Pythagorean Theorem Calculator – Find a Missing Side

What this Pythagorean theorem calculator does

The Pythagorean theorem is the rule that ties the three sides of every right triangle together: the square of the longest side equals the sum of the squares of the two shorter sides. Written as an equation, a² + b² = c². It is the single most-used relationship in elementary geometry — and the one students, carpenters, surveyors and game developers reach for whenever they need a diagonal length.

This calculator does far more than plug numbers into one formula. It brings seven practical jobs into one right triangle calculator:

• Solve — use it as a hypotenuse calculator or missing leg calculator: enter any two sides, or one side plus one angle, and get the answer with step-by-step working, exact radicals where possible, a draggable scaled diagram, adjustable side sliders, toggleable geometric overlays, and companion measurements.
• Check — enter three side lengths and find out whether they form a right triangle, an acute triangle, an obtuse triangle, or no triangle at all, with the squares compared visually.
• Triples — generate Pythagorean triples with Euclid's formula, browse every primitive triple up to c = 300, and send any of them straight into the solver.
• Trig — all six trigonometric ratios of both acute angles, with the SOH CAH TOA ratios labelled, plus the Pythagorean identity sin² θ + cos² θ = 1 verified against your specific numbers.
• Advanced — every geometric quantity of the triangle: area, perimeter, semiperimeter, inradius, circumradius, all three altitudes, all three medians, all three angle bisectors, and the coordinates of all four classical triangle centres (centroid, circumcentre, orthocentre, incentre).
• Distance — calculate the distance between two points in 2D or 3D, follow the formula line by line, drag the 2D endpoints, and orbit, tilt, or zoom the 3D box visual.
• Batch — paste a list of triangles and solve them all at once, with CSV export.

Everything runs in your browser. No sign-up, no upload, no server round trip.

Pythagorean theorem formula and its three rearrangements

In a right triangle, the two sides that meet at the 90° angle are called the legs (conventionally a and b), and the side opposite the right angle is the hypotenuse (c). The hypotenuse is always the longest side. The theorem states:

Depending on which side is missing, the same equation rearranges into three working formulas. This is why a hypotenuse calculator and a missing leg calculator use the same theorem with different rearrangements:

Two rules matter when you use the leg formulas. First, you must subtract the smaller square from the larger one — subtracting a² − c² instead of c² − a² gives a negative number under the root, which has no real answer. Second, the hypotenuse you enter must genuinely be longer than the leg you already know; if it is not, the triangle cannot exist.

Hypotenuse calculator: find side c

To find the hypotenuse, enter the two perpendicular legs and leave side c blank. The calculator squares each leg, adds the two results and takes the positive square root. This is the standard hypotenuse formula, c = √(a² + b²). For example, legs of 9 and 12 give c = √(81 + 144) = √225 = 15.

Missing leg calculator: find side a or b

To calculate a missing leg, enter the hypotenuse and the other leg, leaving the unknown leg blank. The calculator subtracts the known leg's square from the hypotenuse's square before taking the square root. For example, if c = 13 and a = 5, then b = √(169 − 25) = 12. The hypotenuse must be longer than the known leg.

When you know one side and one angle instead of two sides, the sides follow from the trigonometric definitions below — but the underlying geometry is still the same right triangle, and the Pythagorean theorem still relates the three sides once you have them.

How to use the calculator

The Solve tab is designed so that you never have to tell it which side you are looking for. It works out the target from whichever boxes you fill in — and if you know an angle rather than a second side, you can switch to Angle mode.

Solve by two sides

1. Type any two sides. Enter values into leg a, leg b, or hypotenuse c — any combination of two. Leave the third box empty.
2. Watch the result appear. The solved side is shown large, with the formula used, the decimal value, and an exact radical if one exists. The empty box gets a small note showing the solved value.
3. Expand the steps. The step-by-step panel shows the theorem, the substitution, the squaring, the addition or subtraction, and the square root — exactly the working you would write by hand.
4. Toggle diagram overlays. The interactive diagram shows the triangle to scale with the solved side highlighted. Add or remove the incircle, circumcircle, altitude to the hypotenuse, median to the hypotenuse, angle arcs, and all four classical centres. Zoom in and out with the buttons in the corner.
5. Adjust unit and precision. Choose mm, cm, m, in, ft, yd, mi, km, or no unit. Changing between physical units converts the measurements already entered; the no-unit option removes the label without guessing a conversion.

Solve by one side and one angle

1. Click One side + one angle at the top of the Solve tab.
2. Enter the known side length and pick which side it is (leg a, leg b, or hypotenuse c).
3. Enter the known acute angle in degrees or radians, and pick which angle it is (angle A opposite leg a, or angle B opposite leg b).
4. The calculator computes the two unknown sides and the remaining angle from the trig ratios, then shows the full triangle in the same result card and diagram.

📌 Fill in all three sides? The calculator switches to verification mode. If your three sides satisfy a² + b² = c² it confirms it; if they do not, it tells you what the hypotenuse should be and by how much your figure is off. That is usually a typo, not a new geometry.

