Similar Triangles Calculator

How to use the similar triangles calculator

What this calculator does

This calculator tests whether two triangles are similar using Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA), and it can also complete missing corresponding sides when a scale factor is known or can be derived. Similar triangles have the same shape: corresponding angles are equal and corresponding side lengths are proportional. The tool reports the common scale factor, perimeter ratio, area ratio, completed side values, perimeters, and areas where the selected data determine them. It does not prove that labels in a diagram correspond correctly; you must match AB with DE, BC with EF, and AC with DF in the intended vertex order.

When to use it

Use the calculator to check geometry homework, verify measurements from a scale drawing or model, estimate an inaccessible length from a proportional triangle, or confirm that two designs preserve shape after resizing. The underlying proportionality is described in the OpenStax explanation of similar triangles.

How to calculate

1. The calculator opens with a 3 – 4 – 5 cm triangle and a 6 – 8 – 10 cm triangle. The first result and the Excel workbook are immediately available.
2. Choose Type. Select Check similarity to test a theorem, or Find missing side(s) to complete proportional side pairs.
3. For a check, choose Similarity criterion. Enter the fields shown for SSS, SAS, or ASA. For solving, leave genuinely unknown sides blank and enter Known scale factor k, or provide at least one complete corresponding pair from which k can be calculated.
4. Read Similarity result, Scale factor k, the perimeter and area values, and the corresponding-side table. Use Download Excel to save the current validated model.
5. Reset clears the demonstration and every numeric input. It also disables Download Excel until a complete valid state is entered again.

Input guide

Type is required and controls the workflow. Similarity criterion is required only in check mode: SSS uses all six sides; SAS uses AB, AC, Angle A and their counterparts DE, DF, Angle D; ASA uses AB, Angle A, Angle B and DE, Angle D, Angle E. Length unit is required and accepts centimeters, meters, inches, or feet. Changing it converts all entered side values rather than merely relabeling them.

AB (a), BC (b), and AC (c) are the sides of Triangle ABC; DE (d), EF (e), and DF (f) are their corresponding sides in Triangle DEF. Side values are positive decimals using a dot as the decimal separator; grouped values such as 1,250.5 are accepted, while decimal-comma and scientific notation are rejected to avoid ambiguity. In check mode, the sides required by the selected criterion cannot be blank. In solve mode, unknown sides may be blank, but every pair must become resolvable. A common mistake is pairing sides by visual position instead of vertex order.

Angle A and Angle D are the included angles for SAS. Angle A, Angle B, Angle D, and Angle E are used for ASA. Angles are required positive decimal degrees below 180°, and each triangle's two entered ASA angles must sum to less than 180°. Known scale factor k is optional in solve mode, must be positive, and means DEF length divided by ABC length. For example, k = 2 doubles every corresponding side. If both k and complete side pairs are entered, they must agree.

Output guide

Similarity result states whether the chosen theorem passes or whether a missing-side model has been completed. Scale factor k is the linear ratio DEF ÷ ABC. A value above 1 means DEF is larger; below 1 means it is smaller; 1 means equal size. Area ratio DEF ÷ ABC equals k², so it changes faster than side length. ABC perimeter and DEF perimeter are sums of their three sides and scale linearly with k. ABC area and DEF area are calculated from Heron's formula or from two sides and an included angle, depending on the selected criterion. The table lists each corresponding side pair and its ratio; a green “solved” tag marks a value inferred in missing-side mode. These are mathematical estimates based on the entered correspondence and measurement precision.

Worked example

The opening values are AB = 3 cm, BC = 4 cm, AC = 5 cm, DE = 6 cm, EF = 8 cm, and DF = 10 cm. The three ratios are 6 ÷ 3 = 2, 8 ÷ 4 = 2, and 10 ÷ 5 = 2, so the SSS test passes and k = 2. The perimeters are 3 + 4 + 5 = 12 cm and 6 + 8 + 10 = 24 cm. The smaller right triangle has area 6 cm², and the larger area is 6 × 2² = 24 cm². This is why the first-open area ratio is 4 rather than 2.