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Math calculator

Pythagorean Theorem Calculator

Solve a right triangle with a² + b² = c². Enter any two of legs a, b and hypotenuse c to find the third side, plus area and perimeter. Try it free.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (solve-c, solve-b, solve-a, verify, invalid-domain) · Matrix
Numerical scope
93 independent full result packs across4operations and extreme scales; no exhaustive-domain claim
Known limitations
  • Positive sides; solve a|b|c or verify
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Missing side of a right triangle from any two of a, b, c (solve_for), or verification of a²+b²=c² when all three are given; also area and perimeter.
Scope
Right triangle; positive side lengths.
Verification
Engine tested · Source checked · v1.1.6 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.6 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.meta-metrics-v2
Verification revision
2026-10-28.meta-metrics-v2 · 2/2 property · digest 4e661f033102
Legacy regression
16/16 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · Overall CVP state: VERIFIED_STALE under CVP 1.0.0-proposed
CVP status
STALE · attestation snapshot core b4e2795757fd ≠ live cvp_core_sha256 4e661f033102 (2026-10-28.meta-metrics-v2)
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
93/93 golden · 10/10 CVP boundary · 12/12 invalid · 2/2 property · 2/2 metamorphic · 4/4 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Euclid, Elements, Book I, Proposition 47 — In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle
  • Right-triangle area — A = ab/2
Sources
Evidence
6 legacy golden · 10 legacy boundary · legacy regression suite · 93/93 oracle-backed golden · 12/12 invalid · Artifact integrity PASS
STALE · Build schema 1.1.6 matches page · production binding STALE · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Theorema² + b² = c²
Hypotenusec = √(a² + b²)
Leg aa = √(c² − b²)
Leg bb = √(c² − a²)
AreaA = ab / 2
PerimeterP = a + b + c
ic must be the longest side and strictly longer than each leg. Only right triangles. Positive side lengths required. solve_for selects the unknown; mode=verify checks all three sides.

How to use

1

Choose solve or verify

Solve hypotenuse c (default), leg a, or leg b — or switch to Verify when you already have all three sides.

2

Enter the known sides

The solved field is calculated. Share URLs record solve_for=a|b|c, or mode=verify with a, b, and c.

3

Read side, area, and perimeter

The result shows the unknown (or a ✓ when verifying), plus area ab/2 and perimeter. Exact radicals are provided only for safe integer radicands; other results are binary64 approximations.

Example calculations

Common configurations with formula and result.

ϟ

Classic 3-4-5

Legs 3 and 4

c = √(9+16)
5
ϟ

Find a leg

b = 4, c = 5

a = √(25−16)
3
ϟ

Verify 3-4-5

All three sides

3² + 4² = 5²
✓ valid
ϟ

Legs 7 and 9

Hypotenuse

√130
≈ 11.4018

Common Pythagorean triples

Common values at a glance.

abc
345
51213
6810
72425
81517
91215
94041
202129
i Multiples of a primitive triple also work (e.g. 6-8-10 = 2×(3-4-5)).

Pythagorean Theorem calculator specification

Version 1.1.6 · Engine tested

Calculation status

Review policy · Evidence

Definition
For a right triangle with legs a and b and hypotenuse c (opposite the right angle), the Pythagorean theorem states a² + b² = c². Enter any two side lengths to solve for the third; enter all three to verify. The tool also reports area ab/2 and perimeter.
What it calculates
Missing side of a right triangle from any two of a, b, c (solve_for), or verification of a²+b²=c² when all three are given; also area and perimeter.
Inputs
  • Leg a (optional)
  • Leg b (optional)
  • Hypotenuse c (optional)
  • solve_for: a | b | c. If omitted, inferred from which sides are present.
  • mode=verify to check all three sides
Outputs
  • The missing side length, or a verify result
  • Area A = ab/2 when both legs known
  • Perimeter a+b+c when all sides known
  • Exact radical when the radicand is not a perfect square
Formula
a²+b²=c²; A=ab/2; P=a+b+c
Assumptions
  • Right triangle; positive side lengths.
  • c > a and c > b when solving for a leg.
  • Exactly two fields are inputs when solving. Explicit solve_for selects the output. If all three are supplied without solve_for, they are verified rather than overwritten.
  • Interactive evaluation, shared URLs, and REST use the same engine, solve_for contract, and error codes.
  • Share URL is canonical: solve_for=a|b|c forces solve even if a leftover third side is present; mode=verify is required to stay in Verify after a tab change.
Units
  • Any consistent length unit; area in squared units
Boundary conditions
  • Need exactly two known sides to solve the third, or all three to verify
  • Invalid solve_for → INVALID_SOLVE_FOR
  • Fewer than two inputs → MISSING_REQUIRED_INPUT
  • Non-positive side → SIDE_NOT_POSITIVE
  • c ≤ a or c ≤ b when solving a leg → HYPOTENUSE_TOO_SHORT
  • All three without solve_for that do not satisfy Pythagoras → status invalid (not overwritten)
Numerical precision
  • Computation uses IEEE-754 binary64 (JavaScript Number).
  • Displayed results use up to 12 significant digits; trailing zeros are stripped. Scientific notation is used when |value| ≥ 1e12 or 0 < |value| < 1e-6.
  • Verify requires nondegenerate triangle geometry and |a²+b²-c²|/c² <= 1e-6, evaluated with scaled squares. Diagnostic rounding near the threshold can differ by 2e-15. This is not proof of exact equality.
  • Interactive evaluation, shared URLs, and REST use the same formulas, solve_for contract, and error codes.
Example
a=3, b=4 → c=5
Validation cases

