Right Triangle Solver – Pythagorean Theorem & Trigonometry Tutorial

What Is a Right Triangle?

A right triangle is a triangle with one interior angle equal to exactly 90°. This 90° angle is called the right angle; the side opposite it is always the longest side and is called the hypotenuse (denoted c). The other two sides, a and b, are called legs. The two non-right angles α (alpha) and β (beta) always sum to 90°: α + β = 90°.

The Pythagorean Theorem

In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse:

c² = a² + b²
c = √(a² + b²)

c: hypotenuse (longest side); a, b: legs

The same formula gives the legs: a = √(c² − b²) and b = √(c² − a²). The hypotenuse must always be longer than either leg: c > a and c > b.

Which Mode Should I Use?

Choose the mode that matches the values you already know:

Mode	Known values	Computed	Method
A	a and b (both legs)	c (hypotenuse)	Pythagorean theorem
B	a + c or b + c	The other leg	Pythagorean theorem (reversed)
C	a + α or b + α	All sides	Trigonometry

In mode C: if you enter a, α is treated as the angle opposite a (sin α = a/c). If you enter b, α is treated as the angle adjacent to b (cos α = b/c).

Trigonometric Ratios

For a given angle α, the ratios of the sides are constant — they are the same in all similar right triangles. This means knowing the angle lets you find side ratios, and knowing the sides lets you find angles.

sin(α) = opposite / hypotenuse = a / c
cos(α) = adjacent / hypotenuse = b / c
tan(α) = opposite / adjacent = a / b

As α increases: sin(α) increases, cos(α) decreases. tan(α) can be read as slope.

Area and Perimeter

In a right triangle the two legs act as base and height, making area calculation straightforward:

Area = ½ · a · b
Perimeter = a + b + c

Example: The 3-4-5 Triangle

The world's most famous right triangle has a=3, b=4, c=5. Verification:

c = √(3² + 4²) = √(9 + 16) = √25 = 5 ✓
Area = ½ · 3 · 4 = 6
Perimeter = 3 + 4 + 5 = 12
α = arcsin(3/5) ≈ 36.87° → β ≈ 53.13°
sin(α) = 0.6 · cos(α) = 0.8 · tan(α) = 0.75

This triangle has been used in construction and surveying for centuries to verify right angles in the field.

Real-World Applications

Engineering & architecture: Roof pitch, staircase length, diagonal bracing
Mapping & navigation: Straight-line distance between two coordinate points
Physics: Decomposing forces into x–y components, inclined plane problems
Computer graphics: Screen-space distance, vector magnitude, direction calculations
Everyday life: TV screen diagonal, room corner distance, cable length

Things to Watch Out For

The hypotenuse c must always be the longest side: c > a and c > b.
Angles α and β are always strictly between 0° and 90°.
All sides must be in the same unit (cm, m, inches, etc.).
This calculator accepts degrees and converts to radians internally.
A warning appears if the entered sides deviate from the Pythagorean condition by more than 2%.

Note: This calculator is intended for educational and estimation purposes. Use specialist software for critical engineering projects.

Right Triangle Solver – Pythagorean Theorem & Trigonometry – Tutorial