Two triangles are similar when they have the same shape at different sizes: every pair of corresponding angles is equal, and every pair of corresponding sides is in the same ratio. That single ratio is called the scale factor. This similar triangles calculator works in both directions. Give it three sides of one triangle and one side of the other, and it scales the rest. Give it all six sides, and it tests whether the two shapes really are similar, prints the three side ratios so you can see how close they are, and states which criterion the evidence satisfies.
Arb Digital keeps a set of free geometry tools for people who need a number checked rather than a lesson repeated. This page is about the relationship between two triangles. If you want the sides of one right triangle from the other two, our Pythagorean theorem calculator is the direct route, and for the area of a single triangle from its sides the triangle area calculator applies Heron's formula. Neither of those compares two figures, which is the job here.
What This Similar Triangles Calculator Does
It first checks that the sides you gave for triangle 1 can actually form a triangle — the sum of any two sides has to exceed the third — and it says so plainly if they cannot. It then computes all three interior angles by the law of cosines, so you can see the shape you are working with. Because similar triangles share their angles, those same three angles belong to triangle 2 as well.
If you supplied exactly one, or two, sides of the second triangle, the tool takes the scale factor from the first known pair and fills in the rest. If you supplied all three, it computes the three separate ratios a′/a, b′/b and c′/c, compares them against your tolerance, and returns a verdict. The area ratio in the fourth result box is the square of the scale factor, which is the single fact about similarity that most often surprises people: double the sides and the area quadruples.
How to Use It
- Enter all three sides of triangle 1. Any consistent unit works — the scale factor is dimensionless, so the answer is the same in inches or metres.
- Enter what you know of triangle 2. Put a 0 in every side you do not know. Corresponding sides must sit in the same box position: a′ pairs with a, and so on.
- Read the scale factor. Above 1 means triangle 2 is an enlargement; below 1 means a reduction; exactly 1 means the triangles are congruent, not merely similar.
- Widen the tolerance for measured data. Field measurements rarely give three ratios that agree to five decimal places, and 1–3% is a realistic allowance.
- Check the diagram. Both triangles are drawn to scale against each other so you can see the shape as well as the numbers.
The Formula and How It's Calculated
The scale factor is k = a′ / a for any pair of corresponding sides, and similarity requires that a′/a = b′/b = c′/c. The interior angles come from the law of cosines: the angle opposite side a satisfies cos A = (b² + c² − a²) ÷ (2bc). Because that formula uses only ratios of sides, scaling every side by k leaves each angle untouched — which is the algebraic reason similar triangles are equiangular.
Work the default by hand. Triangle 1 has sides 3, 4 and 5. Triangle 2 has a′ = 6 with the other two unknown, so k = 6 ÷ 3 = 2, giving b′ = 8 and c′ = 10. The area ratio is k² = 4, and indeed the 3-4-5 triangle has area 6 while the 6-8-10 triangle has area 24. The angles are 36.87°, 53.13° and 90° in both. Euclid proved the underlying result in Elements Book VI, Proposition 4: in equiangular triangles the sides about the equal angles are proportional.
The Three Similarity Criteria and When Each Applies
AA is the cheapest test. If two angles of one triangle equal two angles of the other, the third pair must match too, because the angles of any triangle sum to 180°. Two angles is therefore enough to prove similarity outright, and no side measurement is needed at all. This is why surveyors and photographers can work with similar triangles from sight lines alone.
SSS is the test this calculator performs when you enter all six sides: if all three pairs of sides are in the same ratio, the triangles are similar. SAS sits between the two — two pairs of sides in the same ratio, with the angle between those sides equal in both triangles, is sufficient, and Euclid establishes it in Elements Book VI, Proposition 6. The angle has to be the included one. Two proportional sides plus a non-included equal angle is the ambiguous case, and it does not prove similarity, which is the single most common error in school geometry proofs.
Why Areas Scale by the Square and Volumes by the Cube
Scale every length in a figure by k and each area scales by k², because area is a product of two lengths. Extend the idea to three dimensions and volume scales by k³. That is Euclid's Book VI, Proposition 19 — similar triangles are in the duplicate ratio of the corresponding sides — and it has consequences well outside geometry.
Print a design at 150% and it uses 2.25 times the ink, not 1.5 times. Double the width of a banner ad while keeping the aspect ratio and you have quadrupled its pixel count and roughly quadrupled the file size. Scale a photograph's dimensions by 0.5 to make a thumbnail and the storage falls to a quarter, not a half. Our aspect ratio calculator handles the two-dimensional version of this problem for screens and images directly.