8 published on this page · 16/16 tests · Production surface contract 6/6 · View evidence

  • a=3, b=4 → c=5
  • solve_for=c, a=3, b=4 → c=5
  • a=5, c=13 → b=12
  • a=7, b=9 → c=√130 ≈ 11.4018
  • mode=verify, a=3, b=4, c=5 → valid
  • a=3, b=4, c=5 (no solve_for) → verify valid (not solve-for-c)
  • mode=verify, a=3, b=4, c=6 → invalid (9+16 ≠ 36)
  • b=4, c=4, solve a → error HYPOTENUSE_TOO_SHORT
Sources
  • Euclid, Elements, Book I, Proposition 47 — In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle · accessed 2026-08-28
    Supports: Classical statement of the Pythagorean theorem used as the geometric identity a² + b² = c².
  • Right-triangle area — A = ab/2
    Supports: Area of a right triangle is half the product of the legs that form the right angle. Perimeter is the sum of the three sides.
Calculation version
1.1.6

Background

Interpretation and common distinctions.

What does this pythagorean theorem calculator compute?

For a right triangle with legs a and b and hypotenuse c (opposite the right angle), the Pythagorean theorem states a² + b² = c². Enter any two side lengths to solve for the third; enter all three to verify. The tool also reports area ab/2 and perimeter.

The published formula or method is a²+b²=c²; A=ab/2; P=a+b+c. The following references support the stated mathematics or method:

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses Legs 3 and 4 with c = √(9+16). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.pythagorean v1.1.6 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

What is the Pythagorean theorem?

For a right triangle (one angle = 90°), if a and b are the legs and c is the hypotenuse (opposite the right angle):

a² + b² = c²

The square on the hypotenuse equals the sum of the squares on the other two sides.

Pythagorean triples

Integer solutions such as 3,4,5, 5,12,13, 8,15,17, and their multiples (6,8,10, …).

Inputs accept positive finite JSON numbers or strict decimal strings of at most 512 characters, including scientific notation. Nonzero decimal text that converts to zero is rejected. Matching aliases are accepted; conflicting aliases and unknown fields reject. Empty fields are absent; null and nonnumeric types reject. All supplied side tokens validate, even when solve_for selects a supplied output. mode=verify cannot be combined with solve_for.

The diagram is schematic and is not to scale. Full binary64 values, input parameters, diagnostic and range statuses are available in CSV/JSON and REST/MCP. Invalid verification does not establish a right angle; ab/2 is not reported as its area. Positive derived values outside binary64 range are null with explicit range status.

Typing calculates locally. Shared URLs, REST and MCP send inputs to the server and may enter server or analytics logs; browser history may retain links and calculations. Logging retention and processing regions are not asserted here.

Near a flat triangle, a small perturbation to c or the known leg can strongly change the missing leg. Decimal inputs first round to binary64; c must remain strictly larger after rounding. A tolerance match is not an exact or symbolic proof.

Other calculators in this family: Annulus Calculator, Arc Calculator, Circle Calculator, Conic Section Calculator, Ellipse Calculator, Hyperbola Calculator, Irregular Polygon Calculator, Parabola Calculator . Explore all Geometry.

→ Annulus Calculator Calculate annulus area and perimeter with complete numerical results, clear input limits, shareable links and CSV export. Runs locally in your browser. › → Arc Calculator Calculate circular arc length from radius and central angle in degrees. See the radian formula, worked examples, units, and mathematical sources. › → Circle Calculator Calculate circle radius, diameter, circumference, area, arc length, and sector area. Explore formulas, worked examples, units, and mathematical sources. › → Conic Section Calculator Explore ellipse, parabola and hyperbola geometry: calculate foci, eccentricity and equations, inspect numerical limits, and share or export complete results. › → Ellipse Calculator Calculate ellipse area, foci, eccentricity and approximate perimeter from two semiaxes. Inspect formulas, numerical limits, figures and complete result exports. › → Hyperbola Calculator Calculate hyperbola foci, eccentricity and asymptote slopes from positive semiaxes. Choose orientation, inspect numerical limits and export complete results. › → Irregular Polygon Calculator Calculate irregular polygon area and perimeter with full results, clear input limits, shareable links and CSV export. Runs locally in your browser. › → Parabola Calculator Find the focus, directrix and equation of a horizontal or vertical parabola from its signed focal parameter. Explore figures, limits and full result exports. ›

Frequently asked questions

Key distinctions behind the calculation.

What is the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².

Which side is the hypotenuse?

The side opposite the right angle — always the longest side, labeled c here.

How do I find the hypotenuse?

Choose Hypotenuse c (solve_for=c). Then c = √(a² + b²) when you know both legs.

How do I find a missing leg?

a = √(c² − b²) or b = √(c² − a²). The hypotenuse must be longer than the known leg. Share URLs use solve_for=a or solve_for=b.

What happens if I enter all three sides?

The calculator verifies a² + b² = c² instead of overwriting a side. Use mode=verify in the share URL. A match shows ✓ Consistent within tolerance. Failed verification has no right-angle diagram or right-triangle area.

What is a Pythagorean triple?

Three positive integers a, b, c that satisfy a² + b² = c², such as 3, 4, 5.

Does this work for non-right triangles?

No. For other triangles use the law of cosines (which reduces to Pythagoras when the included angle is 90°).

How do I find the area and perimeter?

For a right triangle, A = ab/2 using the two legs that form the right angle. Perimeter is P = a + b + c once all three sides are known.