Pythagorean Theorem Calculator (Hypotenuse and Legs of a Right Triangle)

Of the three sides a, b and c of a right triangle (c is the hypotenuse), enter only the two you know. The Pythagorean theorem gives the missing side, and the two acute angles, the area, the perimeter and the altitude to the hypotenuse are shown too.

Leg a

Leg b

Hypotenuse c

Enter only two sides (leave the missing one blank). Lengths must be numbers greater than 0 (decimals are OK), all in the same unit (for example, all in inches).

Result and figure

Enter the two side lengths you know in the fields on the left and press "Calculate". The result will appear here.

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What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter any two of the three sides of a right triangle, and the Pythagorean theorem \(a^2 + b^2 = c^2\) gives you the third side on the spot
Questions like "I only know it is 3 ft one way and 4 ft the other. How long is the diagonal?" are answered in one step
Along with the missing side, it also finds the two acute angles, the area, the perimeter and the altitude to the hypotenuse, with the steps shown
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

This works only for right triangles (triangles with one angle of exactly 90°). The hypotenuse c is the longest side, opposite the right angle. It cannot be used for the sides of a triangle without a right angle.

What is this calculation used for?

Estimating how high a ladder reaches (DIY and construction)

If you lean a 16-foot ladder against a wall with its feet 4 feet from the wall, it reaches \(\sqrt{16^2 - 4^2} = \sqrt{240} \approx 15.49\) feet up the wall (about 15.5 ft).
A ladder is set with its feet away from the wall so it does not tip over (a common safety rule is 1 foot out for every 4 feet of height), so it never reaches as high as its full length. Estimating the reach this way before the job is a basic habit in construction and electrical work.

Squaring a corner with 3-4-5 (carpentry and foundations)

Measure 3 feet along one side and 4 feet along the other. If the distance between those two marks is exactly 5 feet, the corner is square (since \(3^2 + 4^2 = 5^2\), the converse of the Pythagorean theorem says it is a right angle). For bigger layouts, carpenters use 6-8-10 or 9-12-15.
This "3-4-5 method" has been used for thousands of years, from rope stretchers in ancient Egypt to today's foundation and deck layouts. It is the most practical everyday use of the Pythagorean theorem.

Straight-line distance on a map (and in programming)

A place 3 miles east and 4 miles north of you is \(\sqrt{3^2 + 4^2} = 5\) miles away in a straight line.
Finding the distance between two points from the east-west difference and the north-south difference (the distance formula) is the Pythagorean theorem itself. It runs inside almost every program that works with locations, such as GPS navigation, map apps and video games.

Finding the sloped length of stairs and ramps (design and remodeling)

A staircase that runs 12 feet horizontally and rises 9 feet has a sloped length of \(\sqrt{12^2 + 9^2} = \sqrt{225} = 15\) feet.
The length of any sloped part, such as a stair handrail or a ramp board, comes from the horizontal and vertical dimensions on the plan with the Pythagorean theorem. You can use it directly to estimate materials before going to the store.

Formulas and figures

The Pythagorean theorem (how the three sides are related)

Figure

Standard notation (the usual math form)

\(a^2\) \(+\) \(b^2\) \(=\) \(c^2\)

In words (symbols replaced with words)

① \(a^2\): leg \(a\) squared \(+\) ② \(b^2\): leg \(b\) squared \(=\) ③ \(c^2\): hypotenuse \(c\) squared

The formula in words

① Add \(a^2\): leg \(a\) squared

② and \(b^2\): leg \(b\) squared

③ and the sum is exactly equal to \(c^2\): hypotenuse \(c\) squared

Quick example

Checking it with a right triangle whose sides are 3, 4 and 5

leg a squared (3² = 9) \(+\) leg b squared (4² = 16) \(=\) hypotenuse c squared (5² = 25)

\(3^2 + 4^2 = 9 + 16 = 25 = 5^2\)

Key idea

The hypotenuse \(c\) is the side opposite the right angle and the longest side of a right triangle. This relationship holds only for right triangles. It also works the other way: any triangle whose three sides fit this equation is always a right triangle (the converse of the Pythagorean theorem). This is what builders use to check that a corner is square.

