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Pythagorean Theorem Calculator

Solve a² + b² = c² for any missing side with step-by-step work

Side c

5

Area

6.00

Triple

Yes

a² + b² = c²

The hypotenuse (c) must always be the longest side

Triangle Sides

a

3

a² = 9
b

4

b² = 16
c (hyp)

5

c² = 25
Pythagorean Triple: (3, 4, 5)

Measurements

Area

6

(a×b)/2
Perimeter

12

a+b+c

Step-by-Step Solution

1.

a² + b² = c²

2.

3² + 4² = c²

3.

9 + 16 = c²

4.

25 = c²

5.

c = √25 = 5

Common Pythagorean Triples

abca²+b²=c²
3459+16=25
5121325+144=169
7242549+576=625
8151764+225=289
9404181+1600=1681
116061121+3600=3721
202129400+441=841

Formulas Used

Pythagorean Theorem (solve for c)

c = √(a² + b²)

Find the hypotenuse given both legs. Square each leg, add them, and take the square root.

Where:

a= First leg of the right triangle
b= Second leg of the right triangle
c= Hypotenuse (longest side)

Solve for a Leg

a = √(c² - b²)

Find a missing leg given the hypotenuse and the other leg. The hypotenuse must be longer than the known leg.

Where:

a= Missing leg to solve for
b= Known leg
c= Hypotenuse (must be > b)

Example Calculations

1Classic 3-4-5 Triangle (Find c)

Inputs

Side a3
Side b4
Solve forc

Result

Hypotenuse c5
Pythagorean TripleYes
Area6

3² + 4² = 9 + 16 = 25, and √25 = 5. This is the smallest Pythagorean triple, and every multiple (6-8-10, 9-12-15) is also a triple.

2Find a Missing Leg (5-12-13)

Inputs

Side b12
Hypotenuse c13
Solve fora

Result

Side a5
Pythagorean TripleYes
Area30

a² = c² - b² = 169 - 144 = 25, so a = √25 = 5. The (5, 12, 13) triple is a primitive triple (no common factors).

3Non-Integer Result

Inputs

Side a1
Side b1
Solve forc

Result

Hypotenuse c1.414214
Pythagorean TripleNo
Area0.5

1² + 1² = 2, so c = √2 ≈ 1.414214. This is the diagonal of a unit square and forms a 45-45-90 triangle. Not a Pythagorean triple since √2 is irrational.

Frequently Asked Questions

Q

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It only works for right triangles (triangles with one 90° angle).

  • Formula: a² + b² = c² where c is the hypotenuse
  • Only applies to right triangles (one 90° angle)
  • The hypotenuse is always the longest side
  • Named after Greek mathematician Pythagoras (~570–495 BC)
  • Used in construction, navigation, and distance calculations
Known SidesCalculationMissing Side
a=3, b=49 + 16 = 25c = 5
a=5, b=1225 + 144 = 169c = 13
a=8, b=1564 + 225 = 289c = 17
b=7, c=25625 - 49 = 576a = 24
Q

What is a Pythagorean triple?

A Pythagorean triple is a set of three positive integers (a, b, c) where a² + b² = c². The smallest triple is (3, 4, 5). Any multiple of a triple is also a triple: (6, 8, 10) = 2 × (3, 4, 5).

  • All three sides must be positive integers
  • Smallest triple: (3, 4, 5) since 9 + 16 = 25
  • Multiples are also triples: (6,8,10), (9,12,15), etc.
  • Primitive triples have no common factor: (3,4,5), (5,12,13)
  • There are infinitely many Pythagorean triples
Triplea²b²c²
(3, 4, 5)91625
(5, 12, 13)25144169
(8, 15, 17)64225289
(7, 24, 25)49576625
Q

How do you find the hypotenuse of a right triangle?

To find the hypotenuse: square both legs, add them together, then take the square root. c = √(a² + b²). For legs of 6 and 8: c = √(36 + 64) = √100 = 10.

  • Step 1: Square side a (a²)
  • Step 2: Square side b (b²)
  • Step 3: Add the squares (a² + b²)
  • Step 4: Take the square root: c = √(a² + b²)
  • Example: a=6, b=8 → c = √(36+64) = √100 = 10
StepCalculationResult
Square a6²36
Square b8²64
Add36 + 64100
Square root√10010
Q

How do you find a missing leg (not the hypotenuse)?

Rearrange the formula: a = √(c² - b²). Square the hypotenuse, subtract the known leg squared, then take the square root. For c = 13 and b = 5: a = √(169 - 25) = √144 = 12.

