Pythagorean Theorem Calculator
Solve a right triangle from any two of its sides, angles, area or perimeter — drawn to scale, with the working shown, the altitude and the circle radii.
Hypotenuse c: the long side, opposite the right angle
More about this triangle
Type = scalene right triangle — the 3:4:5 Pythagorean triple Altitude to c (h = a·b/c) = 2.4 Inradius (r) = 1 Circumradius (R) = 2.5
The working
c = √(a² + b²) = √(3² + 4²) = 5 A = atan(a / b) = atan(3 / 4) = 36.869898° B = 90° − A = 90° − 36.869898° = 53.130102° Area = a · b / 2 = 3 · 4 / 2 = 6 Perimeter = a + b + c = 3 + 4 + 5 = 12
🔒 Calculated in your browser — nothing is uploaded.
Any two of the seven
Fill in exactly two of side a, side b, hypotenuse c, angle A, angle B, the area and the perimeter. Two sides is the classic case: a² + b² = c², so the third side follows, whether the missing one is a leg or the hypotenuse. One side and either acute angle is the other way a right triangle is given, and the rest follows through sine, cosine and tangent. An area or a perimeter with any one side or angle is solved through the legs' sum and their product — so is an area and a perimeter together. Whichever pair you have, you also get both acute angles, the area, the perimeter, the altitude to the hypotenuse, the inradius, the circumradius, and the name of the triangle: a 3-4-5 is called a 3:4:5 Pythagorean triple and a 45-45-90 is called isosceles.
Drawn to scale
The triangle is drawn from the answer with the right angle marked and all three sides labelled, at the actual proportions of the numbers: a 3-4-5 looks like a 3-4-5, and a triangle that is nearly flat looks nearly flat. This is what makes the labels mean anything — “angle A, opposite a” is a sentence about a picture, and without the picture the reader has to imagine which side is which. The hypotenuse is glossed on the page too: it is the long side, opposite the right angle.
When the numbers cannot be a triangle
Sides have to be positive and the hypotenuse has to be the longest of the three, so a c smaller than a leg is rejected with a note rather than answered. Each acute angle has to be strictly between 0 and 90 degrees, since the third angle is already the right one, and the two angles on their own are refused with a sentence of their own: they fix the shape but not the size. Fill in one box or three and the page says how many it wants instead of guessing which you meant.
The working, in your units
Under the answer the tool writes out each rule with your own numbers substituted into it — c = √(a² + b²) = √(3² + 4²) = 5 — and it only derives what you did not type, so asking from an area never hands the area back. A Unit dropdown labels every length and squares it on the area; the tool converts nothing, it labels, so an area comes back as cm² instead of a figure with no dimension. A Decimals box, 0 to 10, rounds the answer and the working together. Both live in the link along with the seven boxes.
Frequently asked questions
How does the Pythagorean theorem work?
In a right triangle the two legs and the hypotenuse are related by a² + b² = c², where c is the side opposite the right angle. So the hypotenuse is √(a² + b²), and a missing leg is √(c² − other²).
Can I solve from an angle instead of two sides?
Yes, and from either acute angle. Enter one side with angle A or angle B and the rest follows: a = c·sin A, b = c·cos A, and a = b·tan A, with angle B taken as 90 − A first. That covers the case where you measured one length and one angle, whichever angle it was.
Which side is the hypotenuse?
The long one, opposite the right angle. It is always the largest of the three, which is why the tool rejects a c that is shorter than a leg. The diagram labels all three sides so there is nothing to guess.
Does it give the area and perimeter?
Yes, both — and you can go the other way too. The area of a right triangle is half the rectangle its legs make, a·b ÷ 2, and the perimeter is simply a + b + c. Type an area or a perimeter into its own box with any one side or angle, or the two of them together, and the tool finds the triangle: the area fixes the legs' product, the perimeter their sum, and the legs are the two roots.
Does it show the steps?
Yes. Every rule is printed with your own numbers in it and rounded to the same number of decimals as the answer, so the working never disagrees with the block above it. It derives only what you did not already type.
Is anything uploaded?
No. The calculation and the drawing both happen in your browser. Nothing you type is sent anywhere or stored.