Pythagorean Theorem: Find the Missing Side (Hypotenuse and Legs)

This guide covers which side is which, solving for a leg versus the hypotenuse, 3-4-5 and other Pythagorean triples, the link to distance in 2D, mistakes that cost points, and when to open DevOkk’s Pythagorean Theorem Calculator, Right Triangle Calculator, or Distance Calculator. No account. The math stays in the tab.

For quadratics that sometimes appear when you square both sides of a radical equation, see Quadratic formula, discriminant, and roots. For a different “distance over time” story, velocity from distance and time uses displacement along a path-not the Pythagorean shortcut across a grid.

The theorem applies only to right triangles

The Pythagorean theorem is true in Euclidean plane geometry when the triangle has a 90° angle. It is false for acute or obtuse triangles unless you move to the law of cosines.

Given legs a and b and hypotenuse c:

a² + b² = c²

The hypotenuse is always the longest side. It sits opposite the right angle. The two legs form the right angle themselves.

If a problem says “isosceles right triangle” with legs 5, the hypotenuse is 5√2, not 5. If it says “30-60-90,” the side ratios are 1 : √3 : 2-that is a special triangle shortcut, not a contradiction of a²+b²=c².

Which side is the hypotenuse?

Three reliable checks:

1. Right-angle marker - small square at the vertex between the legs. The side across from it is c.
2. Longest side - in a right triangle, c > a and c > b. If your “c” is shorter than another side, you swapped labels.
3. Wording - “side opposite the right angle,” “diagonal of a rectangle,” “hypotenuse” in the prompt.

In a rectangle with width 6 and height 8, the diagonal is the hypotenuse of a right triangle with legs 6 and 8. It is not a leg.

In a ladder problem, the wall and ground are legs (meeting at 90° if the wall is vertical and the ground level). The ladder is c.

Students often assign c to a vertical height when the diagram is not a right triangle with that height as hypotenuse. Read the angle first.

Finding the hypotenuse when you know both legs

If a and b are known:

c = √(a² + b²)

Example: legs 3 and 4.

c = √(9 + 16) = √25 = 5.

Example: legs 7 and 24.

c = √(49 + 576) = √625 = 25.

These are classic Pythagorean triples: (3, 4, 5) and (7, 24, 25). Multiples work too: (6, 8, 10), (10, 24, 26).

When numbers are messy: legs 2 and 5.

c = √(4 + 25) = √29 ≈ 5.39 if decimals are allowed.

The Pythagorean Theorem Calculator takes two sides and whether you are solving for a leg or the hypotenuse. Use it to verify; still show a²+b²=c² on paper if the rubric requires it.

Finding a leg when the hypotenuse is known

Rearrange:

a = √(c² − b²) or b = √(c² − a²)

The leg must be shorter than c. If c² − b² is negative, you swapped hypotenuse and leg.

Example: c = 13, b = 5.

a = √(169 − 25) = √144 = 12. Another 5-12-13 triple.

Example: c = 10, a = 6.

b = √(100 − 36) = √64 = 8.

Example with decimals: c = 15, a = 9.

b = √(225 − 81) = √144 = 12.

3-4-5 triples and scaling

The 3-4-5 triangle is the most common integer right triangle in textbooks.

3² + 4² = 9 + 16 = 25 = 5².

Multiply all sides by k:

• k = 2 → 6-8-10
• k = 3 → 9-12-15
• k = 1.5 → 4.5-6-7.5 (still valid, not integers)

Other primitive triples worth recognizing:

Triple	Check
5-12-13	25+144=169
8-15-17	64+225=289
7-24-25	49+576=625

If a problem gives two legs whose squares sum to a perfect square, the hypotenuse is an integer-do not leave √169 as “13.0” with six decimals unless asked.

Conversely, if sides are 2-3-?, the hypotenuse is √13, not 5. Not every pair of small integers is a leg pair.

Distance in 2D is the same theorem

Place two points on a coordinate plane. Horizontal change Δx and vertical change Δy are legs of a right triangle. Distance d is the hypotenuse:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Example: (1, 2) to (4, 6).

Δx = 3, Δy = 4 → d = √(9 + 16) = 5.

Example: (−2, 1) to (3, −3).

Δx = 5, Δy = −4 → squares kill the sign → d = √(25 + 16) = √41.

The Distance Calculator expects coordinates or equivalent inputs depending on the page layout. It is the Pythagorean theorem with subtraction baked in. It does not replace showing Δx and Δy on a grid problem.

This is not the same as total path length along streets. Manhattan distance |Δx| + |Δy| ignores the diagonal shortcut. Pythagorean distance is “as the crow flies” in 2D.

Right Triangle Calculator: when the page goes beyond one missing side

Pythagorean Theorem Calculator is the narrow tool: two sides known, one unknown, right angle assumed.

