What is the Similar Triangles Calculator?
Two triangles are similar when they have the same shape but possibly different sizes — their corresponding angles are equal and their corresponding sides are in a constant ratio. This calculator finds that ratio, called the scale factor (k), from one matched pair of sides, then uses it to scale the remaining sides and to find how the areas compare.
How to use it
Enter side a from the first triangle and its matching side a' from the second triangle. Then enter the other two sides of the first triangle, b and c. The calculator returns the scale factor and the corresponding sides b' and c', plus the area ratio.
The formula explained
The scale factor is simply \(k = a' / a\). Because all corresponding sides share this ratio, \(b' = k \cdot b\) and \(c' = k \cdot c\). Area is two-dimensional, so it scales by the square of the linear factor: \(A'/A = k^2\). For example, doubling every side (\(k = 2\)) makes the triangle four times larger in area.
$$\begin{gathered} k = \dfrac{\text{Side } a'}{\text{Side } a} \\[1.5em] b' = k \cdot \text{Side } b \qquad c' = k \cdot \text{Side } c \qquad \text{Area Ratio} = k^{2} \end{gathered}$$
Worked example
Suppose triangle 1 has sides \(a = 3\), \(b = 4\), \(c = 5\), and the matching side in triangle 2 is \(a' = 6\). Then \(k = 6 / 3 = 2\). So \(b' = 2 \times 4 = 8\) and \(c' = 2 \times 5 = 10\). The area ratio is \(k^2 = 4\), meaning the second triangle has four times the area of the first.
FAQ
How do I know which sides correspond? Corresponding sides sit opposite equal angles. Pair the sides that play the same role in each triangle.
What if k is less than 1? That means the second triangle is smaller; the area ratio is still \(k^2\) (a value below 1).
Does this prove the triangles are similar? No — it assumes similarity. Confirm similarity first (AA, SSS, or SAS), then use this tool to scale sides.
