²
- b) 40 cm²
- c) 200 cm²
- d) 100 cm²
Correct answer: d) 100 cm² — Right: the new side is 10 cm and 10 × 10 = 100 cm², which is 25 × 2² = 25 × 4.
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