Similar Triangles Calculator - Proportion and Scale Factor

The similar triangles calculator solves the proportion a / b = c / d for any one of the four side lengths and then reports the linear scale factor, the perimeter ratio, and the area ratio. Pick which side is unknown with the Find toggle, type the three known sides in the same unit, and the result updates as you type.

• • Geometry homework: Use a / b = c / d to find the missing side when the problem gives three of the four corresponding sides.
• • Scale models and maps: Convert a measured length on a model or map into the real-world length using the printed scale ratio.
• • Shadows and indirect measurement: Use a stick and its shadow plus a tall object's shadow to estimate the height of trees, buildings, or poles.
• • Surveying and engineering: Reduce a hard-to-measure distance to a similar triangle on a plan, then scale back to the real length.

Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio. The proportion a / b = c / d is the practical form of that statement. The same input also gives the scale factor k = c / a and the area ratio k squared, so doubling the linear ratio quadruples the area. Keep every input in the same unit to avoid mixing feet with meters.

For the rest of a triangle's properties (angles, perimeter, inradius) once the sides are known, the Triangle Calculator extends the same side inputs to a complete triangle solution.

The similar triangles calculator applies the proportion a / b = c / d using the side the Find toggle marks as unknown. Once that side is known, it returns the scale factor k = c / a, the perimeter ratio (also k), and the area ratio (k squared). The default mode assumes side d is the unknown, so a, b, and c are the inputs.

• a, b: Two known sides of the reference triangle; a is the divisor when side d is the unknown.
• c, d: Corresponding sides of the similar triangle; d is the side the calculator most often solves for.
• k (scale factor): Linear ratio of the similar triangle to the reference triangle, equal to c / a or d / b.

The Find toggle controls which side is solved. Setting Find to Side a means a = (b * c) / d; setting it to Side b means b = (a * d) / c. Setting it to Side c means c = (a * d) / b; the default Side d means d = (b * c) / a.

After the missing side is known, k = c / a works for every mode because both a and c are now numbers. The perimeter ratio equals k because perimeter scales linearly, and the area ratio is k squared because area scales with the square of any linear dimension. A 3 m reference triangle and a 9 m similar triangle have a scale factor of 3, not 3 m.

According to Wolfram MathWorld, corresponding sides of similar triangles are in the same ratio k and the ratio of their areas is k squared.

Right triangles are a special case of similar triangles because the 3-4-5, 5-12-13, and 8-15-17 families all share the same shape, so the Pythagorean Triples Calculator is a natural neighbor for the proportion a / b = c / d.

These four ideas decide how the proportion behaves and how to read the scale factor, perimeter ratio, and area ratio.

AA similarity is the most common way to prove two triangles are similar, but the calculator only needs the consequence: the proportion a / b = c / d. The math is the same whether AA, SSS, or SAS was used to justify the similarity.

Once the side lengths of one of the two similar triangles are known, the Scalene Triangle Area Calculator uses Heron's formula to convert those three sides into an actual area, which is the input the area ratio scales.

Pick the side to solve for, enter the three known side lengths, and read the four derived values in the results panel.

1. 1 Choose which side is the unknown: Set the Find dropdown to Side a, Side b, Side c, or Side d. The default is Side d, the most common missing-side problem.
2. 2 Enter the three known side lengths: Type the other three sides in the same length unit. Leave the unknown side at zero and the calculator overwrites it.
3. 3 Read the missing side and the scale factor: The missing side is the primary result in the same unit as the inputs. k = c / a is the linear ratio: k = 1 means the triangles are congruent, k > 1 means the similar triangle is larger, 0 < k < 1 means smaller.
4. 4 Read the perimeter and area ratios: The perimeter ratio equals k. The area ratio is k squared and tells you how many times larger the surface of the similar triangle is than the reference.

A surveyor wants the height of a tree whose shadow is 24 feet. A 5 foot stick casts a 3 foot shadow. The triangles formed by the stick, the tree, and their shadows are similar, so a / b = c / d becomes 5 / 3 = h / 24. Set Find to Side a, enter b = 3, d = 24, and the calculator returns h = 40, the height of the tree in feet.

When the similar triangles in the problem are both right triangles, the Right Triangle Calculator carries the proportion a / b = c / d into a full Pythagorean solution for the second triangle.

Putting the proportion, the scale factor, and the area ratio in one similar triangles calculator keeps the work straight.

• • Solve for any of the four sides: The Find toggle turns the same proportion into four different problems, so a / b = c / d, a / c = b / d, and the other rearrangements all work without rewriting the formula.
• • Three derived values in one pass: Scale factor, perimeter ratio, and area ratio update together, so the linear and the area consequences of a similarity stay in lockstep.
• • Honest input checks: Zero, negative, and non-numeric sides are caught with a clear message, so the proportion never silently returns a wrong number from a typo.
• • Works in any length unit: Inches, feet, centimeters, and meters all work because the proportion only requires the three inputs to share one unit. The derived ratios are unitless.

The scale factor and the area ratio are the two numbers a geometry teacher is most likely to ask for after a missing-side problem. The similar triangles calculator reports them together to remove the second pass that students usually do by hand. A correct missing side is enough to confirm that the user has the correspondence right; if the result looks unreasonable, the issue is usually the pairing of a, b, c, and d, not the arithmetic.

For map and model work, the scale factor k is the same linear ratio the Scale Conversion Calculator uses to move between a model dimension and the real-world length, so the two tools line up directly.

The proportion is exact, but the inputs and the correspondence decision decide how trustworthy the result is.

• • The calculator does not check that the four sides form two valid triangles. A real similarity needs positive sides and triangles that close, which is a separate step from the proportion.
• • The proportion a / b = c / d is silent about angles. Two triangles with the same side lengths can be similar and mirror-reflected, which the proportion treats as identical.

If the result looks too small or too large, the first thing to check is the pairing of a with c and b with d. Keep the same unit across all three inputs.

According to Math Open Reference, sides of similar triangles are in proportion and perimeters share the same ratio, while areas are in the ratio of the linear scale factor squared.

According to Khan Academy, the AA similarity criterion states that two triangles are similar if two pairs of corresponding angles are equal, which is the foundation for the proportion a / b = c / d.

The area ratio k squared only matters once you have a real area to scale, and the Triangle Area Calculator returns area from base and height, three sides, or two sides and the included angle.