Measuring Heights Without Climbing Anything
The classic practical use of similar triangles is indirect measurement. Stand a metre stick vertically in the sun next to a tree. The stick and its shadow form a right triangle; the tree and its shadow form a larger one with the same sun angle, so the two are similar by AA. If the stick is 1 m tall with a 0.8 m shadow, and the tree casts a 12 m shadow, the tree is 12 × (1 ÷ 0.8) = 15 m tall. The same reasoning powers a clinometer, a sextant, and the depth-of-field maths in a camera lens.
The trap is correspondence. The ratio must pair each side with its true counterpart, not simply with the side that happens to be in the same input box. If triangle 2 has been reflected or rotated, or if it was labelled by someone else, check which angle each side faces before you divide. A mismatched pairing gives three inconsistent ratios and a "not similar" verdict on two triangles that are genuinely similar — which is exactly why this tool prints all three ratios rather than only the verdict.
Similar Versus Congruent, and Why the Distinction Matters
Congruent triangles are identical in size and shape; similar triangles agree only in shape. Congruence is the special case of similarity where k = 1. Every congruence criterion has a similarity counterpart: SSS congruence becomes SSS similarity, SAS becomes SAS, and ASA and AAS both collapse into AA, because once two angles match the third is forced and no side information is needed.
There is one asymmetry worth remembering. AAA proves similarity but never congruence — you can have two triangles with identical angles and wildly different sizes. In the other direction, SSA proves neither, in general. Given two sides and a non-included angle there can be two genuinely different triangles that fit, and picking the one you expected rather than checking both is the standard way to get a wrong answer in a trigonometry exercise. The trigonometric functions calculator is useful for working through the ambiguous case by hand.
Working With Measured Rather Than Exact Data
Textbook triangles are exact; surveyed and photographed ones are not. If you measure three sides of each triangle in the field, the ratios will disagree in the third or fourth significant figure even when the shapes really are similar, because every measurement carries error. Insisting on an exact match will reject correct results.
The tolerance box exists for this. Set it to the relative precision of your worst measurement — 1% for a tape measure over a few metres, more for an estimate paced out on foot. Then judge the result on whether the spread of the three ratios sits inside that band, and record the tolerance you used alongside the conclusion. Two people can look at the same six numbers and reach opposite verdicts purely because one demanded five decimal places and the other allowed two, and that disagreement is about the tolerance, not about the geometry. Our percentage difference calculator is a quick way to quantify the gap between two ratios you are trying to reconcile.
Arb Digital designs ad and web assets around a proportional system, so a resize never means a redraw.
See Web Design Services Talk To Our TeamCommon Mistakes to Avoid
- Pairing the wrong sides — corresponding sides face corresponding angles, not whichever box you typed them into.
- Using SSA as a similarity criterion — two proportional sides with a non-included equal angle proves nothing on its own.
- Scaling area by k instead of k squared, which understates an enlargement badly and overstates a reduction.
- Demanding an exact ratio match on measured data, which rejects genuinely similar triangles because of ordinary measurement error.
- Confusing similar with congruent — congruence is only the special case where the scale factor is exactly 1.
Related Free Tools From Arb Digital
Solve a single right triangle with the Pythagorean theorem calculator, get exact radical side lengths for the 30-60-90 and 45-45-90 cases from the special right triangle calculator, find an area from three sides with the triangle area calculator, or simplify a ratio to its lowest terms with the ratio calculator. Everything else is on the free online tools hub.
Frequently Asked Questions
Equal corresponding angles and corresponding sides in a constant ratio. Either condition implies the other for triangles, so proving one is enough.
Two angles are enough on their own under the AA criterion. Otherwise you need all three side ratios for SSS, or two side ratios plus the angle between those sides for SAS.
The ratio of any side of the second triangle to the corresponding side of the first. It is the same number for all three pairs when the triangles are genuinely similar.
Area is a product of two lengths, so scaling every length by k multiplies the area by k times k. Doubling the sides gives four times the area.
Yes. Congruence is similarity with a scale factor of exactly 1, so every congruent pair is similar, but similar triangles are only congruent when their sizes match.
No. Two proportional sides and a non-included equal angle can describe two different triangles, which is why that arrangement is called the ambiguous case.
Usually because the tolerance is tighter than your measurement precision, or because the sides have been paired in the wrong order. Check the three printed ratios before accepting the verdict.
This tool is provided for educational use. Results depend entirely on the correspondence between sides that you enter, so confirm the pairing before relying on the verdict.
