Areas of similar figures

Learning Objectives State the relationship between the ratio of similarity and the ratio of areas for two similar figures. Calculate the ratio of areas given the ratio of corresponding side lengths. Calculate the ratio of corresponding side lengths given the ratio of areas. Solve for a missing area of a figure given a similar figure with a known area and corresponding side lengths. Solve for a missing side length of a figure given its area and a similar figure with a known area and corresponding side length. Apply the area ratio theorem to solve real-world problems involving scaling. Ever wonder how an architect can be sure a building will be sound just by looking at a small blueprint? 📐 It's all about scaling, and the math is more powerful than you think! In this tut...

TermDefinitionExample Similar FiguresTwo geometric figures are similar if they have the same shape but not necessarily the same size. This means their corresponding angles are congruent (equal) and the ratios of their corresponding side lengths are equal.A 3cm x 4cm rectangle is similar to a 6cm x 8cm rectangle because all angles are 90° and the sides are in a 1:2 ratio. Ratio of Similarity (or Scale Factor)The constant ratio of the lengths of any two corresponding sides of similar figures. It is often denoted by the variable 'k'.If Triangle ABC is similar to Triangle XYZ and side AB = 5 and corresponding side XY = 15, the ratio of similarity is 5/15 = 1/3. Corresponding SidesSides that are in the same relative position in two similar figures.In two similar triangles, the side o...

Ratio of Perimeters Theorem If the ratio of similarity of two similar figures is a : b, then the ratio of their perimeters is also a : b. Use this rule to relate the side lengths of similar figures directly to their perimeters. If you know the scale factor, you know the perimeter ratio. Ratio of Areas Theorem If the ratio of similarity of two similar figures is a : b, then the ratio of their areas is a^2 : b^2. This is the core rule for this lesson. To find the ratio of areas, you must square the ratio of the corresponding sides. This works for all similar polygons, circles, and other 2D figures. Finding an Unknown Area (Side_1 / Side_2)^2 = Area_1 / Area_2 This formula is a direct application of the Ratio of Areas Theorem. Set up a proportion with the square of the...