Pythagorean Theorem Calculator

Right-triangle side solver

Find the missing side without guessing which value to leave blank

Choose the side you need, enter the two known positive lengths, and see the answer, triangle, and substitution together.

For right triangles onlya² + b² = c²c is always the hypotenuse

1Choose the missing side
2Enter two known lengths
3Calculate and check the diagram

Result

Your completed right triangle

Step 3

Missing hypotenusec = 5in the same length unit as your inputs

Triangle area6square units

Perimeter12units

a legb legc hypotenuse

See the formula with your numbers

Start with the theorema² + b² = c²
Substitute the known legsc = √(3² + 4²)
Add, then take the positive rootc = √(9 + 16) = √25 = 5

Right-triangle length check

Identify the hypotenuse before squaring any side

The Pythagorean theorem solves a missing length only after the geometry proves there is a right angle and the longest, opposite side has been assigned to c.

Step-by-step guide · Right-triangle length check

A wheelchair-ramp check separates sloped length from horizontal run

Applicability: right triangles only
Leg relation: a² + b² = c²
Hypotenuse: c lies opposite the 90° angle

Square the legs in one unit
Use meters throughout: 0.75²=0.5625 and 9²=81. Mixing centimeters for one leg would make the sum dimensionally meaningless.
Add areas, then take one root
c²=81.5625, so c=√81.5625≈9.0312 meters. The equation adds squared lengths; √0.5625+√81 is not an equivalent shortcut.
Perform a magnitude check
The ramp is slightly longer than 9 meters and far shorter than 9.75 meters, which fits the diagram. For project compliance, a geometric length alone is insufficient; applicable construction rules remain outside this calculator.

Identify the hypotenuse before squaring any side: A sketch shows a vertical rise of 0.75 meter and a horizontal run of 9 meters. The direct ramp surface is the hypotenuse because it sits opposite the right angle made by vertical and horizontal directions.

The right-angle mark determines the entire side naming scheme

The hypotenuse is not whichever side happens to be drawn horizontally or assigned the largest letter. It is the side opposite the 90° angle and must be the longest side. The other two sides are legs. With that labeling, a²+b²=c² states that the areas of squares on the legs add to the area of the square on the hypotenuse. Without a right angle, this equation is not licensed; use the general triangle calculator for other side-and-angle configurations.

Solving for c requires c=√(a²+b²). Solving for a leg requires subtraction first: a=√(c²−b²). The radicand for a valid missing leg cannot be negative, and the result cannot exceed c. Those constraints offer powerful pre-calculation checks that a generic square-root button does not know.

A wheelchair-ramp check separates sloped length from horizontal run

A sketch shows a vertical rise of 0.75 meter and a horizontal run of 9 meters. The direct ramp surface is the hypotenuse because it sits opposite the right angle made by vertical and horizontal directions.

Square the legs in one unit
Use meters throughout: 0.75²=0.5625 and 9²=81. Mixing centimeters for one leg would make the sum dimensionally meaningless.
Add areas, then take one root
c²=81.5625, so c=√81.5625≈9.0312 meters. The equation adds squared lengths; √0.5625+√81 is not an equivalent shortcut.
Perform a magnitude check
The ramp is slightly longer than 9 meters and far shorter than 9.75 meters, which fits the diagram. For project compliance, a geometric length alone is insufficient; applicable construction rules remain outside this calculator.

Three nearby calculators answer three distinct geometry questions

Distance from perpendicular components

Pythagoras combines two orthogonal lengths. In coordinate work, those are Δx and Δy; the finding slope calculator instead divides them to report direction of change.

Area from base and perpendicular height

Triangle area is bh/2 and does not require the hypotenuse unless a missing height must first be derived. The area calculator also handles other plane shapes and squared-unit conversion.

Principal root from a bare radicand

The exponent calculator can evaluate a principal inverse power such as n^(1/2), but it cannot know which side is the hypotenuse, whether a right angle exists, or which quantities share a unit.

Impossible right-triangle inputs announce themselves early

If solving for a leg and the entered hypotenuse is not longer than the known leg, the geometry is invalid or the labels are reversed.
If no 90° condition is stated or inferable, do not assume one merely because a diagram looks rectangular.
Convert every length to a common unit before squaring; conversion after adding unlike squared units cannot repair the setup.
Retain extra digits through the square root and round only the final length to the precision justified by the measurements.

