Right Triangle Calculator — Find Missing Side and Angle
Finding the missing side or angle couldn't be easier than with our great tool — right triangle side and angle calculator. Choose two given values, type them into the calculator, and the calculator will determine the remaining unknowns in a blink of an eye! If you are wondering how to find the missing side of a right triangle, keep scrolling, and you'll find the formulas behind our calculator.
How to Find the Sides of a Right Triangle
There are a few methods of obtaining right triangle side lengths. Depending on what is given, you can use different relationships or laws to find the missing side:
1. Given Two Sides (Pythagorean Theorem)
If you know two other sides of the right triangle, it's the easiest option; all you need to do is apply the Pythagorean theorem:
a² + b² = c²
- If leg a is the missing side:
a = √(c² − b²) - If leg b is unknown:
b = √(c² − a²) - For hypotenuse c missing:
c = √(a² + b²)
2. Given an Angle and the Hypotenuse
Apply trigonometry to find the right triangle side lengths:
a = c × sin(α)ora = c × cos(β)b = c × sin(β)orb = c × cos(α)
3. Given an Angle and One Leg
Find the missing leg using trigonometric functions:
a = b × tan(α)b = a × tan(β)c = a / sin(α)c = b / cos(α)
4. Given the Area and One Leg
As we remember from the basic triangle area formula, we can calculate the area by multiplying the triangle height and base and dividing the result by two. A right triangle is a special case where one leg is the height when the second leg is the base:
area = (a × b) / 2
For example, if we know only the right triangle area and the length of leg a, we can derive the equation for the other sides:
b = 2 × area / ac = √(a² + (2 × area / a)²)
How to Find the Angle of a Right Triangle
Since one angle is always 90°, the sum of the other two angles equals 90°. If you know one of the acute angles, the other is simply:
β = 90° − α
If you know two sides, use the inverse trigonometric functions:
α = arcsin(a / c)α = arccos(b / c)α = arctan(a / b)
How Do You Solve a Right Angle Triangle with Only One Side?
With only one side, you cannot uniquely solve a right triangle unless you also know at least one angle (other than the 90° right angle). With one side and one acute angle, use:
- Hypotenuse + angle α →
a = c × sin(α),b = c × cos(α) - Leg a + angle α →
b = a / tan(α),c = a / sin(α) - Leg b + angle α →
a = b × tan(α),c = b / cos(α)
Example: Finding the Missing Side
Suppose you know leg a = 3 ft and leg b = 4 ft (imperial system). Using the Pythagorean theorem:
- c = √(3² + 4²) = √(9 + 16) = √25 = 5 ft
- α = arctan(3 / 4) ≈ 36.87°
- β = 90° − 36.87° ≈ 53.13°
- Area = (3 × 4) / 2 = 6 ft²
- Perimeter = 3 + 4 + 5 = 12 ft
The same calculation in the metric system: leg a = 3 m, leg b = 4 m → hypotenuse c = 5 m.
Units of Measurement
Our calculator supports both measurement systems:
- Imperial (US) system: feet (ft), inches (in), yards (yd) — commonly used in the United States
- Metric system: meters (m), centimeters (cm), millimeters (mm) — used in most countries worldwide
Simply select your preferred unit system before entering values. The results (sides, perimeter, area) will be displayed in the chosen unit.
Frequently Asked Questions (FAQs)
What is a right triangle?
A right triangle (also called a right-angled triangle) is a triangle in which one angle equals exactly 90°. The side opposite the right angle is called the hypotenuse — it is always the longest side. The other two sides are called legs.
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the two legs: a² + b² = c². This is one of the most fundamental theorems in mathematics and geometry.
What are the most common right triangle ratios?
The most well-known right triangle ratios (Pythagorean triples) include:
- 3-4-5 triangle: a=3, b=4, c=5
- 5-12-13 triangle: a=5, b=12, c=13
- 8-15-17 triangle: a=8, b=15, c=17
- 45-45-90 triangle: a=b, c=a√2
- 30-60-90 triangle: a=1, b=√3, c=2
How do I find the hypotenuse?
If you know both legs (a and b), use the Pythagorean theorem: c = √(a² + b²). If you know one leg and an angle, use: c = a / sin(α) or c = b / cos(α).
How do I calculate the area of a right triangle?
The area of a right triangle is half the product of its two legs: Area = (a × b) / 2. This is because the two legs form the base and height of the triangle.