Right Triangle Calculator
Find the missing sides, angles, area and perimeter of a right triangle.
The right triangle
A right triangle has one 90° angle. The side opposite it is the hypotenuse, and the other two are the legs. Knowing any two measurements (besides the right angle) is enough to find all the rest using the Pythagorean theorem and basic trigonometry.
Formulas
- Pythagorean theorem: a² + b² = c².
- sin A = a ÷ c, cos A = b ÷ c, tan A = a ÷ b.
- The two non-right angles add up to 90°.
Example
On the first tab (two legs) with the starting values shown (Leg a 3; Leg b 4), Hypotenuse c comes out as 5. Other results include angle a: 36.8699; angle b: 53.1301; area: 6.
The key relationships
- Pythagorean theorem: a² + b² = c², where c is the hypotenuse.
- SOH-CAH-TOA: sin A = opposite ÷ hypotenuse, cos A = adjacent ÷ hypotenuse, tan A = opposite ÷ adjacent.
- Angles: the two acute angles add up to 90°.
- Area: ½ × a × b. The altitude to the hypotenuse is a × b ÷ c.
Worked examples
| You know | Result |
|---|---|
| Legs 3 and 4 | Hypotenuse 5, angles 36.87° and 53.13°, area 6, altitude 2.4 |
| Hypotenuse 10 and leg 6 | Other leg 8, angles 36.87° and 53.13°, area 24 |
| Leg 5 and angle A = 30° | Other leg 8.66, hypotenuse 10, angle B = 60°, area 21.65 |
Special right triangles
- 45-45-90: the sides are in the ratio 1 : 1 : √2.
- 30-60-90: the sides are in the ratio 1 : √3 : 2, as in the third example above.
Next steps: Pythagorean Theorem Calculator, Triangle Calculator, Area Calculator.
Frequently asked questions
How do I solve a right triangle?
Use the Pythagorean theorem to find a missing side, and sine, cosine or tangent to find the angles.
What is the hypotenuse?
The longest side, opposite the right angle.
How do I find an angle in a right triangle?
Use the inverse trigonometric function, such as A = arctan(opposite ÷ adjacent).
Is there a shortcut for 30-60-90 triangles?
Yes. If the short leg is x, the long leg is x√3 and the hypotenuse is 2x.
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