Right Triangle Calculator often handles more triangle data-sometimes an angle and a side, or area and perimeter relations, depending on the implementation. Open it when the worksheet mixes trigonometry (sin, cos, tan) with sides, or when you need another derived quantity after you find the missing length.

If the only ask is “find c given legs 9 and 12,” the dedicated Pythagorean page is enough. If the ask is “find all angles and the area,” the broader right-triangle tool may match the form fields.

Common student errors (and how to lose zero points)

1. Hypotenuse on the wrong side

Writing a² + c² = b² when b is the hypotenuse gives nonsense when you solve for a leg. Always put c alone on the side opposite the single largest square: legs squared sum to hypotenuse squared.

2. Forgetting the square root

After c² = 25, c = 5, not 25. After a² = 144, a = 12, not 144.

3. Adding instead of squaring

(a + b)² is not a² + b². The theorem uses squares of sides, not the square of their sum.

4. Using the theorem on non-right triangles

A triangle with sides 2, 3, 4 is obtuse. a²+b²=c² with c=4 gives 4+9=16, which fails (16 ≠ 16 is false for 2²+3² vs 4²: 4+9=13≠16). You need law of cosines for general triangles.

5. Unit mix-ups

Legs in centimeters and hypotenuse requested in meters-convert at the end, not mid-square.

6. Rounding too early

If a = √2 ≈ 1.414, and you need c with a = b = √2, compute c² = 2 + 2 = 4 exactly when possible. Round the final answer once.

7. Confusing leg with height in 3D

A rectangular box diagonal uses √(l² + w² + h²)-that is three dimensions, not the planar theorem alone. A single right triangle in a cross-section still uses a²+b²=c² for that slice.

Word problems: translate words to legs and c

Ladder: 12 ft ladder, base 5 ft from wall. How high?

Legs: 5 (ground), h (wall). c = 12.

h = √(144 − 25) = √119 ≈ 10.9 ft.

Diagonal of a screen: 16:9 aspect, width 32 in. Treat width and height as legs; diagonal is c.

If height is 18 in (because 32/16×9), c = √(32² + 18²) = √(1024 + 324) = √1348.

Navigation sketch: east 3 km, north 4 km. Displacement magnitude 5 km northeast-ish-use distance formula, not 3+4.

Draw the right angle explicitly. Word problems hide it in “vertical wall,” “level ground,” or “north then east.”

Converse of the Pythagorean theorem

If side lengths satisfy a² + b² = c² with c the longest side, the triangle is right-angled.

Use this to check answers: sides 7, 24, 25 work; sides 7, 24, 26 do not (49+576=625, 676≠625).

Some tests ask “is this a right triangle?” without giving a right-angle mark. Square the longest side and compare to the sum of squares of the other two.

Special right triangles versus general Pythagorean

45-45-90: legs equal, hypotenuse leg×√2.

30-60-90: sides 1 : √3 : 2 (short leg opposite 30°).

These are derived from the Pythagorean theorem once and memorized for speed. They are not separate magic- they are shortcuts when you recognize the angles.

Homework workflow

1. Mark the 90° angle on the sketch.
2. Label c opposite it; label legs a and b.
3. Write a² + b² = c² with the unknown on one side algebraically before numbers.
4. Substitute, simplify, square root last.
5. Check with the converse or a quick calculator pass on Pythagorean Theorem Calculator.
6. For coordinate distance, use Δx and Δy then Distance Calculator if allowed.

Show units in the final line. Box the side name the question asked for, not just a naked number.

Connection to algebra and quadratics

Solving for a leg gives a² = c² − b², then a = √(c² − b²). If you square both sides of √(x+3) = 5 in another unit, you may land on a linear equation-or a quadratic with extraneous roots. When that happens, the algebra path crosses Quadratic formula, discriminant, and roots. The geometry picture here stays linear in the squares.

3D caution in one paragraph

Room diagonal, box space diagonal, and face diagonal are three different segments. Face diagonal of a 3×4 rectangle is 5 in the plane of that face. Space diagonal of a 3×4×12 box is √(9+16+144)=√169=13. Draw which right triangle you are in before applying a²+b²=c² once.

Privacy

Side lengths typed into DevOkk calculators run locally in the browser. No account is required to find a missing side. The tool does not prove your diagram was a right triangle-that remains your job from the problem statement.

The missing side is one square root away

Identify the hypotenuse first. Put a² + b² = c² in standard form. Recognize 3-4-5 family triples when integers appear. Use the same logic for distance in 2D between coordinates.

Open Pythagorean Theorem Calculator for two known sides and one unknown. Use Right Triangle Calculator when the assignment adds angles or extra triangle data. Use Distance Calculator for coordinate plane distance. Cross-check with Quadratic formula when algebra takes over, and keep legs, hypotenuse, and square root straight on the page you turn in.