How to Find Similar Figures?

Two shapes are called similar when both have the same properties, but may not be the same. For example, the sun and the moon may seem the same size, but in reality, they are different sizes.

Related Topics

• How to Find Similar and Congruent Figures?

A step-by-step guide to finding similar figures

In geometry, when two shapes like triangles, polygons, quadrilaterals, etc. Have common dimensions or proportions but the size or length is different, they are considered similar figures. For example, two circles (of any radius) have the same shape but different sizes because they are similar. Look at the picture below.

The symbol for expressing similar figures is the same symbol for congruence, i.e., “\(∼\)”, but it similarly does not mean the same in size. When the ratios of the corresponding sides are equal, the shapes are considered similar; that is, when dividing each set of corresponding side lengths, the number obtained is the scale factor. This number helps increase or decrease the size of the figures, but not in shape, leaving them looking like similar figures.

Similarity of triangles

Two triangles will be similar if the angles are equal (corresponding angles) and the sides are in equal proportion (corresponding sides). Similar triangles may have different lengths of sides of triangles, but their angles must be equal, and their corresponding ratio of the lengths of the sides, or the scale factor, must be the same.

If two triangles are similar, that means,

• All corresponding angle pairs of triangles are equal.
• All corresponding sides of triangles are proportional.

Similarity of polygons

Similar polygons have the same shape but differ in size. There will be certain uniform ratios in similar polygons. In other words, the corresponding angles are congruent, but the corresponding sides are proportional.

There are two important properties of similar polygons:

• The corresponding angles are equal/congruent. (Both interior and exterior angles are the same)
• The ratio of the corresponding sides is the same for all sides. Hence, the perimeters are different.

Quadrilaterals are polygons that have four sides. The sum of the interior angles of a quadrilateral is \(360\) degrees. Two quadrilaterals are similar when the three corresponding angles are the same and the two adjacent sides have equal ratios.

Exercises for Finding Similar Figures

Each pair of figures is similar. Find the missing side.

1. \(\color{blue}{100}\)
2. \(\color{blue}{9}\)
3. \(\color{blue}{12}\)

Frequently Asked Questions

How do you find similar figures in math?

Two figures are similar when all corresponding angles are equal and all corresponding sides share the same ratio. To check, match up the sides in order and divide each pair. Rectangles of 3 × 5 and 6 × 10 give 6/3 = 2 and 10/5 = 2, so they are similar with a scale factor of 2. A 3 × 5 and a 6 × 9 rectangle are not, because 2 ≠ 1.8.

How do you solve similar figures and proportions?

Write a proportion using matching sides, then cross-multiply. Suppose two similar triangles have sides 4 and 6 that match sides 10 and x. Set up 4/10 = 6/x, so 4x = 60 and x = 15. Keep the order the same in both ratios: if the smaller figure is on top in one ratio, keep it on top in the other.

How do you find similar triangles?

You only need one of three tests. AA: two pairs of matching angles are equal. SSS: all three pairs of sides have the same ratio. SAS: two pairs of sides have the same ratio and the angles between them are equal. For example, a triangle with angles of 40° and 65° is similar to any other triangle with those angles, because both third angles must be 75°.

How do you find the missing side of similar figures?

Find the scale factor from a pair of matching sides you know, then apply it to the missing side. If a small trapezoid has a base of 8 cm and the matching base on a similar trapezoid is 20 cm, the scale factor is 20 ÷ 8 = 2.5. A 6 cm side on the small trapezoid then matches 6 × 2.5 = 15 cm on the large one.

How do you find the area of similar figures?

Square the scale factor. If two figures are similar with scale factor k, their areas are in the ratio k². So if a photo is enlarged by a scale factor of 3 and the original covers 20 square inches, the enlargement covers 20 × 9 = 180 square inches. The perimeter, on the other hand, grows by k, so it only triples.

How do you find the volume of similar figures?

Cube the scale factor. For similar solids with scale factor k, the volumes are in the ratio k³. A box that is twice as long, wide and tall as a smaller box holds 2³ = 8 times as much. Working backward, if two similar cylinders have volumes of 27 and 64 cubic units, the scale factor is ∛(27/64) = 3/4.

Related Lessons You May Like

• How to find similar figures
• How to solve proportional ratios
• How to find the scale factor of a dilation
• How to use the Pythagorean Theorem
• How to solve ratio word problems

If you’d like a full workbook on similar figures, scale factors, and related geometry, Geometry for Beginners walks the topic from first principles to coordinate-plane work. For the algebra you’ll lean on when setting up the proportions, Pre-Algebra for Beginners fills in the foundations gently.