What is a Pythagorean theorem calculator?
A Pythagorean theorem calculator finds the missing side of a right triangle when you know the other two. Choose whether to find the hypotenuse c or a leg (a or b), enter the two known sides, and this online Pythagorean theorem calculator gives the answer with steps, the exact value in simplest radical form, both acute angles in degrees, the area and the perimeter. It works in any unit – inches, feet, metres or mm – as long as both sides use the same one.
What is the Pythagorean theorem?
The Pythagorean theorem (Pythagoras theorem) says that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The hypotenuse is the longest side, opposite the 90° angle; the two shorter sides that form the right angle are called the legs. The rule was known in ancient India too: the Baudhayana Sulba Sutra stated it centuries before Pythagoras.
Pythagorean theorem formula
a² + b² = c²
- a and b = the legs (the sides forming the right angle)
- c = the hypotenuse (the side opposite the right angle)
Rearranged for each missing side:
- Find c (hypotenuse): c = √(a² + b²)
- Find a: a = √(c² − b²)
- Find b: b = √(c² − a²)
How to use this calculator
- In Find, choose the side you do not know.
- Enter the two sides you do know. When solving for a leg, the hypotenuse must be the longest side, or the triangle cannot exist.
- Read the missing side at the top and the working below it. If the square under the root is a whole number, the calculator also shows the exact radical form, such as 5√2.
Worked examples with steps
Find c: legs 3 and 4
c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5. This is the famous 3-4-5 triangle.
Find b: a = 5, hypotenuse c = 13
b² = 13² − 5² = 169 − 25 = 144, so b = √144 = 12. Choose “Leg b” in the calculator to check it.
Find a: b = 6, c = 10
a² = 10² − 6² = 100 − 36 = 64, so a = 8.
Simplest radical form: legs 5 and 5
When a and b are equal (an isosceles right triangle), c² = 25 + 25 = 50 and c = √50 = √(25 × 2) = 5√2 ≈ 7.0711. The calculator simplifies the radical by pulling out every perfect-square factor. Our square root calculator shows the same simplification for any number.
Common Pythagorean triples
A Pythagorean triple is a set of three whole numbers that fits a² + b² = c². Remembering a few saves time in non-calculator questions.
| a | b | c | Multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6-8-10, 9-12-15, 30-40-50 |
| 5 | 12 | 13 | 10-24-26 |
| 8 | 15 | 17 | 16-30-34 |
| 7 | 24 | 25 | 14-48-50 |
| 20 | 21 | 29 | – |
Pythagorean theorem calculator with angles, area and perimeter
Once all three sides are known, the calculator uses trigonometry to find the two acute angles: α = tan⁻¹(a ÷ b) and β = 90° − α. For the 3-4-5 triangle, α ≈ 36.87° and β ≈ 53.13°. The area is ½ × a × b (6 for 3-4-5), and the perimeter is a + b + c (12). The height from the right angle to the hypotenuse is a × b ÷ c. For other shapes, use the area calculator.
If you know one side and an angle instead of two sides, the Pythagorean theorem alone is not enough; use sine, cosine or tangent (for example, a = c × sin α) on a scientific calculator, then this tool to check the third side.
What is the Pythagorean theorem used for in real life?
- Construction and carpentry – builders use the 3-4-5 rule to check a corner is square: mark 3 ft on one wall and 4 ft on the other; if the diagonal is 5 ft, the corner is exactly 90°. For bigger layouts use 6-8-10 or 9-12-15.
- Ladders and ramps – a ladder leaning 4 m up a wall with its foot 1.5 m out needs to be √(16 + 2.25) ≈ 4.27 m long.
- Screen and room diagonals – a TV that is 48 in wide and 27 in tall has a diagonal of about 55 in.
- Distance between two points – the distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²) is the Pythagorean theorem on coordinates. From (1, 2) to (4, 6): √(9 + 16) = 5.
- Baseball and cricket – the throw from home plate to second base on a 90 ft baseball diamond is √(90² + 90²) ≈ 127.3 ft.
Pythagorean theorem calculator for feet and inches
To use the Pythagorean theorem calculator in feet and inches, convert everything to one unit first. Either use inches (6 ft 3 in = 75 in) or decimal feet (6 ft 3 in = 6.25 ft). Convert the answer back by multiplying the decimal part by 12: 7.5 ft = 7 ft 6 in.
Does it work for a non-right triangle?
No. a² + b² = c² is true only for right triangles. For any other triangle use the law of cosines, c² = a² + b² − 2ab cos C. You can, however, use the theorem to test whether a triangle is right-angled: if the squares of the two shorter sides add up to the square of the longest, it is. Also, 1-1-√2 is the isosceles right triangle and 1-√3-2 is the 30-60-90 triangle, the half of an equilateral triangle.
How to solve with the Pythagorean theorem calculator
How to solve Pythagorean theorem calculator problems: decide which side is missing, choose it in the Find box, and enter the other two sides in the same unit. The formula is a² + b² = c², where c is the hypotenuse. It only works for right triangle problems. The calculator then shows each line of working, step by step.
Pythagorean theorem calculator: find b (solve for b)
To solve for b when you know leg a and hypotenuse c, use b = √(c² − a²). With a = 8 and c = 17: b = √(289 − 64) = √225 = 15. Choose "Leg b" above to do this.
Pythagorean theorem calculator given hypotenuse: when you have c
Knowing c alone is not enough: the hypotenuse only fixes the size, and you also need one leg or one angle. Given hypotenuse c and leg a, the calculator finds the other leg. Given c and an angle A, use a = c × sin A and b = c × cos A.
