Last updated: 2026-05-19

The right triangle solver

A right triangle contains one 90° angle. Given any two non-right measurements — two legs, one leg and the hypotenuse, or a side and an acute angle — you can determine every other measurement through a handful of trigonometric identities. This calculator automates all four solve paths so you get sides, angles, area, and perimeter in one step.

The four solve modes

The calculator offers four modes based on what you already know:

The Pythagorean theorem

The foundation of right-triangle arithmetic is the Pythagorean theorem:

a² + b² = c²

where a and b are the two legs and c is the hypotenuse (the side opposite the right angle). Given both legs, the hypotenuse is $c = \sqrt{a^2 + b^2}$. Rearranged, any leg can be recovered: $a = \sqrt{c^2 - b^2}$.

Worked example: A ramp rises 1.2 m vertically over a horizontal run of 3.5 m. The ramp length (hypotenuse) is $\sqrt{1.2^2 + 3.5^2} = \sqrt{1.44 + 12.25} = \sqrt{13.69} \approx 3.70$ m.

SOH-CAH-TOA

The three primary trigonometric ratios link angles to side ratios in a right triangle:

• Sin(A) = Opposite / Hypotenuse = a / c
• Cos(A) = Adjacent / Hypotenuse = b / c
• Tan(A) = Opposite / Adjacent = a / b

Use these to find angles from sides (using inverse trig) or sides from angles:

• Angle A from sides: A = arctan(a / b) — or arcsin(a / c) when the hypotenuse is known.
• Leg from hypotenuse and angle: a = c · sin(A), b = c · cos(A).
• Leg from other leg and angle: b = a / tan(A).

These ratios follow directly from the unit circle: rotating by angle A maps the unit radius to coordinates (cos A, sin A), which are exactly the adjacent and opposite ratios for a right triangle inscribed in that circle.

Area and perimeter

Once all three sides are known:

• Area = ½ · a · b (half the product of the two legs, since the legs form the base and height)
• Perimeter = a + b + c

Worked example: A triangular garden plot has legs 7 m and 24 m. Its area is ½ × 7 × 24 = 84 m², its hypotenuse is $\sqrt{49 + 576} = \sqrt{625} = 25$ m, and its perimeter is 7 + 24 + 25 = 56 m.

Choosing a mode

Two Legs is the everyday Pythagorean case — ladders against walls, screen diagonals, diagonal distances on maps.

Leg + Hypotenuse occurs when you can measure the slant length and one side but not the other — e.g., a roof slope where you know the run and the rafter length.

Leg + Angle is common in surveying and construction: you know the angle of inclination and a horizontal or vertical distance. For instance, a 30° inclined conveyor with a 5 m horizontal run has a belt length of 5 / cos 30° ≈ 5.77 m and a rise of 5 · tan 30° ≈ 2.89 m.

Hypotenuse + Angle appears in navigation and physics: a vector of magnitude c at angle A has horizontal component c · cos A and vertical component c · sin A.