📏 Enter known values

📐 Accepted units: mm, cm, dm, m, dam, hm, km, in, ft, yd, mi

Formula Reference

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Worked example

Data
Leg A
\(b = {\color{#6c5ce7}{3\,\text{cm}}}\)
Leg B
\(c = {\color{#6c5ce7}{4\,\text{cm}}}\)
Hypotenuse
\(a = \,?\)
Formula
\[a = \sqrt{{{b}}^{2}+{{c}}^{2}}\]
Steps
1Substitute the values
\[a = \sqrt{{\left({\color{#6c5ce7}{3\,\text{cm}}}\right)}^{2}+{\left({\color{#6c5ce7}{4\,\text{cm}}}\right)}^{2}}\]
Each letter is replaced by its value.
2Square
\[a = \sqrt{9\,\text{cm}^{2}+16\,\text{cm}^{2}}\]
\(3 \times 3 = 9\)\(4 \times 4 = 16\)\(\text{cm} \times \text{cm} = \text{cm}^{2}\)
3Add
\[a = \sqrt{25\,\text{cm}^{2}}\]
\(9+16 = 25\)
4Take the square root
\[a = \sqrt{25\,\text{cm}^{2}} = 5\,\text{cm}\]
\(\sqrt{25} = 5\quad (5 \times 5 = 25)\)\(\sqrt{\text{cm}^{2}} = \text{cm}\)
Hypotenuse: 5 cm
Check

If we use the result to calculate Leg A, we should get the value you entered:

\[b = \sqrt{{{a}}^{2}-{{c}}^{2}}\]
\[b = \sqrt{\left(5\,\text{cm}\right)^{2}-\left(4\,\text{cm}\right)^{2}} = 3\,\text{cm}\]
It gives 3 cm, the same as the value entered.
Calculate the Hypotenuse
Please fill in the fields:
Leg_A Leg_B
And leave empty
Hypotenuse
Calculate the Leg_A
Please fill in the fields:
Hypotenuse Leg_B
And leave empty
Leg_A
Calculate the Leg_B
Please fill in the fields:
Hypotenuse Leg_A
And leave empty
Leg_B

The Pythagorean theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs:

\[ c^2 = a^2 + b^2 \]

So the hypotenuse and each leg are:

\[ \text{Hypotenuse} = \sqrt{a^2 + b^2} \]

\[ a = \sqrt{c^2 - b^2} \]

It only works for right triangles, the ones with a 90° angle. The hypotenuse is the side opposite the right angle and is always the longest side.

How to use the calculator

  1. Enter two of the three sides.
  2. Leave empty the one you want to find.
  3. Press Calculate. To find a leg, the hypotenuse must be longer than the other leg.

Worked examples

1. Legs of 3 and 4

\[ c = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \]

2. Missing leg with a hypotenuse of 13 and a leg of 5

\[ b = \sqrt{13^2 - 5^2} = \sqrt{144} = 12 \]

3. Diagonal of a TV screen

A screen is 121 cm wide and 68 cm tall. Its diagonal is \( \sqrt{121^2 + 68^2} = \sqrt{19{,}265} \approx 138.8 \) cm, about 55 inches.

4. Checking a right angle on site

Builders mark 3 m along one wall and 4 m along the other. If the distance between the marks is exactly 5 m, the corner is square. On large slabs, 6, 8 and 10 m give a more precise check.

Pythagorean triples

Whole numbers that satisfy the theorem. Any multiple of a triple is also a triple: 6, 8, 10 is 3, 4, 5 times 2.

Leg aLeg bHypotenuse c
345
51213
81517
72425
202129
123537
94041
116061

Is the triangle right, acute or obtuse?

With the three sides, compare the square of the longest side \( c \) with the sum of the squares of the other two:

  • \( c^2 = a^2 + b^2 \): right triangle.
  • \( c^2 < a^2 + b^2 \): acute triangle.
  • \( c^2 > a^2 + b^2 \): obtuse triangle.

When Do You Need to Use the Pythagorean Theorem?

📺 Screen and Monitor Sizes

Screens are sold by their diagonal. With the width and the height you get the diagonal, and with the diagonal you can check the real size that fits on your wall or desk.

Check that a TV fits before buying it
🏗️ Squaring Corners in Construction

The 3-4-5 rule checks that foundations, walls and decks are square. A diagonal that does not match the theorem means the corner is not 90°.

Avoids crooked walls and tiles that do not line up
🪜 Ladders Against a Wall

With the height to reach and the safe distance from the wall, the theorem gives the ladder length. With the ladder length, it tells how high it reaches.

Choose a ladder that reaches safely
🗺️ Straight-Line Distances

On a map or a grid, the direct distance between two points is the hypotenuse of the horizontal and vertical displacements. It is the basis of the distance formula.

Distance between two points on a plane
📦 Fitting Long Objects

The longest rod that fits flat in a box or a car trunk lies along the diagonal. The theorem tells you whether a pole, a shelf or a pipe will fit.

Know if it fits before moving it
📚 Geometry and Trigonometry

The theorem is the base of trigonometry, the distance formula and many proofs. It appears in almost every math course from middle school on.

A must-know tool for students

Common Mistakes

⚠️ Adding the Sides Without Squaring
Common Error: Writing c = a + b. For legs of 3 and 4 this gives 7 instead of 5.
✅ Solution: Square, add, then take the square root
⚠️ Forgetting the Square Root
Common Error: Stopping at c² = 25 and answering 25 instead of 5.
✅ Solution: The last step is always the square root
⚠️ Adding When Finding a Leg
Common Error: Using √(c² + b²) to find a leg. The leg is shorter than the hypotenuse, so you subtract.
✅ Solution: Leg = √(c² − b²)
⚠️ Using It on a Non-Right Triangle
Common Error: The theorem only holds with a 90° angle. On other triangles it gives a wrong side.
✅ Solution: For other triangles, use the law of cosines
⚠️ Choosing the Wrong Hypotenuse
Common Error: Taking one of the legs as the hypotenuse. The hypotenuse is opposite the right angle and is the longest side.
✅ Solution: Find the right angle first: the side in front of it is c
⚠️ Squaring a Sum
Common Error: Computing (a + b)² instead of a² + b². (3 + 4)² = 49, not 25.
✅ Solution: Square each leg separately, then add

Applications by Industry

Construction
  • Layout: 3-4-5 checks for square corners
  • Roofs: Rafter length from the rise and the run
  • Stairs: Stringer length from the total rise and run
Technology & Electronics
  • Screens: Diagonal size from the width and the height
  • Images: Distance between pixels in image processing
  • Circuit boards: Length of diagonal traces
Navigation & Mapping
  • Distances: Straight-line distance from north and east displacements
  • Maps: Short distances on projected maps
  • Orienteering: Direct route across a field
Physics & Engineering
  • Vectors: Magnitude of a vector from its components
  • Forces: Resultant of two perpendicular forces
  • Electricity: Impedance from resistance and reactance in AC circuits
Games & Graphics
  • Collisions: Distance between two objects in a 2D game
  • 3D graphics: Length of vectors for movement and lighting
  • Pathfinding: Cost of diagonal moves on a grid
Education
  • Geometry: Proofs and right-triangle problems
  • Algebra: Distance formula between two points
  • Number theory: Pythagorean triples

External Resources

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