Solve Similar Triangles Calculator

How to use this similar triangles calculator

What this calculator does

This calculator first reconstructs one complete Euclidean triangle from a valid set of measurements, then uses one corresponding side of a second triangle to determine the similarity scale factor. It returns the three second-triangle side lengths, the three shared angles, both perimeters, both areas, and the area scale factor. It is designed for ordinary plane-geometry problems in which corresponding vertices are already matched: a corresponds to A, b to B, and c to C. It does not prove that two unrelated measured shapes are similar, and it does not solve the ambiguous side-side-angle case, which can produce zero, one, or two triangles.

When to use it

Use it to estimate an inaccessible height from a shadow or reference object, enlarge or reduce a triangular drawing while preserving shape, check scale-model dimensions, or convert a solved triangle into another triangle with the same angles. Similar triangles have equal corresponding angles and proportional corresponding sides; the similar-triangle relationship explains why one side ratio determines every other length ratio.

How to calculate

1. The calculator opens with a complete demonstration: γ = 90°, a = 5, b = 8, and B = 24. The results and a validated Excel workbook are ready immediately.
2. Choose Given information. Use Angle and 2 adjacent sides (SAS), Three sides (SSS), or Two angles and 1 side (ASA/AAS).
3. Replace the sample values in the visible method panel. All side lengths must be positive, each angle must be between 0° and 180°, and two known angles must total less than 180°.
4. Select Known second-triangle side and enter its Known second-triangle length. Keep every side value in the same physical unit.
5. Read Similarity scale factor, k, Side A, Side B, Side C, the angle chips, perimeter and area cards, and the correspondence table. Use Download Excel to export the current canonical values.
6. Reset clears the demonstration and all computed content. Download Excel is then disabled until a complete valid state is entered again.

Input guide

Given information is required and selects the mathematical method. In SAS mode, Included angle chooses α, β, or γ; Angle measure accepts a decimal degree value strictly between 0 and 180; and the two dynamically named adjacent side fields accept positive decimal lengths. For example, γ = 90, side a = 5, and side b = 8 form a right triangle. A common mistake is selecting an angle that is not between the two entered sides. SAS uses the Law of Cosines to obtain the opposite side.

In SSS mode, Side a, Side b, and Side c are all required positive decimals. They must satisfy the triangle inequality: each pair must sum to more than the remaining side. Values 3, 4, and 5 are a valid example; 2, 3, and 5 are not. In ASA/AAS mode, First known angle and Second known angle must be different labels, their two measures must be positive and sum to less than 180°, and Known side with Known side length supplies the scale of the first triangle. The remaining sides are calculated through the Law of Sines.

Known second-triangle side identifies A, B, or C, and Known second-triangle length is a required positive decimal. Dividing this value by the matching first-triangle side produces k. A larger second-side value gives k > 1 and an enlargement; a smaller value gives 0 < k < 1 and a reduction. Do not mix feet with inches, centimeters with meters, or any other units without converting first.

Output guide

Similarity scale factor, k is an exact proportional relationship derived from the entered measurements, subject only to displayed rounding. Side A, Side B, and Side C are estimated decimal lengths in the same unit as the inputs. The α, β, and γ chips are the shared corresponding angles; they always total 180° in a valid plane triangle. First perimeter and Second perimeter are the sums of their three sides and therefore change in direct proportion to k. First area and Second area use Heron's formula after the sides are solved. Area scale factor equals k², so doubling every side multiplies area by four. The summary pills repeat the active method, k, and k², while the Corresponding measurements table shows the same canonical model values row by row.

Worked example

In the startup example, SAS uses γ = 90°, a = 5, and b = 8. The opposite side is c = √(5² + 8²) = √89 ≈ 9.4340. Because the known second side is B = 24 and its corresponding first side is b = 8, k = 24 ÷ 8 = 3. Therefore A = 5 × 3 = 15.0000, B = 24.0000, and C = √89 × 3 ≈ 28.3019. The first area is 5 × 8 ÷ 2 = 20.0000, and the second area is 20 × 3² = 180.0000. The displayed angles are α ≈ 32.0054°, β ≈ 57.9946°, and γ = 90.0000°.