Worked examples

Example 1 — find the hypotenuse

A right triangle has legs of 3 cm and 4 cm. What is the hypotenuse?

The answer is exactly 5 — a whole number, because 3-4-5 is a Pythagorean triple. Every multiple of it works the same way: 6-8-10, 9-12-15, 30-40-50.

Example 2 — find a missing leg

A right triangle has a hypotenuse of 10 m and one leg of 6 m. What is the other leg?

This is the 3-4-5 triangle scaled by 2, which is why the answer is a clean integer.

Example 3 — when the answer is not a whole number

Legs of 3 and 5:

34 has no square factor, so √34 cannot be simplified — that is the exact answer. The decimal is a rounded approximation. The calculator shows both, and labels which is which.

Example 4 — an answer that simplifies

Legs of 2 and 6:

Here the largest perfect-square factor of 40 is 4, so the radical simplifies to 2√10. The calculator pulls out the largest square factor automatically — the same procedure you would do by hand.

Example 5 — decimals that land exactly

Legs of 2.5 and 6:

Decimal inputs often produce exact decimal answers, and the calculator reports those as exact rather than as a rounded value.

Example 6 — solving from one side and one angle

A ramp has a horizontal run of 12 ft and rises at an angle of 14°. How long is the ramp surface (the hypotenuse), and how high does it rise?

The ramp is about 12.367 ft long and rises about 2.992 ft. Using unrounded values, the theorem confirms the result: 12² + (12 tan 14°)² = (12 / cos 14°)² ≈ 152.9517.

The converse — right, acute, or obtuse?

The theorem runs in both directions. If three side lengths satisfy a² + b² = c² — where c is the longest side — then the triangle must be a right triangle. This is the converse of the Pythagorean theorem, and it is what the Check tab tests.

Sort the three sides so that a ≤ b ≤ c, then compare a² + b² with c²:

The last row is the triangle inequality: any two sides of a triangle must add up to more than the third. If they do not, no triangle exists at all, and testing the Pythagorean relationship is meaningless. The Check tab tests the inequality first, before it says anything about right, acute or obtuse.

💡 Why the comparison works. Fix two sides of length a and b with a hinge between them. Opening the hinge widens the angle opposite the third side and lengthens that side. When the hinge is exactly at 90°, the third side is √(a² + b²). Open it further and the side grows past that value; close it and the side shrinks below it. That is the whole intuition behind the three cases.

Pythagorean triples

A Pythagorean triple is a set of three positive whole numbers that satisfy a² + b² = c² exactly, with no rounding. The smallest and most famous is 3-4-5: 9 + 16 = 25. A triple is primitive when the three numbers share no common factor other than 1 — 3-4-5 is primitive, while 6-8-10 is not, because it is simply 3-4-5 doubled.

Triples are not rare curiosities. Carpenters use the 3-4-5 ratio to square a corner: measure 3 units along one edge, 4 along the other, and adjust until the diagonal is exactly 5. If it is, the corner is a true 90°. The same trick, scaled up, lays out foundations, decks and fence lines.

Euclid's formula

Every primitive triple can be generated from two whole numbers m and n with m > n > 0:

With m = 2 and n = 1: a = 4 − 1 = 3, b = 2 × 2 × 1 = 4, c = 4 + 1 = 5. With m = 3 and n = 2: a = 9 − 4 = 5, b = 12, c = 13. The triple is primitive when m and n are coprime and exactly one of them is even — which is why the Triples tab flags primitives.

To get every triple, primitive or not, multiply a primitive triple by any whole number k. That is why 3-4-5 generates 6-8-10, 9-12-15, 12-16-20 and infinitely many more.

The Triples tab works as a Pythagorean triples calculator: it applies Euclid's formula as you change m and n, then lists every primitive triple with c ≤ 300 so you can generate or look up a result without doing the arithmetic by hand.

Special right triangles

Two right triangles appear so often in textbooks and exams that they have their own shortcut rules. Both are direct consequences of the Pythagorean theorem.

Check the 45-45-90 case against the theorem: if both legs are 1, then c² = 1 + 1 = 2, so c = √2. Check the 30-60-90 case: if the short leg is 1, the long leg is √3, and c² = 1 + 3 = 4, so c = 2. Both fall straight out of a² + b² = c².

The calculator detects both patterns automatically and shows a badge when your triangle matches one, plus a badge when all three sides are integers (a Pythagorean triple).

Trigonometry and the Pythagorean identity

The Pythagorean theorem is the parent of every basic trigonometric ratio. For an acute angle θ in a right triangle, with opposite the leg across from θ and adjacent the leg touching θ:

Divide the theorem a² + b² = c² through by c² and substitute the definitions:

That is the Pythagorean trigonometric identity — the same fact as the geometric theorem, expressed in trigonometric language. Dividing sin² θ + cos² θ = 1 by cos² θ gives tan² θ + 1 = sec² θ; dividing by sin² θ gives 1 + cot² θ = csc² θ. Both are Pythagorean identities by another name.