Important limitations

Pythagorean Theorem Calculator is limited to the finite inputs, modes, and outputs stated for the Pythagorean theorem solver method. Check solve for, leg a, leg b, and hypotenuse c, validation messages, the equation, and the stated display precision before relying on solved side, leg a, leg b, and hypotenuse c.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Right-triangle length check

Identify the hypotenuse before squaring any side

The Pythagorean theorem solves a missing length only after the geometry proves there is a right angle and the longest, opposite side has been assigned to c.

Step-by-step guide · Right-triangle length check

A wheelchair-ramp check separates sloped length from horizontal run

Applicability: right triangles only
Leg relation: a² + b² = c²
Hypotenuse: c lies opposite the 90° angle

Square the legs in one unit
Use meters throughout: 0.75²=0.5625 and 9²=81. Mixing centimeters for one leg would make the sum dimensionally meaningless.
Add areas, then take one root
c²=81.5625, so c=√81.5625≈9.0312 meters. The equation adds squared lengths; √0.5625+√81 is not an equivalent shortcut.
Perform a magnitude check
The ramp is slightly longer than 9 meters and far shorter than 9.75 meters, which fits the diagram. For project compliance, a geometric length alone is insufficient; applicable construction rules remain outside this calculator.

The right-angle mark determines the entire side naming scheme

A wheelchair-ramp check separates sloped length from horizontal run

Square the legs in one unit
Use meters throughout: 0.75²=0.5625 and 9²=81. Mixing centimeters for one leg would make the sum dimensionally meaningless.
Add areas, then take one root
c²=81.5625, so c=√81.5625≈9.0312 meters. The equation adds squared lengths; √0.5625+√81 is not an equivalent shortcut.
Perform a magnitude check
The ramp is slightly longer than 9 meters and far shorter than 9.75 meters, which fits the diagram. For project compliance, a geometric length alone is insufficient; applicable construction rules remain outside this calculator.

Three nearby calculators answer three distinct geometry questions

Distance from perpendicular components

Pythagoras combines two orthogonal lengths. In coordinate work, those are Δx and Δy; the finding slope calculator instead divides them to report direction of change.

Area from base and perpendicular height

Triangle area is bh/2 and does not require the hypotenuse unless a missing height must first be derived. The area calculator also handles other plane shapes and squared-unit conversion.

Principal root from a bare radicand

The exponent calculator can evaluate a principal inverse power such as n^(1/2), but it cannot know which side is the hypotenuse, whether a right angle exists, or which quantities share a unit.

Impossible right-triangle inputs announce themselves early

If solving for a leg and the entered hypotenuse is not longer than the known leg, the geometry is invalid or the labels are reversed.
If no 90° condition is stated or inferable, do not assume one merely because a diagram looks rectangular.
Convert every length to a common unit before squaring; conversion after adding unlike squared units cannot repair the setup.
Retain extra digits through the square root and round only the final length to the precision justified by the measurements.

Pythagorean Theorem Calculator

Find the missing side without guessing which value to leave blank

Which side is missing?

Enter these two sides

Your completed right triangle

Identify the hypotenuse before squaring any side

A wheelchair-ramp check separates sloped length from horizontal run

Square the legs in one unit

Add areas, then take one root

Perform a magnitude check

The right-angle mark determines the entire side naming scheme

A wheelchair-ramp check separates sloped length from horizontal run

Three nearby calculators answer three distinct geometry questions

Distance from perpendicular components

Area from base and perpendicular height

Principal root from a bare radicand

Impossible right-triangle inputs announce themselves early

Important limitations

Find the missing side without guessing which value to leave blank

Which side is missing?

Enter these two sides

Your completed right triangle

Identify the hypotenuse before squaring any side

A wheelchair-ramp check separates sloped length from horizontal run

Square the legs in one unit

Add areas, then take one root

Perform a magnitude check

The right-angle mark determines the entire side naming scheme

A wheelchair-ramp check separates sloped length from horizontal run

Three nearby calculators answer three distinct geometry questions

Distance from perpendicular components

Area from base and perpendicular height

Principal root from a bare radicand

Impossible right-triangle inputs announce themselves early