Formula for the hypotenuse \(c\)

Figure

Standard notation (the usual math form)

\(c\) \(=\) \(\sqrt{a^2 + b^2}\)

In words (symbols replaced with words)

② \(c\): hypotenuse \(=\) ① square root of the sum of squares \(a^2 + b^2\)

The formula in words

① Find the square root of the sum of squares \(a^2 + b^2\) (square each leg, add them, then take the square root)

② and you get the \(c\): hypotenuse

Quick example

When the two legs are 3 and 4, the hypotenuse is

\(c\): hypotenuse \(=\) square root of the sum of squares (3² + 4² = 25)

\(c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

Key idea

This is \(a^2 + b^2 = c^2\) solved for \(c\). A common mistake is to write \(\sqrt{a^2} + \sqrt{b^2}\) (which is just \(a + b\)). First do all of the addition inside the square root, then take the square root only once at the end. The correct answer is \(\sqrt{9 + 16} = 5\), not \(\sqrt{9} + \sqrt{16} = 7\).

Formula for a missing leg

Figure

Standard notation (the usual math form)

\(b\) \(=\) \(\sqrt{c^2 - a^2}\)

In words (symbols replaced with words)

② \(b\): missing leg \(=\) ① square root of the difference of squares \(c^2 - a^2\)

The formula in words

① Find the square root of the difference of squares \(c^2 - a^2\) (subtract the square of the known leg from the square of the hypotenuse, then take the square root)

② and you get the \(b\): missing leg

Quick example

When the hypotenuse is 13 and one leg is 5, the other leg is

\(b\): missing leg \(=\) square root of the difference of squares (13² − 5² = 144)

\(b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12\)

Key idea

Watch the order of the subtraction. Always subtract the square of the known leg from the square of the hypotenuse, the longest side. To find leg \(a\) instead, use the same form: \(a = \sqrt{c^2 - b^2}\). If the subtraction gives 0 or less, the side you called the hypotenuse is not the hypotenuse (the hypotenuse is always longer than either leg).

Altitude to the hypotenuse

Figure

Standard notation (the usual math form)

\(h\) \(=\) \(a\) \(\times\) \(b\) \(\div\) \(c\)

In words (symbols replaced with words)

④ \(h\): altitude to the hypotenuse \(=\) ① \(a\): leg \(\times\) ② \(b\): leg \(\div\) ③ \(c\): hypotenuse

The formula in words

① Multiply \(a\): leg

② by \(b\): leg

③ divide by the \(c\): hypotenuse

④ and you get the \(h\): altitude to the hypotenuse

Quick example

In a right triangle with sides 3, 4 and 5, the height measured from the hypotenuse (the side of length 5) as the base is

altitude \(h\) \(=\) leg a (3) \(\times\) leg b (4) \(\div\) hypotenuse c (5)

\(h = 3 \times 4 \div 5 = 12 \div 5 = 2.4\)

Key idea

This formula comes from writing the area in two ways. Using the two legs, the area of a right triangle is \(a \times b \div 2\). Using the hypotenuse as the base, it is \(c \times h \div 2\). The area is the same, so \(a \times b = c \times h\). Solve this for \(h\) and you get \(h = a \times b \div c\).

The Pythagorean theorem says that the sum of the squares of the two legs equals the square of the hypotenuse, and it holds only for right triangles. If you know two sides, you find the third in two steps - square and add (for the hypotenuse) or square and subtract (for a leg), then take the square root.

