2026-08-07 10 min

Pythagorean Theorem Calculator

Free online Pythagorean Theorem calculator. Calculate the hypotenuse or legs of a right triangle using a² + b² = c² with steps, area and perimeter.

Calculator

Solve for

a (leg) b (leg) c (hypotenuse)

Side a Help

Leg

Side b Help

Leg

Side c Help

Solved — edit others

Results update automatically

Results update live as you type

Status:a=3.0000 b=4.0000 c=5.0000A=6.0000 P=12.0000

Pythagoras Summary

Sidesa=3.0000 b=4.0000 c=5.0000

Check25.0000 = 25.0000 ✓

Area & Perimeter6.0000 · 12.0000

Results

Optimal

Side a (leg)

3.0000

Leg adjacent to right angle

Side b (leg)

4.0000

Other leg

Side c (hypotenuse)

5.0000

Longest side, opposite 90°

Area

6.0000sq units

½ab

Perimeter

12.0000

a + b + c

Verification

25.0000 = 25.0000

a² + b² vs c²

Right-triangle identity a² + b² = c² with hypotenuse-longest validation. Two positive sides required; verification shows both square sums.

Visualizations

Sides Compared

Leg Squares Share (a² vs b²)

a²36%

b²64%

Hypotenuse Share of Perimeter

c / P42%

a²+b² vs c²

a²+b²25

Formula

Squares on the legs sum to the square on the hypotenuse — right triangles only. Rearranged forms isolate each side.

a² + b² = c²

Alternative Form

c = √(a²+b²); a = √(c²−b²); b = √(c²−a²)

Variables

Symbol	Name	Unit	Description
a,b	Legs	units	Sides forming 90°
c	Hypotenuse	units	Opposite 90°, longest
A	Area	sq units	½ab
P	Perimeter	units	a+b+c

Explanation

Geometric area fact with algebraic power: distance, construction squaring, and vector lengths all reduce to it.

Derivation

(a+b)² = 2ab + c² ⇒ a²+b² = c² (rearrangement proof).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Square leg a.

a²

a:3

Intermediate: 3² = 9

Result: a² = 9.0000

Step 2

Square leg b.

b²

b:4

Intermediate: 4² = 16

Result: b² = 16.0000

Step 3

Add squares for c².

c² = a² + b²

Intermediate: 9.0000 + 16.0000 = 25.0000

Result: c² = 25.0000

Step 4

Root for hypotenuse c.

c = √(c²)

Intermediate: √(25.0000)

Result: c = 5.0000

Step 5

Verify the equality.

a² + b² = c²

Intermediate: 9.0000 + 16.0000 = 25.0000

Result: 25.0000 = 25.0000 ✓

Step 6

Compute area and perimeter.

A = ½ab; P = a+b+c

Result: A = 6.0000, P = 12.0000

Final Answer

3.0000

Worked Examples

See how the calculator works with different scenarios.

Scenario

Legs 3 and 4, solve c.

Inputs

a:3

b:4

Calculation

c = √(9+16) = √25

Result

c = 5.0000

Interpretation

Smallest integer triple — the default view.

Understanding Your Result

Your Result: Side a (leg): 3.0000

Typical range: c > each leg; c < a+b (triangle inequality); triples give integer c.

What This Means

The solved side completes the only triple satisfying a²+b²=c²; verification confirms equality.

Why It Matters

Squaring corners, rafters, GPS legs, and graphics distances all depend on it.

Professional Interpretation

Check c longest and a²+b²=c² within tolerance before cutting or coding.

Recommendations

•Pick solve-target first
•Keep hypotenuse longest
•Verify squares match
•Use triples to estimate
•Compute area/perimeter after
•Draw the squares view

Educational Content

Introduction

Pythagoras links side squares in every right triangle: the hypotenuse square equals the leg-squares sum.

Concept Explanation

Area rearrangement proves it; algebra rearranges it for any missing side.

Applications

▸Distance formula

▸Square corners (3-4-5)

▸Rafter and stair math

▸GPS and surveying

▸Force/velocity resultants

▸Collision and rendering

Benefits

•Any-side solving
•Instant verification
•Area + perimeter bundled
•Triple intuition
•Visual squares proof

Limitations

•Right triangles only
•Flat geometry only
•Needs two valid sides
•Hypotenuse must be longest

Best Practices

•Label hypotenuse first
•Square before adding
•Positive root only
•Test converse for squareness
•Prefer triples for layout

Common Misconceptions

Myth: It works on any triangle.

Truth: Only right triangles; otherwise use law of cosines.

Myth: c is just a+b.

Truth: c = √(a²+b²), strictly less than a+b.

Myth: Triples are the only solutions.

Truth: Most solutions are irrational — e.g., 1,1,√2.

