What is a right triangle?
A right triangle is a triangle with one right angle (90°). The side opposite the right angle is the hypotenuse, always the longest side; the other two are the legs. The other two angles always add up to 90°.
Two things about it, at least one a side, are enough to work out the rest: two sides with a² + b² = c², a side and an angle with sine, cosine and tangent.
Source OpenStax · Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem ↗OpenStax · Algebra and Trigonometry 2e, 7.2 Right Triangle Trigonometry ↗Common Core State Standards · High School Geometry, solve right triangles (HSG-SRT.C.8) ↗
How to solve a right triangle
Here it is with your numbers:
- c² = 3² + 4²
- c² = 25
- c = 5
- A = tan⁻¹(3 ÷ 4)
- A = 36.8699°
- B = 90° − 36.8699°
- B = 53.1301°
It means sides a, b and c are 3, 4 and 5 long, in the unit you typed.
For an angle, the opposite side is across from it and the adjacent side is the leg next to it. To get an angle back from two sides, use the inverse: tan⁻¹, sin⁻¹ or cos⁻¹.
Source OpenStax · Precalculus 2e, 6.3 Inverse Trigonometric Functions ↗
Special right triangles: 30-60-90 and 45-45-90
Two right triangles have sides in fixed proportions, so their sides are exact square roots. Multiply the shortest side s by these to get the others. Tap a row to try it.
Measuring a tree: standing 30 ft from it, with its top 57° up, the tree is 30 × tan 57°, which is 46.1959 ft tall. In a 30-60-90 triangle with a short leg of 5, the other leg is 5√3, which is 8.6603, and the hypotenuse is 10. A square cut corner to corner gives two 45-45-90 triangles: a 1 ft square has a diagonal of √2 ft, which is 1.4142 ft.
FAQs
How to find the 3 sides of a right triangle?
You need two measurements, at least one of them a side. From two sides, the third comes from a² + b² = c². From a side and an angle, use sine, cosine or tangent: the side opposite an angle is the hypotenuse × its sine.
Source OpenStax · Algebra and Trigonometry 2e, 7.2 Right Triangle Trigonometry ↗
How do you find the missing angle of a triangle?
A triangle’s angles add up to 180°, so in a right triangle the other two add up to 90°: if one is 30°, the other is 60°. From two legs, angle A is tan⁻¹(a ÷ b).
Source OpenStax · Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem ↗
Is a right triangle always 45 45 90?
No. It only needs one 90° angle; the other two can be any pair adding up to 90°, like 36.8699° and 53.1301° in the 3, 4, 5 triangle.
Source OpenStax · Algebra and Trigonometry 2e, 7.2 Right Triangle Trigonometry ↗
Does 30 40 50 make a right triangle?
Yes. 30² + 40² is 900 + 1,600, which is 2,500, the same as 50². It is the 3, 4, 5 triangle ten times bigger.
Source OpenStax · Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem ↗
Sources
- OpenStax: Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
- OpenStax: Algebra and Trigonometry 2e, 7.2 Right Triangle Trigonometry
- OpenStax: Precalculus 2e, 6.3 Inverse Trigonometric Functions
- Common Core State Standards: High School Geometry, solve right triangles (HSG-SRT.C.8)