Right Triangle Calculator
A right triangle calculator instantly finds the missing side lengths, acute angles, total area, perimeter, and altitude of any 90 degree triangle. Enter any two known values, such as two side lengths or one side and one acute angle, to calculate all remaining properties. The tool uses fundamental geometry rules including the Pythagorean theorem, trigonometric ratios, and triangle area formulas to give accurate, step by step results.
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What Is a Right Triangle?
A right triangle, also called a right angled triangle, is a three sided polygon with one internal angle equal to exactly 90 degrees. The two sides that form the 90 degree angle are called the legs. The longest side, located directly across from the right angle, is the hypotenuse.
In geometric notation, the legs are usually labeled a and b, while the hypotenuse is labeled c. The angles opposite sides a and b are represented by the Greek letters α (alpha) and β (beta). Because the total interior angles of any triangle always equal 180 degrees, the two acute angles α and β must sum to 90 degrees. This makes them complementary angles.
| Element | Notation | Description |
|---|---|---|
| Leg a | a | First short side adjacent to the right angle |
| Leg b | b | Second short side adjacent to the right angle |
| Hypotenuse | c | Longest side, opposite the 90 degree angle |
| Angle Alpha | α | Acute angle opposite to side a |
| Angle Beta | β | Acute angle opposite to side b |
| Right Angle | γ | The fixed 90 degree angle opposite to side c |
Understanding these relationships allows you to solve geometry problems using our pythagorean theorem calculator for quick length calculations.
How to Find the Missing Sides of a Right Triangle
To solve for an unknown side in a right triangle, choose the formula that matches the values you already know.
Method 1: Given Two Sides (Pythagorean Theorem)
When two side lengths are known, use the Pythagorean theorem formula a² + b² = c²:
- Find the hypotenuse (c): c = √(a² + b²)
- Find leg a: a = √(c² - b²)
- Find leg b: b = √(c² - a²)
Method 2: Given One Side and One Acute Angle
When you know one side length and one acute angle, use standard trigonometric functions.
Given hypotenuse c and angle α:
- a = c × sin(α)
- b = c × cos(α)
Given leg a and angle α:
- b = a / tan(α)
- c = a / sin(α)
Given leg b and angle α:
- a = b × tan(α)
- c = b / cos(α)
For more details on how these ratios work across various geometric shapes, check out our trigonometric ratios guide.
How to Find the Angles of a Right Triangle
Finding the missing angles depends on whether you have a known angle or two side measurements.
Method 1: Given One Acute Angle
Subtract the known angle from 90 degrees to find the remaining angle:
- α = 90° - β
- β = 90° - α
Method 2: Given Two Sides
Apply inverse trigonometric functions (arcsin, arccos, or arctan) to calculate the angles in degrees or radians:
- Using sides a and c: α = arcsin(a / c)
- Using sides b and c: α = arccos(b / c)
- Using sides a and b: α = arctan(a / b)
Once α is determined, subtract it from 90 degrees to get β. If you need to convert your angle units, you can refer to our radians to degrees converter.
Calculating Area, Perimeter, and Altitude
Beyond sides and angles, a right triangle has three key structural metrics:
Area (A): The total space enclosed by the three sides. Because the legs serve as the perpendicular base and height, the formula simplifies to:
A = (a × b) / 2
Perimeter (P): The distance around the outside edge of the shape:
P = a + b + c
Altitude (h): The perpendicular height drawn from the 90 degree vertex down to the hypotenuse:
h = (a × b) / c
If you need to calculate the space inside non-right shapes, try our triangle area calculator for additional methods.
Special Right Triangles
Special right triangles have fixed angle ratios or side ratios that make manual calculations much faster.
45-45-90 Triangle (Isosceles Right Triangle)
This triangle has two equal legs and two 45 degree angles. The side lengths always follow a ratio of 1 : 1 : √2.
If a leg length is x, the hypotenuse is x√2.
30-60-90 Triangle
This scalene right triangle has angles measuring 30, 60, and 90 degrees. The side lengths follow a strict ratio of 1 : √3 : 2.
- The shortest leg (opposite 30°) is x.
- The longer leg (opposite 60°) is x√3.
- The hypotenuse is 2x.
Pythagorean Triples
Pythagorean triples are sets of three positive integers that fit the formula a² + b² = c² perfectly. Common triples include:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
To analyze non-right or irregular three-sided shapes, use our scalene triangle calculator.
Frequently Asked Questions
Can a right triangle have two right angles?
No. Every triangle has an interior angle total of 180 degrees. Two 90 degree angles would equal 180 degrees by themselves, leaving 0 degrees for the third angle, which is impossible.
How do you solve a right triangle with only 1 side length?
You cannot solve a right triangle with only one side length unless you also know at least one acute angle or have extra constraints like an area value or a special triangle ratio (such as a 45-45-90 triangle).
What is the hypotenuse leg theorem?
The hypotenuse leg theorem states that two right triangles are congruent if their hypotenuses and one corresponding leg are equal in length.
Can a right triangle be equilateral?
No. An equilateral triangle must have three equal angles of 60 degrees each. A right triangle must have one 90 degree angle, so it cannot be equilateral.
How do you find the height of a right triangle?
If the base is one leg, the height is simply the other leg. If the base is the hypotenuse, calculate height using the altitude formula h = (a × b) / c.
What is the difference between sine, cosine, and tangent?
Sine is the ratio of the opposite leg to the hypotenuse. Cosine is the ratio of the adjacent leg to the hypotenuse. Tangent is the ratio of the opposite leg to the adjacent leg.