Pythagorean theorem calculator where a and b are equal
If you only know c and the two legs are equal (a 45-45-90 triangle), then a = b = c ÷ √2. With only c = 10 known, each leg is 10 ÷ 1.4142 = 7.071.
Pythagorean theorem calculator angle formula
Once you know all three sides, the angles follow from trigonometry: tan A = a ÷ b, sin A = a ÷ c, and B = 90° − A. For a 3-4-5 triangle, A = tan⁻¹(3 ÷ 4) = 36.87° and B = 53.13°.
Pythagorean theorem calculator with variables and fractions
Fractions work the same way. Legs 1/2 and 2/3: c² = 1/4 + 4/9 = 25/36, so c = 5/6. Enter them as decimals (0.5 and 0.6667) to check. With variables, legs x and 2x give c² = x² + 4x² = 5x², so c = x√5.
Pythagorean theorem calculator for isosceles triangle and equilateral triangle
For an isosceles triangle with equal sides s and base b, the height splits it into two right triangles: h = √(s² − (b ÷ 2)²). Sides 10, 10 and base 12 give h = √(100 − 36) = 8. For an equilateral triangle with side s, the height is s√3 ÷ 2: side 6 gives 5.196.
Pythagorean theorem calculator construction guide
Builders use the 3-4-5 rule to check a square corner: mark 3 m along one wall and 4 m along the other; if the diagonal is exactly 5 m, the corner is 90°. For bigger layouts use 6-8-10 or 9-12-15. A rafter over a 4 m run with 1.5 m rise is √(16 + 2.25) = 4.27 m long. Work in meters or mm consistently: the Pythagoras theorem calculator in mm gives 4,272 mm for a 4,000 mm run and 1,500 mm rise.
Pythagorean theorem calculator baseball example
Baseball bases are 90 ft apart, so the throw from home plate to second base is √(90² + 90²) = 90√2 ≈ 127.3 ft. In softball with 60 ft bases it is about 84.9 ft.
Pythagorean theorem calculator for a non right triangle
The theorem only works for right triangles. For a non-right triangle, use the law of cosines: c² = a² + b² − 2ab cos C. With a = 7, b = 9 and C = 60°: c² = 49 + 81 − 63 = 67, so c = 8.19.
Pythagoras theorem formula online calculator vs non calculator questions
This Pythagoras theorem formula online calculator gives decimals instantly. In exams you will also meet Pythagoras theorem non calculator questions, usually built on triples or left in surd form:
- Legs 6 and 8. Hypotenuse? √100 = 10.
- Hypotenuse 25, leg 7. Other leg? √(625 − 49) = √576 = 24.
- Legs 2 and 3. Hypotenuse in surd form? √13.
- A 5 m ladder reaches 4 m up a wall. How far is its foot from the wall? 3 m.
Frequently asked questions
What is the Pythagorean theorem formula?
The formula is a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse. To find c use √(a² + b²); to find a leg use √(c² − other leg²).
How do I find the hypotenuse?
Square both legs, add them and take the square root. With legs 6 and 8: 36 + 64 = 100, and √100 = 10.
How do I find a missing leg (solve for b)?
Subtract the square of the known leg from the square of the hypotenuse and take the square root. With c = 13 and a = 5: 169 − 25 = 144, so b = 12.
What is the Pythagorean theorem used for?
It is used to find distances and diagonals: checking square corners in construction, ladder lengths, screen sizes, map distances and the distance formula in coordinate geometry.
How do I write the answer in simplest radical form?
Factor out the largest perfect square from the number under the root. √50 = √(25 × 2) = 5√2 and √72 = √(36 × 2) = 6√2. The calculator does this automatically for whole-number squares.
Can I use the Pythagorean theorem for a non-right triangle?
No. It only applies to right triangles. For other triangles use the law of cosines, c² = a² + b² − 2ab cos C.
How do you do the Pythagorean theorem on a calculator?
Type a² + b², press equals, then press the square root key. For a leg, type c² − b² and take the square root. This online calculator does all the steps and shows them.
Is this Pythagorean theorem calculator free?
Yes. It is free, requires no sign-up and works on phones and computers.
Can I find the legs if I only know the hypotenuse?
Only if you know one more thing: a leg, an angle, or that the legs are equal (then each leg = c ÷ √2).
How far is it from home plate to second base?
About 127.3 ft, because the bases form a 90 ft square and 90 × √2 ≈ 127.3.
Is there a Pythagorean theorem calculator with hypotenuse only?
With the hypotenuse only, you also need one leg or angle. Knowing c and that both legs are equal, each leg is c ÷ √2, so this Pythagorean theorem calculator with only c works for 45-45-90 triangles.
Can I use the Pythagorean theorem calculator in meters or feet and inches?
Yes. Any unit works as long as all sides use the same one. For the Pythagorean theorem calculator in feet and inches, convert inches to decimal feet first.
Does the Pythagorean theorem calculator find the missing leg?
Yes. Choose leg a or leg b and enter the other leg and the hypotenuse. As a Pythagorean theorem calculator for a missing leg, it uses √(c² − a²).
Can the Pythagorean theorem calculator give the exact value?
Yes. It shows the Pythagorean theorem calculator exact value in simplest radical form, such as 5√2 for legs 5 and 5, alongside the decimal.
What is Pythagorean theorem formula?
a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse. What is Pythagorean theorem used for? Finding a missing side, checking right angles and measuring diagonals.
Last updated 2026-09-18