Ratio of Areas

Copies of the original

A square 2 units by 2 units covers 2 × 2 = 4 unit squares.

Multiply each of its sides by 3, and it becomes a square 6 by 6, which covers 6 × 6 = 36 unit squares. That is 36 ÷ 4 = 9 times as much, not 3 times as much. The large square holds 9 copies of the small one, in 3 rows of 3.

A 2 by 2 square covers 4 unit squares.

Every side multiplied by 3. The lines cut the 6 by 6 square into 3 rows of 3 copies of the 2 by 2 square, and each copy covers 4 unit squares between the dots: 9 × 4 = 36.

A triangle does the same

Squares fit together neatly, but the count works for a triangle too. Enlarge a triangle by a scale factor of 2, and the large triangle holds exactly 4 copies of the small one: three the same way up, and one turned round in the middle.

Enlarge it by 3, and the large triangle holds 9 copies. Count them in rows from the top corner: 1, then 3, then 5, and 1 + 3 + 5 = 9.

A right-angled triangle with sides of 2 and 2, enlarged by 2. The large triangle, with sides of 4 and 4, is cut into 4 copies of the small one.

The same small triangle enlarged by 3, with sides of 6 and 6. It holds 9 copies: 1 in the top row, 3 in the middle row and 5 in the bottom row.

Why the scale factor is squared

Any flat shape can be covered with small squares, as closely as you like if the squares are small enough. Enlarging the shape by a scale factor k turns each small square into a square k times as wide and k times as tall, which holds $k\times k=k^2$k × k = k² of the small squares. Every piece of the area is multiplied by $k^2$k², so the whole area is multiplied by $k^2$k².

The area formulas show it too. The area of a rectangle is length × width, and both are multiplied by k. The area of a triangle is ½ × base × height, and both the base and the height are multiplied by k. The area of a circle is $\pi r^2$πr², and a circle with radius 3r has area $\pi \times 3r\times 3r=9\pi r^2$π × 3r × 3r = 9πr².

So similar shapes whose lengths are in the ratio a : b have areas in the ratio $a^2$a² : $b^2$b². Lengths in the ratio 2 : 3 give areas in the ratio 4 : 9.

Scale factors that are not whole numbers

The rule holds for every scale factor. A triangle at half scale has ${\left(\frac{1}{2}\right)}^2=\frac{1}{4}$(1/2)² = 1/4 of the area, which the midpoint theorem showed as four equal triangles. A scale factor of 1.5 multiplies an area by $1.5^2=2.25$1.5² = 2.25.

Two similar triangles have matching sides of 4 cm and 6 cm, so the scale factor is 6 ÷ 4 = 1.5. If the smaller triangle has an area of 20 cm², the larger has an area of 20 × 2.25 = 45 cm². In ratio form: the lengths are in the ratio 4 : 6 = 2 : 3, so the areas are in the ratio 4 : 9, and $20\times \frac{9}{4}=45$20 × 9/4 = 45.

Working back to the lengths

Given the areas of two similar shapes, take the square root of their ratio to find the ratio of their lengths. Two similar photos have areas of 150 cm² and 600 cm². The areas are in the ratio 150 : 600 = 1 : 4, so the lengths are in the ratio 1 : 2, because $\sqrt{1}=1$√1 = 1 and $\sqrt{4}=2$√4 = 2. If the small photo is 10 cm wide, the large one is 20 cm wide.

The ratio does not have to be a perfect square. A sheet of A3 paper is similar to a sheet of A4 and has twice its area. So the sides of A3 are $\sqrt{2}$√2 times as long as the sides of A4, and $\sqrt{2}\approx 1.41$√2 ≈ 1.41. A4 measures 210 mm by 297 mm, and A3 measures 297 mm by 420 mm: $297÷210\approx 1.41$297 ÷ 210 ≈ 1.41 and $420÷297\approx 1.41$420 ÷ 297 ≈ 1.41, rounded to two decimal places.

A triangle cut off by a parallel line

In the figure below, a line parallel to the base cuts a small triangle off the top of a large one. The two triangles share the top angle, and the parallel line makes equal corresponding angles with the sides, so they are similar.

The small triangle's base is 6 units and the large triangle's is 9, so the scale factor is 9 ÷ 6 = 1.5. The areas are multiplied by $1.5^2=2.25$1.5² = 2.25. In ratio form, the lengths are in the ratio 2 : 3, so the areas are in the ratio 4 : 9, since $2^2=4$2² = 4 and $3^2=9$3² = 9.

Check by counting. The small triangle has a base of 6 and a height of 4, so its area is ½ × 6 × 4 = 12 square units. The large one has a base of 9 and a height of 6, so its area is ½ × 9 × 6 = 27, and 12 × 2.25 = 27. The strip between the line and the base is a trapezium, and its area is what is left: 27 − 12 = 15. In the ratio 4 : 9, the trapezium takes the other 9 − 4 = 5 parts: $27\times \frac{5}{9}=15$27 × 5/9 = 15.

The line across is parallel to the base and 4 units below the top corner. It is 6 units long; the base is 9 units long, and 6 units below the top corner.

The usual mistakes

Multiplying the area by the scale factor itself. With a scale factor of 3, an area of 5 cm² becomes $5\times 3^2=5\times 9=45$5 × 3² = 5 × 9 = 45 cm², not 5 × 3 = 15 cm².

Using $k^3$k³, the rule for volume. A flat area has two dimensions, so the scale factor is used twice: 5 × 27 = 135 is not the answer.

Going back from areas without the square root. Areas in the ratio 1 : 4 come from lengths in the ratio 1 : 2, because $2^2=4$2² = 4.