What Are Similar Triangles?
Two triangles are similar when they have the same shape but possibly different sizes. Their corresponding angles are equal and their corresponding sides are all in the same ratio. That ratio is called the scale factor, written as \(k\). This calculator takes the three sides of one triangle and one known side of a second similar triangle, then computes \(k\) and every remaining side automatically.
How to Use This Calculator
Enter sides a, b, and (optionally) c of the first triangle. Then enter the corresponding side a′ of the second triangle. The tool divides a′ by a to find the scale factor \(k\), then multiplies your other sides by \(k\) to produce b′ and c′. If you only have two sides, leave c blank and c′ will simply be zero.
The Formula Explained
Because corresponding sides are proportional, the ratios are all equal:
$$\frac{a}{a^{\prime}} = \frac{b}{b^{\prime}} = \frac{c}{c^{\prime}} = \frac{1}{k}$$
Rearranged for the scale factor, $$k = \frac{a^{\prime}}{a}.$$ Once \(k\) is known, every side of the larger triangle is the matching side of the smaller one multiplied by \(k\): $$b^{\prime} = b \times k \qquad c^{\prime} = c \times k$$
Worked Example
Suppose triangle 1 has sides a = 3, b = 4, c = 5, and the corresponding side a′ = 6 on triangle 2. The scale factor is $$k = \frac{6}{3} = 2.$$ Therefore $$b^{\prime} = 4 \times 2 = 8 \qquad c^{\prime} = 5 \times 2 = 10.$$ Triangle 2 has sides 6, 8, 10 — exactly double triangle 1.
FAQ
What does a scale factor greater than 1 mean? The second triangle is larger than the first. A factor less than 1 means it is smaller, and exactly 1 means the triangles are congruent.
Do I need all three sides? No. You need at least one full pair of corresponding sides to get \(k\). Enter as many sides of triangle 1 as you have; unknown ones can be left blank.
Does this work for areas? The ratio of areas of similar triangles equals \(k^{2}\), not \(k\). This calculator returns the linear scale factor for sides.
