Similar Triangles Calculator

What Are Similar Triangles?

In Euclidean geometry, two triangles are said to be similar ($\triangle ABC \sim \triangle DEF$) if they have the exact same shape, even if they have different sizes. Specifically, two triangles are similar if and only if:

Their corresponding interior angles are congruent (equal in measure).
Their corresponding side lengths are proportional (they share a constant ratio, known as the scale factor $k$).

Similar triangles are fundamental in trigonometry, surveying, computer graphics, and architectural scale models. For specialized right-angled triangles, check our dedicated Similar Right Triangles Calculator and Scalene Triangle Calculator.

Theorems for Proving Triangle Similarity

You do not need to measure all six parts (three sides and three angles) to prove that two triangles are similar. Mathematicians rely on three core similarity postulates:

1. Angle-Angle (AA) Similarity

If two angles of one triangle are congruent to two angles of another triangle, then the third angle must also be equal (since triangle angles always sum to 180 degrees), and the two triangles are guaranteed to be similar.

2. Side-Side-Side (SSS) Similarity

If all three corresponding sides of two triangles are proportional, then the triangles are similar:

$$\frac{a_2}{a_1} = \frac{b_2}{b_1} = \frac{c_2}{c_1} = k$$

3. Side-Angle-Side (SAS) Similarity

If two sides of one triangle are proportional to two sides of another triangle and their included angles (the angle between the two measured sides) are congruent, then the triangles are similar.

Scale Factor, Perimeter, and Area Relationships

The constant ratio between corresponding side lengths of two similar triangles is called the linear scale factor ($k$):

$$k = \frac{\text{Side of Triangle 2}}{\text{Side of Triangle 1}}$$

Understanding how scale factor impacts dimensions is crucial:

Perimeters scale linearly: The perimeter of the second triangle is multiplied by $k$: $$P_2 = k \times P_1$$
Areas scale quadratically: The surface area of the second triangle is multiplied by the square of the scale factor: $$A_2 = k^2 \times A_1$$

How to Find Missing Sides in Similar Triangles

Identify corresponding sides that match between the two figures.
Compute the scale factor $k = \text{known side on Triangle 2} / \text{corresponding side on Triangle 1}$.
To find an unknown side on Triangle 2, multiply the matching side on Triangle 1 by $k$.
To find an unknown side on Triangle 1, divide the matching side on Triangle 2 by $k$.

Frequently Asked Questions

What is the difference between similar triangles and congruent triangles?

Congruent triangles have both the same shape and the exact same size (scale factor k = 1). Similar triangles have the same shape with identical angles, but their sizes can differ by any positive scale factor k. All congruent triangles are similar, but not all similar triangles are congruent.

Are all equilateral triangles similar?

Yes. Every equilateral triangle has three interior angles of exactly 60 degrees. By the Angle-Angle (AA) similarity criterion, all equilateral triangles in Euclidean space are similar to one another regardless of side length.

If the scale factor between two similar triangles is 3, what is the ratio of their areas?

The ratio of their areas is equal to the square of the linear scale factor: k² = 3² = 9. The larger triangle has nine times the area of the smaller triangle.

How do you find the scale factor if you only know the areas of two similar triangles?

Divide the area of the second triangle by the area of the first triangle to find the area ratio (k²), then take the square root to determine the linear scale factor: k = √(A₂ / A₁).

What is the AAA theorem in triangle geometry?

The AAA (Angle-Angle-Angle) condition proves that two triangles are similar, but it does NOT prove that they are congruent. Because knowledge of two angles automatically fixes the third angle, this is commonly referred to simply as the AA similarity shortcut.

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