Similar Triangles - ratio of areas

 

In two similar triangles, the ratio of their areas is the square of the ratio of their sides.

Try this The two triangles below are similar. Drag any orange dot at P,Q,R. Note the ratio of the two corresponding sides and the ratio of the areas.

In two similar triangles, the ratio of their areas is the square of the ratio of their sides. In the figure above, the left triangle LMN is fixed, but the right one PQR can be resized by dragging any vertex P,Q or R. As you drag, the two triangles will remain similar at all times.

Notice that the ratios are shown in the upper left. Press "reset" and note how the ratio of the areas is 4, which is the square of the ratios of the sides (2). As you drag the triangle PQR, notice how that one ratio is always the square of the other. Only one side pair is shown for clarity, but any pair of corresponding sides could have been used.

As can be seen in Similar Triangles - ratios of parts, the perimeter, sides, altitudes and medians are all in the same ratio. Therefore, the area ratio will be the square of any of these ratios too.

See also Similar Triangles - ratios of parts

Related topics

Similar Polygons