What this calculator does
Two triangles are similar when their angles match and their sides are in the same proportion, even if the triangles are different sizes. This similar triangles calculator takes the three sides of one triangle and one known side of a second, similar triangle, and works out the scale factor between them along with the two remaining sides.
The underlying idea is that once you know how much bigger or smaller the second triangle is, in the form of a single scale factor, every side scales by that same amount. Perimeter scales the same way, and area scales by the square of the factor, which is the part people most often get wrong.
The formula
The scale factor k is the known side of triangle 2 divided by the corresponding side of triangle 1 (a2 ÷ a1). Every other side of triangle 2 is then that same corresponding side of triangle 1 multiplied by k. This is the similar triangles formula behind the whole calculation: once k is fixed by one pair of sides, it applies to all three.
| Term | Meaning |
|---|---|
| Scale factor (k) | How many times larger triangle 2 is than triangle 1, along matching sides: k = a2 ÷ a1. |
| Corresponding sides | Sides in the same relative position in each triangle, opposite the matching angle. |
| Perimeter ratio | The ratio of the two perimeters, which is always equal to k for similar shapes. |
| Area ratio | The ratio of the two areas, equal to k², since area scales with the square of a linear dimension. |
The inputs explained
| Field | What to enter |
|---|---|
| Triangle 1: side a | One side of the first (reference) triangle. |
| Triangle 1: side b | A second side of the first triangle. |
| Triangle 1: side c | The third side of the first triangle. |
| Triangle 2: side corresponding to a | The side of the second triangle that corresponds to side a1 of the first triangle. |
When to use it
Scaling a drawing or model
A blueprint, map or scale model is a similar-triangle relationship in disguise: knowing one real measurement and its drawn equivalent gives the scale factor for every other length on the same drawing.
Solving a geometry problem with a shadow or mirror
Classic height problems, such as finding a tree or building height from a shadow and a known-height stick, work by setting up two similar triangles and solving for the missing side, exactly what this similar triangles calculator does.
Checking a scaled design keeps proportions
When resizing a triangular panel, gusset or truss for a different job, applying the same scale factor to every side keeps the new piece the same shape, not just the same rough size.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the scale factor changes as the known side of triangle 2 changes
A fixed reference triangle with sides 3, 4 and 5, against a range of values for the corresponding side of triangle 2.
| Triangle 2: side a | Scale factor (k) | Triangle 2: side b | Triangle 2: side c | Area ratio |
|---|---|---|---|---|
| 6 | 2.000× | 8.000 | 10.000 | 4.000× (k²) |
| 9 | 3.000× | 12.000 | 15.000 | 9.000× (k²) |
| 12 | 4.000× | 16.000 | 20.000 | 16.000× (k²) |
| 15 | 5.000× | 20.000 | 25.000 | 25.000× (k²) |
| 18 | 6.000× | 24.000 | 30.000 | 36.000× (k²) |
How the missing sides change as triangle 1 shrinks, for a fixed triangle 2
The corresponding side of triangle 2 held fixed at 12, against a range of values for side a1 of the smaller, reference triangle.
| Triangle 1: side a | Scale factor (k) | Triangle 2: side b | Triangle 2: side c |
|---|---|---|---|
| 2 | 6.000× | 24.000 | 30.000 |
| 3 | 4.000× | 16.000 | 20.000 |
| 4 | 3.000× | 12.000 | 15.000 |
| 6 | 2.000× | 8.000 | 10.000 |
| 8 | 1.500× | 6.000 | 7.500 |
Questions
What does it mean for two triangles to be similar?
Their corresponding angles are equal and their corresponding sides are all in the same ratio to each other. The triangles can be different sizes and different orientations; only the shape has to match.
How is the similar triangles formula different from congruence?
Congruent triangles are identical in both shape and size. Similar triangles share the same shape but can differ in size, related by a single scale factor applied to every side.
Do I need to know all three sides of both triangles to use a similar triangles calculator?
No. Because similar triangles share one fixed scale factor across every side, one known pair of corresponding sides is enough to work out the other two sides of the second triangle, provided you already know all three sides of the first.
Why does area scale by k² rather than k?
Area is the product of two linear measurements, such as base and height, and both of those scale by k. Multiplying two quantities that each scale by k gives a result that scales by k², so a triangle twice the linear size has four times the area, not twice.
For the angles and area of a single triangle from its side lengths, see the triangle calculator. If you already know one angle and want the sides using trigonometry rather than a similarity ratio, the triangle trigonometry calculator covers that case.