Similar Triangles Calculator

Similar triangles are triangles that have the same shape but possibly different sizes — equal corresponding angles, and corresponding sides in the same ratio. Triangle similarity is the foundation of the entire concept of geometric similarity and the basis for indirect measurement (finding the height of a building from its shadow, scaling architectural blueprints, and more). This tutorial covers the three similarity postulates (AA, SSS-sim, SAS-sim), the scale factor, how to find missing sides via proportions, and the relationship between similarity and congruence.

What "similar" means

Two triangles △ABC and △DEF are similar (written △ABC ~ △DEF) when BOTH of the following hold:

1. Corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.
2. Corresponding sides are proportional: AB/DE = BC/EF = CA/FD = k (the scale factor).

Either condition implies the other for triangles (this is the special property of triangles that makes similarity easy to test — see the postulates below). For more complex polygons (quadrilaterals, pentagons, etc.), both conditions must be verified separately.

The three similarity postulates

For triangles, you only need to verify one of these three conditions to conclude similarity:

AA is the most commonly used because it requires the least information. If two angles match, the third is automatically equal (all three sum to 180°), and once all three angles match the side ratios are forced.

The scale factor

The scale factor k from △ABC to △DEF is the ratio of corresponding sides:

k = DE / AB = EF / BC = FD / CA

All three ratios MUST be equal for the triangles to be similar — that's the definition.

• k = 1: the triangles are congruent (same shape AND same size)
• k > 1: △DEF is an enlargement of △ABC
• 0 < k < 1: △DEF is a reduction of △ABC

Worked example — finding a missing side using AA

Triangle 1 (△ABC) has sides AB = 5, BC = 8, with angle ∠B = 50°.
Triangle 2 (△DEF) is similar to △ABC, with corresponding side DE = 7.5. Find EF.

Step 1: confirm similarity via AA (angles ∠B and ∠E are corresponding; if they're both 50° in matching positions, the AA condition is met — assumed here since the problem states similarity).

Step 2: scale factor: k = DE / AB = 7.5 / 5 = 1.5.

Step 3: find EF using k: EF = BC × k = 8 × 1.5 = 12.

This is the universal method: identify the scale factor from any one pair of corresponding sides, then multiply.

Worked example — proving similarity via AA

△ABC has angles 50°, 60°, 70°. △DEF has angles 70°, 60°, 50°. Are they similar?

Both triangles have the same set of three angles. So yes, they are similar by AA (two pairs of equal angles → all three angles match).

Note: the matching must be at corresponding vertices. If △ABC has ∠A = 50°, ∠B = 60°, ∠C = 70° and △DEF has ∠D = 70°, ∠E = 60°, ∠F = 50°, then ∠A corresponds to ∠F (both 50°), ∠B to ∠E (both 60°), ∠C to ∠D (both 70°). So the correct similarity statement is △ABC ~ △FED, NOT △ABC ~ △DEF.

How similarity differs from congruence

Every pair of congruent triangles is similar (with k = 1). Most similar triangles are NOT congruent — they share shape but not size.

Why is there no "AAA" similarity?

"AAA" isn't needed because once two angles match, the third is determined by the 180° rule. AA is sufficient; the third A is redundant.

There is also no "AAA" or "AAA-sim" because three matching angles only prove SIMILARITY, not CONGRUENCE. Two triangles can have the same angles and very different sizes (a small 30-60-90 and a giant 30-60-90 are similar but not congruent).

Why is there no "ASA-sim"?

Because ASA-sim collapses to AA-sim. If two angles are equal, the third is automatically equal — so adding a "side" requirement actually upgrades you to congruence (the included side fixes the scale). ASA on its own is a congruence postulate, not a similarity one.

The area and perimeter ratios

For similar triangles with scale factor k:

• Corresponding sides are in ratio k
• Perimeters are in ratio k
• Areas are in ratio k²

This is the central "scale factor for length scales the square for area" rule. Doubling all sides quadruples the area; halving all sides quarters the area.

Example: △ABC has area 12 and side AB = 4. △DEF is similar with side DE = 6. Scale factor k = 6/4 = 1.5. Area of △DEF = 12 × k² = 12 × 2.25 = 27.

Real-world applications — indirect measurement

Classic problem: find the height of a tree (or building, flagpole, etc.) without climbing it.

Setup: place a meter stick vertically on the ground in sunlight. The stick casts a shadow of measurable length. The tree, in the same sunlight at the same time, casts a longer shadow.

The tree and the stick form similar triangles with the ground and the sun's rays. By similar-triangle proportions:

tree height / tree shadow = stick height / stick shadow

If the stick is 1 m and casts a shadow of 0.8 m, and the tree casts a shadow of 12 m: tree height = (1 / 0.8) × 12 = 15 m.

This technique is still used today for surveying and astronomy (the same idea was used by Eratosthenes around 240 BC to estimate the Earth's circumference to within 2% — using the shadow lengths of vertical sticks in two distant cities at the same moment).

Common mistakes

• Matching the wrong vertices. The similarity statement △ABC ~ △DEF specifies that A corresponds to D, B to E, C to F (in order). Get the order wrong and the side ratios are also wrong.
• Forgetting area scales as k², not k. Doubling sides quadruples area.
• Assuming "similar" means "looks alike". Similarity is a precise mathematical relationship — equal angles AND proportional sides. Two figures that visually appear similar may not satisfy the conditions exactly.
• Using SSA-similarity. SSA is not a similarity postulate any more than it is a congruence postulate. Two sides plus a non-included angle do not lock down a triangle, similar or otherwise.