The Trig tab shows all six functions for both acute angles of your triangle and verifies sin² θ + cos² θ = 1 against your specific numbers.

💡 Why this identity matters. It lets you find sin θ from cos θ (and vice versa) with a single square root, without knowing the triangle. That is the single most-used identity in trigonometry, and it exists because the Pythagorean theorem does.

The four triangle centres

Every triangle has four classical centres. In a right triangle each has a clean closed-form location, which makes the right triangle a favourite example in geometry courses.

The Advanced tab computes all four centres with the actual triangle, and the interactive diagram draws them as coloured dots when you enable the overlay.

Every metric this calculator computes

Beyond the three sides and two acute angles, the Advanced tab produces every standard right-triangle metric:

Real-world applications

The theorem shows up wherever a right angle and a diagonal appear together — which is most of the physical world.

• Ladders. A ladder leaning against a wall forms a right triangle with the wall and the ground. If the base is 5 ft from the wall and the ladder must reach 12 ft up, the ladder needs to be √(5² + 12²) = 13 ft long.
• TV and monitor sizes. Screen sizes are quoted as the diagonal. A 16:9 panel with a 48-inch width has a height of 27 inches, so the advertised diagonal is √(48² + 27²) ≈ 55.1 inches.
• Construction and carpentry. The 3-4-5 method squares corners on decks, foundations and fences. Any multiple works — 6-8-10, 9-12-15 — so you can scale the check to the size of the job.
• Roofs. A rafter is the hypotenuse of the triangle formed by the roof's rise and its horizontal run. A 12 ft rise over a 16 ft run gives a rafter of √(144 + 256) = 20 ft.
• Ramps and roads. The length of a ramp is the hypotenuse of its rise and run; so is the distance along a sloped road.
• Navigation and surveying. When you travel east then north, the straight-line distance from the start is the hypotenuse of those two legs.
• Computer graphics and games. Distance between two points on screen, collision detection, and normalising a vector all rely on the same square-root-of-sum-of-squares pattern.
• The distance formula. In coordinate geometry, the distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²) — the Pythagorean theorem applied to the horizontal and vertical gaps. In 3D, add a third squared difference under the root.

📌 Why the 3D version still works. Take the two points and draw the rectangular box they define. The diagonal of the box's base is a 2D distance; that diagonal and the box's height then form a second right triangle, whose hypotenuse is the 3D distance. Applying the theorem twice gives √(dx² + dy² + dz²).

Common mistakes

• Applying the theorem to a triangle that is not right-angled. For any other triangle you need the Law of Cosines, c² = a² + b² − 2ab·cos C. The Pythagorean relationship is a special case that only holds when C = 90°.
• Subtracting the squares in the wrong order. When finding a leg, it must be c² − a², never a² − c². The second version gives a negative number under the square root.
• Adding the sides before squaring. √(a + b) is not the same as √(a² + b²). With legs 3 and 4, the first gives √7 ≈ 2.65 and the second gives 5. Square first, then add.
• Assuming the longest number you typed is the hypotenuse. That is only true for a genuine right triangle. If you are solving for a leg, the hypotenuse is whichever side you designated as c, and it must exceed the known leg.
• Rounding in the middle of a calculation. Keep full precision until the final step. Rounding √2 to 1.41 and then squaring it gives 1.9881, not 2.
• Confusing the area with the hypotenuse. The area of a right triangle is ½ab; the hypotenuse is √(a² + b²). They are unrelated quantities that happen to be small numbers in the 3-4-5 case (area 6, hypotenuse 5).
• Forgetting that exact radicals exist. √50 is not "7.0711" — it is exactly 5√2. In an exam answer, the exact form is usually what is being marked.
• Mixing degrees and radians. If the Angle mode expects degrees and you enter 1.57 (≈π/2 radians), the calculator returns nonsense. Always check the unit selector.
• Assuming the inradius equals the circumradius. They are wildly different: for 3-4-5, r = 1 while R = 2.5. The inradius is inside the triangle; the circumradius passes through the vertices.

Limitations

• The theorem applies only to right triangles. For oblique triangles, use the Law of Cosines or the Law of Sines.
• Exact radical output is available when the squared result is an integer that can be simplified. Exact integer or terminating-decimal results are shown when they can be identified reliably; other decimal inputs receive a decimal approximation.
• The unit selector changes the label only. It does not convert between units — entering 3 feet and 4 inches will not give a correct answer.
• The diagram is drawn to scale within a fixed box, so extremely lopsided triangles are compressed. It illustrates the relationship, it is not a precision drawing.
• Results are computed in standard double-precision floating point (about 15–17 significant digits), so very large inputs can lose precision in the last digits.
• The Angle mode accepts one side and one acute angle. It does not accept the right angle (90° would give no unique triangle) or obtuse angles (which would contradict the right-triangle assumption).
• The Batch tab treats two numbers as legs and three numbers as a full triple to verify. Two-number rows are always interpreted as the two perpendicular legs. For a three-number row, enter any three side lengths; the calculator sorts them, checks the triangle inequality, and classifies the triangle before reporting its area and angles.

Frequently asked questions

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