Symbols and terms

Symbols

\(a\), \(b\)	a, b	The lengths of the two legs, the sides that form the right angle. It does not matter which one you call a; the result is the same.
\(c\)	c	The length of the hypotenuse, the longest side of a right triangle, opposite the right angle.
\(a^2\)	a squared	A number multiplied by itself (\(a^2 = a \times a\)). The small raised 2 is an exponent that says "use it as a factor twice". (Example - \(3^2 = 3 \times 3 = 9\))
\(\sqrt{\phantom{x}}\)	square root (radical sign)	The symbol for a square root. \(\sqrt{25}\) is the positive number that gives 25 when squared, which is 5. It can also be a number that never ends as a decimal, like \(\sqrt{2}\) (about 1.414).
\(\alpha\), \(\beta\)	alpha, beta	Greek letters often used to name angles. On this page, \(\alpha\) is the angle opposite leg a and \(\beta\) is the angle opposite leg b, and \(\alpha + \beta = 90°\).
\(\arcsin\)	arcsine	The inverse of the sine ratio \(\sin\) (opposite side ÷ hypotenuse). It works back from the ratio to the angle. This calculator finds the angles with it. On a scientific calculator it is the sin⁻¹ key.
\(h\)	h	The altitude to the hypotenuse - the height from the right-angle vertex to the hypotenuse, with the hypotenuse as the base. It is the first letter of "height".

Terms

Pythagorean theorem	The theorem that the three sides of a right triangle satisfy \(a^2 + b^2 = c^2\) (c is the hypotenuse). It is named after the ancient Greek mathematician Pythagoras (around the 6th century BC). It is taught in Grade 8 math.
leg	One of the two sides of a right triangle that form the right angle. The two legs are usually called a and b, and they are always shorter than the hypotenuse.
right triangle	A triangle in which one of the three angles is exactly 90° (a right angle). The Pythagorean theorem works only for this shape.
hypotenuse	The side of a right triangle opposite the right angle. It is the longest of the three sides, and in the Pythagorean theorem its square always stands alone on one side of the equation.
square root	A number that gives the original number when squared. The positive square root of 25 is 5, and the square root of 2 is \(\sqrt{2} \approx 1.414\). Square roots are taught in Grade 8.
Pythagorean triple	A set of three whole numbers that satisfy the Pythagorean theorem. (3, 4, 5), (5, 12, 13) and (8, 15, 17) are well known. They show up often in practice problems and when builders lay out square corners.
converse of the Pythagorean theorem	The theorem that a triangle whose three sides satisfy \(a^2 + b^2 = c^2\) is a right triangle. By measuring only the sides, you can check whether an angle is a right angle, so it is used on real construction sites.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.

Squares and exponents (Grade 6)	Knowing that squaring a number means multiplying it by itself, as in \(3^2 = 3 \times 3 = 9\) Being able to square a decimal too (for example, \(1.5^2 = 2.25\))
Square roots (Grade 8)	Being able to restate "the positive number that gives 25 when squared" as \(\sqrt{25} = 5\) Knowing that a square root can be a number that never ends as a decimal, like \(\sqrt{2}\)
Triangle basics (Grades 4–6)	Being able to find the right angle (90°) in a figure Knowing that the area of a triangle is "base × height ÷ 2"
The Pythagorean theorem (Grade 8)	Being able to tell which side of a right triangle is the hypotenuse (opposite the right angle, the longest side) Keeping in mind that \(c\) in \(a^2 + b^2 = c^2\) is always the hypotenuse

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to check for a right triangle

Leg a	3
Leg b	4
Hypotenuse c	5
Value of a²+b²	=B1^2+B2^2
Value of c²	=B3^2

Table to find the hypotenuse c

Leg a	3
Leg b	4
Hypotenuse c	=SQRT(B1^2+B2^2)

Table to find the missing leg b

Hypotenuse c	13
Known leg a	5
Missing leg b	=SQRT(B1^2-B2^2)

Table to find the altitude to the hypotenuse h

Leg a	3
Leg b	4
Hypotenuse c	5
Altitude h	=B1*B2/B3

After pasting, the upper rows are your inputs and the formula in the last row is calculated automatically.
"^2" squares a number, and "SQRT(…)" is the Excel function for the square root.
In the first table, if the value of a²+b² (B4) and the value of c² (B5) are equal, the triangle is a right triangle (for the example 3, 4, 5, both are 25).
In the second table, B3 shows the hypotenuse, 5. Just replace the input numbers with your own side lengths.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to check for a right triangle

Leg a	3
Leg b	4
Hypotenuse c	5
Value of a²+b²	=B1^2+B2^2
Value of c²	=B3^2

Table to find the hypotenuse c

Leg a	3
Leg b	4
Hypotenuse c	=SQRT(B1^2+B2^2)

Table to find the missing leg b

Hypotenuse c	13
Known leg a	5
Missing leg b	=SQRT(B1^2-B2^2)

Table to find the altitude to the hypotenuse h

Leg a	3
Leg b	4
Hypotenuse c	5
Altitude h	=B1*B2/B3

The same formulas as in Excel (the SQRT function and ^2) work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with your own side lengths.

