Free Similar Triangles Calculator

Understanding Similar Triangles

Similar triangles are geometric figures that share the same shape but may differ in size. Their corresponding angles are equal, and the lengths of corresponding sides are proportional by a constant factor called the scale factor ($k$). For instance, if $\triangle ABC \sim \triangle DEF$, then:

$$k = \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$$

and $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$. This free similar triangles calculator lets you verify similarity conditions and find missing side lengths or scale factors instantly.

Three Core Criteria for Checking Triangle Similarity

1. SSS (Side-Side-Side)

If all three pairs of corresponding sides are in the same proportion, the triangles are similar. This is the most straightforward rule—you only need the side lengths of both triangles. The calculator checks whether:

$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = k$$

2. SAS (Side-Angle-Side)

If two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal, the triangles are similar. This test is especially useful when you know only two sides and the angle between them:

$$\frac{AB}{DE} = \frac{BC}{EF},\quad \angle B = \angle E$$

It works well when the triangles share a common angle, reducing the amount of data needed.

3. ASA (Angle-Side-Angle)

If two angles of one triangle equal the corresponding angles of another triangle and the sides between those angles are in proportion, similarity holds. Because the sum of interior angles is always $180^{\circ}$, knowing two angles effectively gives you the third. However, to determine the scale factor $k$, you still need at least one side length:

$$\angle A = \angle D,\ \angle B = \angle E,\ \frac{AB}{DE} = k$$

Finding the Missing Side of a Similar Triangle

Once similarity is established, the scale factor $k$ unlocks unknown side lengths. Follow these steps:

1. Compute $k$ as the ratio of a known side on the larger triangle to its corresponding side on the smaller triangle: $k = \dfrac{\text{larger side}}{\text{smaller side}}$.
2. Determine whether the triangle with the missing side is larger or smaller:
   • If the missing side belongs to the smaller triangle, divide the corresponding large side by $k$.
   • If the missing side belongs to the larger triangle, multiply the corresponding small side by $k$.

Example: A larger triangle has a side of length 10; the smaller triangle’s corresponding side is 5, so $k = 2$. If the smaller triangle has another side of 3, the corresponding larger side is $3 \times 2 = 6$.

Relationship Between Area and Scale Factor

The areas of similar triangles scale by the square of the scale factor. If $A_{\text{large}}$ and $A_{\text{small}}$ denote the areas of the larger and smaller triangles, then:

$$\frac{A_{\text{large}}}{A_{\text{small}}} = k^{2}$$

Thus, when you know the area of one triangle and the scale factor, you can find the other area: $A_{\text{small}} = \dfrac{A_{\text{large}}}{k^{2}}$ or $A_{\text{large}} = A_{\text{small}} \times k^{2}$.

How to Use This Similar Triangles Calculator

This online geometry calculator offers two main modes:

• Check Triangle Similarity: Choose between SSS, SAS, or ASA, input the required side/angle data, and the tool immediately reports whether the triangles are similar.
• Find Missing Side of Similar Triangle: Provide known side lengths, areas, or the scale factor, and the calculator automatically computes the unknown value.

Whether you are a student verifying homework or an engineer solving design proportions, this triangle proportion calculator simplifies the process and eliminates manual errors.