How do you calculate the area of a similar triangle?

To calculate the area of a similar triangle, use the ratio of the areas which is the square of the ratio of their sides.

When dealing with similar triangles, it's important to remember that their corresponding angles are equal, and their corresponding sides are proportional. This means that if you know the ratio of the lengths of two corresponding sides, you can find the ratio of their areas.

Let's say you have two similar triangles, Triangle A and Triangle B. If the ratio of the lengths of their corresponding sides is \( \frac{a}{b} \), then the ratio of their areas will be \( \left(\frac{a}{b}\right)^2 \). For example, if the sides of Triangle A are twice as long as the sides of Triangle B, the area of Triangle A will be \( 2^2 = 4 \) times the area of Triangle B.

To put this into practice, suppose Triangle A has an area of 50 square centimetres and the sides of Triangle A are 1.5 times the length of the sides of Triangle B. The ratio of the sides is \( \frac{3}{2} \). Therefore, the ratio of the areas is \( \left(\frac{3}{2}\right)^2 = \frac{9}{4} \). If you want to find the area of Triangle B, you would set up the equation \( \frac{Area\ of\ Triangle\ A}{Area\ of\ Triangle\ B} = \frac{9}{4} \). Solving for the area of Triangle B, you get \( Area\ of\ Triangle\ B = \frac{4}{9} \times 50 = 22.22 \) square centimetres.

Understanding this relationship helps you quickly determine the area of similar triangles without needing to measure all sides and angles.