How to calculate it in Python

import math

side_a = 3.0   # leg a
side_b = 4.0   # leg b

hypotenuse = math.sqrt(side_a ** 2 + side_b ** 2)   # hypotenuse c (math.hypot(side_a, side_b) gives the same)
area = side_a * side_b / 2                          # area
perimeter = side_a + side_b + hypotenuse            # perimeter
height = side_a * side_b / hypotenuse               # altitude to the hypotenuse
angle_alpha = math.degrees(math.asin(side_a / hypotenuse))   # angle opposite leg a (degrees)

print(f"Hypotenuse c: {hypotenuse}")
print(f"Area: {area}")
print(f"Perimeter: {perimeter}")
print(f"Altitude to the hypotenuse: {height}")
print(f"Angle alpha: {angle_alpha}")

# If you know the hypotenuse c and leg a, find the missing leg b with
# side_b = math.sqrt(hypotenuse ** 2 - side_a ** 2)

Runs with only the standard math library. "**" raises to a power (here, squaring) and "math.sqrt" is the square root. Change the two side lengths at the top and run it.

How to write it in LaTeX and other math languages (copy and paste)

The Pythagorean theorem (how the three sides are related)

a² + b² = c²

a^2 + b^2 = c^2

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msup><mi>a</mi><mn>2</mn></msup>
    <mo>+</mo>
    <msup><mi>b</mi><mn>2</mn></msup>
    <mo>=</mo>
    <msup><mi>c</mi><mn>2</mn></msup>
  </mrow>
</math>

a^2 + b^2 = c^2

a^2 + b^2 == c^2

a^2 + b^2 = c^2;

a^2 + b^2 == c^2

a^2 + b^2 = c^2

Formula for the hypotenuse \(c\)

c = √(a² + b²)

c = \sqrt{a^2 + b^2}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>c</mi>
    <mo>=</mo>
    <msqrt>
      <mrow>
        <msup><mi>a</mi><mn>2</mn></msup>
        <mo>+</mo>
        <msup><mi>b</mi><mn>2</mn></msup>
      </mrow>
    </msqrt>
  </mrow>
</math>

c = sqrt(a^2 + b^2)

Sqrt[a^2 + b^2]

c := sqrt(a^2 + b^2);

c = sqrt(a^2 + b^2);

c = √(a^2 + b^2)

Formula for a missing leg

b = √(c² − a²)

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

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>b</mi>
    <mo>=</mo>
    <msqrt>
      <mrow>
        <msup><mi>c</mi><mn>2</mn></msup>
        <mo>&#x2212;</mo>
        <msup><mi>a</mi><mn>2</mn></msup>
      </mrow>
    </msqrt>
  </mrow>
</math>

b = sqrt(c^2 - a^2)

Sqrt[c^2 - a^2]

b := sqrt(c^2 - a^2);

b = sqrt(c^2 - a^2);

b = √(c^2 - a^2)

Altitude to the hypotenuse

h = a × b ÷ c

h = \frac{ab}{c}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>h</mi>
    <mo>=</mo>
    <mfrac>
      <mrow><mi>a</mi><mi>b</mi></mrow>
      <mi>c</mi>
    </mfrac>
  </mrow>
</math>

h = (a b)/c

a*b/c

h := a*b/c;

h = a*b/c;

h = ab/c

How to have ChatGPT do the calculation

You are a geometry calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

In a right triangle, the two legs are a = 3 and b = 4.
Find each of the following:
1. The length of the hypotenuse c (Pythagorean theorem c = √(a² + b²))
2. Angle α (opposite leg a) and angle β (opposite leg b) in degrees
3. The area and the perimeter
4. The altitude to the hypotenuse h = ab ÷ c

Show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula