MultiConvers

Right Triangle Calculator

When the corner between the legs is already 90°, two legs — or one leg and the hypotenuse — recover the missing side, both acute angles, the area and the perimeter without a third generic side.

How to use

  1. Choose “both legs” when you know the two sides that meet at the right angle, or “one leg and the hypotenuse” when the longest side is already measured.
  2. Enter those lengths in one unit. In the second mode the hypotenuse must be strictly longer than the known leg, or there is no real third side.
  3. Read the missing side as the headline (the hypotenuse when you started from the legs). Details list the other leg if needed, both acute angles, area and perimeter.
  4. Leave this page if no angle is 90°. An acute or obtuse triangle needs three sides, base-and-height or SAS on the general triangle calculator.

How it works

A wheelchair ramp, a stair stringer, a roof rafter against a wall and a TV listed by diagonal all assume a right angle. The legs meet at 90°; the hypotenuse is the side across from that corner, √(a² + b²). Legs 3 and 4 give hypotenuse 5, area ½×3×4 = 6, perimeter 12 and complementary acute angles of about 36.87° and 53.13°. Starting from a leg of 3 and a hypotenuse of 5 recovers the other leg as 4.

That is narrower than the general triangle tool. You never type a third side or an included angle that is not 90°. You also never ask this page to accept an obtuse 3-4-6: those three lengths do not contain a right angle, so they belong on the SSS triangle page (or they fail Pythagoras here). The bargain is extra outputs — both acute angles, area and perimeter — that the Pythagorean page refuses to compute.

Angle opposite leg a is arctan(a/b); the other acute angle is 90° minus that, so they are complementary. Area is ½ab because the legs are already perpendicular — you do not need a separately measured altitude. Sides stay in the length unit; area is that unit squared.

This is not a trig-identity playground beyond those two arctangents. It does not solve from two angles and a side (AAS/ASA), from the altitude to the hypotenuse, or from an angle plus one non-leg length except the implemented leg-and-hypotenuse pair. It is not a slope tool: a rise and run can form a right triangle, but steepness as a ratio lives on the slope page.

A hypotenuse equal to a known leg would force the other leg to zero, which is not a triangle, so that input is rejected. Keep both given lengths positive and in the same unit before you trust the diagonal of a screen or the run of a ramp.

Formula

c = √(a² + b²)
∠A = arctan(a / b),  ∠B = 90° − ∠A
Area = ½ab
Perimeter = a + b + c

Worked examples

  • Legs 3 and 4

    a = 3, b = 4 → c = 5, area 6, angles ≈ 36.87° and 53.13°

  • Leg and hypotenuse

    leg 3, hypotenuse 5 → other leg 4

Useful notes

  • This assumes the 90° angle is between a and b.
  • For a non-right triangle, use the triangle calculator.
  • The headline result is the hypotenuse; legs, angles, area and perimeter appear as details.
  • Pythagoras alone — a missing side with no angles — is the Pythagorean theorem page.
  • A known hypotenuse must exceed the known leg; equality is not a triangle.

FAQ

How do I solve a right triangle from two legs?

Hypotenuse is √(a²+b²). Area is ½ab. The acute angles are arctan(a/b) and its complement.

Why are area and sides in different units?

Sides are lengths. Area is length times length. A 3 cm by 4 cm triangle has a 5 cm hypotenuse and a 6 cm² area.

Is this the same as the Pythagorean page?

Pythagoras finds one missing side. This page also reports angles, area and perimeter.

Can I use this for an acute triangle that is not right-angled?

No. This page builds in a 90° corner between the legs. Use the triangle calculator with three sides, base and height, or SAS.

Which acute angle is opposite which leg?

The angle opposite leg a is arctan(a/b). The angle opposite leg b is 90° minus that, so the two acute angles are complementary.

Guides

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