Right Triangle Calculator
Solve any right triangle instantly. Enter two known values — two sides, a side and an angle, or the hypotenuse and an angle — and get every missing side, angle, the area, and the perimeter using the Pythagorean theorem and basic trigonometry.
What is a right triangle?
A right triangle (also called a right-angled triangle) is any triangle containing exactly one 90° angle. That single fixed angle is what makes right triangles special: once you know just two more pieces of information — two side lengths, or one side and one of the remaining angles — the entire shape is locked into place. Every other measurement, from the missing sides to the missing angle to the area and perimeter, can be worked out with certainty. This is the mathematical backbone behind the Pythagorean theorem and trigonometry, and it's why right triangles show up constantly in construction, navigation, physics, and design.
In the naming convention this calculator uses, the right angle always sits at point C. The two shorter sides that form that right angle are called the legs — labelled a and b — and the longest side, which sits opposite the right angle, is the hypotenuse, labelled c. Angle A is the angle opposite side a, and angle B is the angle opposite side b. Because the three interior angles of any triangle always sum to 180°, and one of those angles is already fixed at 90°, angles A and B must always add up to exactly 90° between them.
Right triangle formulas
This calculator relies on two families of formulas: the Pythagorean theorem, which relates the three sides to each other, and basic trigonometric ratios (sine, cosine, and tangent), which relate the sides to the angles. Depending on which two values you enter, the calculator automatically picks the right formula to solve for everything else. Here are the core relationships it uses:
Pythagorean theorem: a² + b² = c²
Angle A: A = arcsin(a ÷ c) = arctan(a ÷ b)
Angle B: B = 90° − A
Area: Area = (a × b) ÷ 2
Perimeter: Perimeter = a + b + c
The Pythagorean theorem only works when you already know two of the three sides. When you instead know a side and an angle, the calculator switches to trigonometry: sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. Rearranging these ratios lets the calculator solve for any missing side once one side and one non-right angle are known.
Step-by-step: how to use this calculator
- Choose what you already know: two legs, a leg and the hypotenuse, a leg and an angle, or the hypotenuse and an angle.
- Enter the two known values in the fields — they update automatically to match your chosen mode.
- Click Solve triangle to get every side, angle, the area, and the perimeter.
- Pick a unit label if you want your results displayed in cm, m, in, or ft.
Worked examples
- 3-4-5 triangle: a = 3, b = 4 → c = √(3² + 4²) = 5. Angle A ≈ 36.87°, Angle B ≈ 53.13°, Area = 6, Perimeter = 12.
- Leg + hypotenuse: a = 5, c = 13 → b = √(13² − 5²) = 12. This is the classic 5-12-13 triangle.
- Leg + angle: a = 10, ∠A = 30° → c = 10 ÷ sin(30°) = 20, b = 10 ÷ tan(30°) ≈ 17.32.
- Hypotenuse + angle: c = 10, ∠A = 45° → a = 10 × sin(45°) ≈ 7.07, b = 10 × cos(45°) ≈ 7.07 (an isosceles right triangle).
Real-world uses for a right triangle calculator
Right triangles aren't just a classroom topic — they're one of the most practical shapes in everyday problem-solving. Builders and roofers use the Pythagorean theorem to check that a wall, deck frame, or foundation is perfectly square, using the well-known 3-4-5 method: if one side measures 3 units, an adjacent side measures 4 units, and the diagonal between their endpoints measures exactly 5 units, the corner is a true right angle. Carpenters use the same logic when cutting rafters and roof trusses, where the rise, run, and rafter length form a right triangle and the pitch angle determines how steep the roof will be.
Anyone setting up a ladder against a wall is unknowingly working with a right triangle: the wall forms one leg, the ground forms the other, and the ladder itself is the hypotenuse. Knowing the ladder's length and the safe angle of inclination lets you calculate exactly how far the base should sit from the wall. The same relationship applies to ramps, staircases, and wheelchair-accessible slopes, where building codes often specify a maximum angle or a rise-to-run ratio that needs converting into real measurements on site.
In navigation and surveying, right triangles are used to calculate distances that can't be measured directly — for example, the height of a building or tree using the angle of elevation from a known distance away, or the shortest path between two points when only north-south and east-west distances are known. Pilots, sailors, and surveyors rely on exactly this kind of trigonometry to convert bearing and distance data into usable coordinates. Physics problems involving projectile motion, inclined planes, and force vectors also break down into right-triangle components, since any vector can be split into perpendicular horizontal and vertical parts.
Students and teachers use a right triangle calculator to check homework, explore how changing one value affects the others, and build intuition for the relationship between sides and angles before tackling more complex trigonometry and geometry problems. Because this tool shows the formula used for every calculation, it doubles as a way to see the working, not just the final answer.
Common right triangle side ratios
| Triangle | Side a | Side b | Hypotenuse c |
|---|---|---|---|
| 3-4-5 | 3 | 4 | 5 |
| 5-12-13 | 5 | 12 | 13 |
| 7-24-25 | 7 | 24 | 25 |
| 8-15-17 | 8 | 15 | 17 |
| 45-45-90 (isosceles) | 1 | 1 | √2 ≈ 1.414 |
| 30-60-90 | 1 | √3 ≈ 1.732 | 2 |
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Frequently asked questions
A right triangle calculator solves the unknown sides and angles of a triangle that contains a 90° angle. Enter any two known values — such as two sides, or a side and an angle — and it uses the Pythagorean theorem and trigonometric ratios to find the rest.
Use the Pythagorean theorem: c = √(a² + b²), where a and b are the two legs. For example, if a = 6 and b = 8, then c = √(36 + 64) = √100 = 10.
Since one angle is always 90°, the other two angles add up to 90°. If you know one non-right angle, subtract it from 90° to find the other. If you know two sides, use inverse trig functions such as arctan(opposite ÷ adjacent) or arcsin(opposite ÷ hypotenuse).
Area = (a × b) ÷ 2, where a and b are the two legs (the sides that meet at the right angle). The hypotenuse is not used directly in this formula.
Yes. Because the right angle is already known, any one side plus any one other angle gives enough information to solve the entire triangle using sine, cosine, and tangent ratios.
The two legs are the sides that form the right angle. The hypotenuse is the side opposite the right angle and is always the longest side of a right triangle.
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