Geometry
Missing sides and angles of a right triangle from its two legs — hypotenuse, angles, area, and perimeter.
The two sides meeting at the right angle. All lengths in the same unit. Results update as you type. All calculations run locally in your browser.
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Pythagoras: hypotenuse c = √(a² + b²). The acute angles come from the sides: the angle opposite leg a is arctan(a ÷ b), and the angle opposite leg b is 90° minus that angle — the two acute angles always add to 90°. Area = ½ × a × b, and perimeter = a + b + c.
Legs 3 and 4: hypotenuse = √(3² + 4²) = √25 = 5. The angle opposite the side of 3 is arctan(3 ÷ 4) = 36.87°, so the other acute angle is 90 − 36.87 = 53.13°. Area = ½ × 3 × 4 = 6, and perimeter = 3 + 4 + 5 = 12. The 3–4–5 triangle is the smallest right triangle with whole-number sides, which is why it shows up everywhere from building sites to textbook problems.
This page takes the two legs of a right triangle. It does not solve triangles given a hypotenuse and one leg, or given an angle and a side — for those, use Pythagoras or trigonometry directly on the same formulas above. It assumes the angle between the two entered sides is exactly 90°; for any other triangle, these formulas do not apply. Angles display in degrees to two decimals.
It is Pythagoras' theorem: in a right triangle, the square built on the hypotenuse has the same area as the two squares built on the legs combined. It holds for every right triangle, not just whole-number ones.
The smallest right triangle with whole-number sides: 3² + 4² = 9 + 16 = 25 = 5². Builders have used knotted ropes in 3–4–5 proportions for millennia to lay out exact right angles, since any multiple (6–8–10, 9–12–15) works the same way.
From the ratio of the legs: the angle opposite leg a satisfies tan(angle) = a ÷ b, so angle = arctan(a ÷ b). The second acute angle is 90° minus the first, because a triangle's angles sum to 180° and the right angle already takes 90°.
Not on this page — it expects the two legs. If you know the hypotenuse c and one leg a, the other leg is √(c² − a²), which you can compute and then enter here.
Only that both legs use the same unit: the hypotenuse, area, and perimeter then come out in that unit and its square. Mixing metres and centimetres gives a wrong answer, so convert first.
Pythagoras does not apply. General triangles need the law of cosines (c² = a² + b² − 2ab·cos C) or the triangle area calculator's Heron formula, depending on what you know.
Results are estimates only and do not constitute professional advice.