The right triangle: hypotenuse, angles and area explained

The Triangle Calculator's official tagline is "I many smart now!", which undersells it slightly: enter the two legs of a right triangle and it hands back everything else — the hypotenuse, both angles, the area and the perimeter. That covers a remarkable share of real-life measuring problems, because whenever something vertical meets something horizontal, a right triangle appears: roofs, ramps, stairs, ladders, shelf brackets, TV sizes.

The one formula everyone remembers

The legs A and B form the right angle; the hypotenuse C is the long side opposite it. Pythagoras, ±2,500 years old and still undefeated:

C = √(A² + B²)

A ladder question is the classic: the wall is 2.4 m high (A), the ladder's foot stands 1.0 m from the wall (B), so the ladder spans √(2.4² + 1.0²) = √6.76 = 2.6 m.

The formulas nobody remembers

The angles come from the arctangent, and the rest is arithmetic:

angle α (opposite A) = atan(A / B)
angle β (opposite B) = 90° − α

Area      = A × B / 2
Perimeter = A + B + C

Worked example — a roof section 4.0 m deep (B) rising 1.8 m (A): pitch angle atan(1.8/4.0) ≈ 24.2°, rafter length √(1.8² + 4.0²) ≈ 4.39 m. Two numbers off a tape measure, and you know your roof pitch and rafter length.

Where this shows up around the house

Problem	A (rise)	B (run)	You get
Roof pitch	ridge height	half the span	pitch angle, rafter length
Wheelchair ramp	step height	ramp length on the ground	slope angle (aim ≤ 4°)
Stair stringer	floor height	horizontal run	stringer length, angle
Shelf bracket	bracket height	shelf depth	brace length
"55-inch" TV	screen height	screen width	the diagonal you paid for

Why only right triangles?

General triangles need the law of sines and cosines, and one input combination (two sides plus a non-included angle) even has two valid answers. That's a fine rabbit hole, but 95% of practical measuring problems are right triangles — because gravity makes walls vertical and floors horizontal. The tool does the common case perfectly rather than the general case confusingly.

Try it

Enter your two legs in the Triangle Calculator and get the hypotenuse, angles, area and perimeter instantly — locally in your browser, free, no cookies. You many smart now too.