Myth: Longest side can be a leg.

Truth: Hypotenuse is always longest.

Practical Tips

•Solve-target first
•Square legs then add
•Hypotenuse must be longest
•Verify a²+b²=c²
•Use 3-4-5 to estimate
•Area is ½ab
•Test squareness via converse

Common Mistakes

Adding sides not squares

Square each side first — c ≠ a+b.

Solving with c shorter than a leg

Swap labels; hypotenuse must be longest.

Applying to non-right triangles

Use law of cosines with the included angle.

Dropping the root

c² is area units — root back to length.

Negative lengths

Lengths are positive; reject ≤ 0.

Assumptions

This calculator makes the following assumptions to provide estimates:

Right triangle: One exact 90° angle.

Euclidean plane: Flat geometry, no curvature.

Positive lengths: Sides must exceed zero.

Hypotenuse longest: c > a and c > b enforced.

Consistent units: All sides share one unit.

Limitations

Be aware of these limitations when using this calculator:

Right-only: Oblique triangles need law of cosines.

Two inputs needed: One side alone is underdetermined.

Ideal shape: No thickness or measurement error.

Flat only: Spherical triangles differ.

Frequently Asked Questions

Answers to common questions about this calculator.

What is the Pythagorean theorem?

In a right triangle, hypotenuse² = leg₁² + leg₂² (a² + b² = c²). The square on the hypotenuse equals the sum of squares on the legs.

Who discovered the Pythagorean theorem?

Named for Pythagoras (~570–495 BC) but known to Babylonians (Plimpton 322, ~1800 BC), Indians (Baudhayana Sulba Sutra), and Chinese (Zhoubi Suanjing) far earlier.

What is the formula for the Pythagorean theorem?

a² + b² = c². Hypotenuse c = √(a²+b²); leg a = √(c²−b²); leg b = √(c²−a²).

How do you find the hypotenuse?

Square both legs, add, root: c = √(a²+b²). For 3 and 4: √(9+16) = √25 = 5.

How do you find a missing leg?

Subtract squares then root: b = √(c²−a²). For c=13, a=5: √(169−25) = √144 = 12.

Does the Pythagorean theorem work on all triangles?

No — right triangles only. General triangles use the law of cosines: c² = a²+b²−2ab·cos(C), which reduces to Pythagoras at 90°.

What are Pythagorean triples?

Integer solutions like (3,4,5), (5,12,13), (7,24,25), (8,15,17). Multiples (6,8,10) also work. Euclid’s formula generates all primitives.

What is the converse of the Pythagorean theorem?

If a² + b² = c² then the triangle is right-angled opposite c — used to test squareness without a protractor.

Where is the Pythagorean theorem used in real life?

Construction (3-4-5 square corners, rafters, stairs), GPS/surveying, physics vectors, graphics collision and rendering.

How is it related to the distance formula?

Distance d = √((x₂−x₁)² + (y₂−y₁)²) is Pythagoras with Δx, Δy as legs and d as hypotenuse.

What is the difference from the law of cosines?

Law of cosines handles any triangle; Pythagoras is its 90° special case where cos(90°) = 0.

Can sides be irrational?

Yes — e.g., legs 1,1 give hypotenuse √2 ≈ 1.414. The calculator shows decimals for non-perfect squares.

How do you calculate area and perimeter?

Area = ½ab from legs; perimeter = a+b+c after solving the third side. Both update live here.

What is Euclid’s formula for triples?

For m>n coprime, not both odd: a = m²−n², b = 2mn, c = m²+n². m=2,n=1 → (3,4,5); m=3,n=2 → (5,12,13).

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Squaring with 3-4-5

Layout without a square.

Proofs you can draw

Rearrangement in minutes.

Triples cheat sheet

Common integer sets.

Beyond right triangles

When to use law of cosines.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Hypotenuse: Longest side opposite 90°.
Leg: Side forming the right angle.
Right triangle: Triangle with one 90° angle.
Pythagorean triple: Integers with a²+b²=c².
Converse: a²+b²=c² ⇒ right-angled.
Law of cosines: General c² = a²+b²−2ab·cos(C).
Euclid’s formula: m,n generator of triples.
Distance formula: Pythagoras on coordinates.
Square on a side: Area a² visualized as a square.
Primitive triple: Coprime a,b,c triple.

Authoritative References

Trusted sources and standards referenced by this calculator.

Pythagorean Theorem — Wikipedia Right Triangle — Wikipedia Pythagorean Triple — Wikipedia Khan Academy: Pythagorean theorem Math is Fun: Pythagoras Cut-the-Knot: 122 proofs

Applicable Standards

Common Core 8.G.B — PythagorasNCTM Geometry expectations

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Page Information

Last Updated:

2026-08-07

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.