Pythagorean Theorem Calculator
Enter any two known sides of a right triangle to solve the missing side, verify a²+b²=c², and review area and perimeter in one calculator.
Pythagorean Theorem Calculator
This calculator helps you solve the missing side of a right triangle when you already know the other two sides, and it keeps the a² + b² = c² verification in view on the same screen. Choose whether you want to solve for the hypotenuse, the base, or the height and calculate right away.
When you know the two legs a and b, use c = √(a² + b²) to solve the hypotenuse.
The horizontal leg along the bottom of the right triangle.
The vertical leg on the left side of the right triangle.
This is the leg you already know.
The hypotenuse must be longer than the known leg.
This is the leg you already know.
The hypotenuse is always the longest side.
Use these presets to test familiar Pythagorean triples and preview the result instantly.
- In the modes that include the hypotenuse, hypotenuse c must be longer than the leg for a valid right triangle.
- You can use cm, m, in, or any other unit, but every input should stay in the same unit.
- The verification values under the result card help you confirm whether a² + b² and c² are actually equal.
- If you also want the angles, area, and perimeter in one place, continue with the Right Triangle Calculator.
Check your inputs and press Calculate to update the result.
If the two legs are 3 and 4, the hypotenuse c becomes 5 and the Pythagorean theorem checks out exactly.
| Calculation mode | Solve hypotenuse |
|---|---|
| Base a | 3 |
| Height b | 4 |
| Hypotenuse c | 5 |
| Solved value | Hypotenuse c = 5 |
| Verification formula | 3² + 4² = 5² |
| Verdict | Pythagorean theorem holds |
The left and right sides match, so the right-triangle relationship checks out correctly.
- a² = 9 and b² = 16, so a² + b² = 25.
- Then c = √25 = 5, which gives the hypotenuse.
- That also gives an area of 6 and a perimeter of 12.
What is the Pythagorean Theorem Calculator?
The Pythagorean Theorem Calculator helps you find the missing side of a right triangle by using the relationship between the two perpendicular legs and the hypotenuse. The core formula is a² + b² = c². If you know both legs, you can solve the hypotenuse. If you know one leg and the hypotenuse, you can work backward to recover the remaining leg.
Instead of showing only one answer, this tool also lays out the formula steps and the verification result. That makes it useful for homework, rough design work, and on-site dimension checks where you need to understand why the result makes sense.
When this calculator is useful
The Pythagorean theorem shows up in school math, but also in real measurements such as slope length estimates, ladder checks, and right-angle corner checks. If you already know two values, this tool helps you solve the remaining side immediately and cuts down on manual calculation.
- When you want the hypotenuse or the missing leg right away in a right-triangle problem
- When you need to back-solve a hard-to-measure length such as a ladder, slope, or diagonal
- When you want to verify classic triples such as 3-4-5 or 5-12-13 quickly
- When you want to copy both the answer and the verification formula into notes or assignments
Key features
This calculator keeps the layout focused on the Pythagorean theorem itself. The top result card highlights the solved value first, while the verification values, squared values, area, and perimeter stay together underneath so the calculation flow is easy to follow.
- Three calculation modes – solve for the hypotenuse, the base, or the height
- Pythagorean check card – compare
a² + b²andc²side by side - Squared values, area, and perimeter – keep the follow-up measurements close to the main answer
- Quick example buttons – test well-known integer triples instantly
- Copy result – copy the formula and the key values in one step
How to use it
Start by choosing which side you want to solve, then enter the two sides you already know. After you calculate, the solved value, verification formula, and squared values update together so you do not need to recheck everything by hand.
- Choose the mode – decide whether you want to solve for hypotenuse c, base a, or height b.
- Enter the known lengths – provide the two sides that match the selected mode and keep them in the same unit.
- Set the decimal places – choose how many decimal places you want to read.
- Press Calculate – the solved value, verification formula, area, and perimeter update together.
- Review and copy the result – use the copy button if you want to share the output right away.
Pythagorean theorem formulas
If you label the base as a, the height as b, and the hypotenuse as c, the core formula is a² + b² = c². That means you can solve the hypotenuse with c = √(a² + b²), the base with a = √(c² - b²), and the height with b = √(c² - a²). If you want to check the square-root part separately, the Root Calculator is a helpful companion.
The key rule is that the hypotenuse must always be the longest side. In the modes that use one leg plus the hypotenuse, the hypotenuse value has to stay larger for a real right triangle. If it is equal to or shorter than the leg, then c² - a² or c² - b² becomes 0 or negative and the missing side cannot be solved in the real-number range.
The result area also shows the squared-value comparison and the verification formula. That means you can do more than read the answer — you can confirm that the left and right sides really match. If you also want angles and trig ratios, move on to the Right Triangle Calculator for a broader view.
- Hypotenuse formula – apply the square root to the sum of the two leg squares.
- Leg formula – subtract the other leg square from the hypotenuse square, then apply the square root.
- Verification rule – in the end,
a² + b²must matchc². - Units – use any length unit you want, but keep every input and every interpretation in the same unit.
Frequently asked questions
Which formula does the Pythagorean Theorem Calculator use?
The core formula is a² + b² = c². Use c = √(a² + b²) for the hypotenuse, and use √(c² - known side²) when you need the remaining leg.
Why does the calculator reject a hypotenuse that is shorter than a leg?
Because the hypotenuse is always the longest side in a right triangle. If it is equal to or shorter than a leg, then c² - a² or c² - b² becomes 0 or negative, which does not produce a valid real side length.
Can I use decimal lengths?
Yes. You can use decimal lengths as well as whole numbers. Adjust the decimal setting so the result matches the level of detail you want in notes, assignments, or design work.
How is this different from the Right Triangle Calculator?
This tool stays focused on solving one missing side with the Pythagorean theorem. The Right Triangle Calculator goes further and also covers angles, area, perimeter, and trig ratios.
Can I check integer triples such as 3-4-5?
Yes. The quick examples include familiar triples such as 3-4-5, 5-12-13, and 8-15-17, and you can also enter your own integer triples to verify them immediately.
Where is this result useful?
It works well for math assignments, carpentry or interior dimension checks, ladder or slope estimates, and quick field-note reviews whenever you need to confirm a length relationship fast. You can also share the formula and result instantly with the copy button.
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