  • Formula: a = √(c² - b²) or b = √(c² - a²)
  • The hypotenuse must be longer than the known leg
  • Example: c=13, b=5 → a = √(169-25) = √144 = 12
  • Example: c=25, a=7 → b = √(625-49) = √576 = 24
  • If c² - b² is negative, the inputs are invalid
KnownCalculationMissing Leg
c=13, b=5√(169-25) = √144a = 12
c=25, a=7√(625-49) = √576b = 24
c=17, b=15√(289-225) = √64a = 8
Q

What are real-world applications of the Pythagorean theorem?

The Pythagorean theorem is used in construction to ensure walls are square (3-4-5 rule), in navigation to calculate straight-line distances, in architecture for roof pitches, and in computer graphics for distance between two points on a screen.

  • Construction: 3-4-5 rule to verify 90° angles
  • Navigation: straight-line distance = √(x² + y²)
  • Architecture: roof rafter length from rise and run
  • TV screens: diagonal = √(width² + height²)
  • Computer graphics: distance between pixels
ApplicationWhat You MeasureFormula
TV diagonalWidth 48", Height 27"√(2304+729) = 55"
Roof rafterRun 12ft, Rise 5ft√(144+25) = 13ft
Ladder reachWall 12ft, Base 5ft√(144+25) = 13ft

Pythagorean Theorem: Complete Guide with Examples

1

The Formula and How to Use It

a² + b² = c² connects the three sides of every right triangle: the two legs (a and b) and the hypotenuse (c). For legs of 6 and 8: 36 + 64 = 100, so c = √100 = 10. To find a missing leg when you know c and one leg: a = √(c² − b²). With c = 13 and b = 12: a = √(169 − 144) = √25 = 5.

The theorem only applies to right triangles (one 90° angle). For non-right triangles, the law of cosines generalizes the formula: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. When C = 90°, cos(90°) = 0 and the formula reduces to the Pythagorean theorem. The Trigonometry Calculator computes the cosine values needed for the generalized form.

The distance formula in coordinate geometry is a direct application: the distance between points (x₁,y₁) and (x₂,y₂) is d = √((x₂−x₁)² + (y₂−y₁)²). Between (1,2) and (4,6): d = √(9+16) = √25 = 5. This extends to 3D: d = √(Δx² + Δy² + Δz²).

a² + b² = c²a = 8b = 6c = 10Step-by-step:a² = 64b² = 36a² + b² = 100c = √100 = 10Area = ½ × 8 × 6 = 24Perimeter = 8+6+10 = 24Legs (a, b)Hypotenuse (c)
2

Pythagorean Triples and Patterns

A Pythagorean triple is three positive integers (a, b, c) satisfying a² + b² = c². The smallest is (3, 4, 5): 9 + 16 = 25. Every multiple is also a triple: (6, 8, 10), (9, 12, 15), (30, 40, 50). A primitive triple has no common factor among all three numbers — there are infinitely many, including (3,4,5), (5,12,13), (8,15,17), (7,24,25), and (20,21,29).

Euclid’s formula generates all primitive triples: a = m² − n², b = 2mn, c = m² + n², where m > n > 0, m and n are coprime, and one is even. For m=2, n=1: a = 3, b = 4, c = 5. For m=3, n=2: a = 5, b = 12, c = 13. This formula proves there are infinitely many Pythagorean triples, a fact that has been known since antiquity.

Fermat’s Last Theorem states that no three positive integers satisfy aⁿ + bⁿ = cⁿ for any integer n > 2. This was conjectured by Pierre de Fermat in 1637 and not proven until Andrew Wiles’ landmark 1995 proof. The case n = 2 (Pythagorean theorem) is the only power for which integer solutions exist.

Visual Proof: a² + b² = c²a²=93×3b²=164×4+=c²=255×59 + 16 = 25(3, 4, 5) — smallest Pythagorean triplea² = 9b² = 16c² = 25
Primitive Pythagorean triples generated by Euclid’s formula
Primitive Triplea² + b²c²Euclid (m,n)
(3, 4, 5)9 + 16 = 2525(2, 1)
(5, 12, 13)25 + 144 = 169169(3, 2)
(8, 15, 17)64 + 225 = 289289(4, 1)
(7, 24, 25)49 + 576 = 625625(4, 3)
(20, 21, 29)400 + 441 = 841841(5, 2)
3

Construction and Real-World Applications

The 3-4-5 rule is the most common field application. To verify a 90° corner: measure 3 feet along one wall, 4 feet along the other, and confirm the diagonal is exactly 5 feet. Multiplying by 2 (6-8-10) or 3 (9-12-15) accommodates larger structures while maintaining integer measurements for easy verification with a tape measure.

Roof rafter length is calculated directly: rafter = √(run² + rise²). A roof with 12-foot run and 5-foot rise needs rafters of √(144 + 25) = √169 = 13 feet. The rafter adds about 2 feet for overhang, so a 15-foot rafter length is ordered. The Right Triangle Calculator can compute all triangle properties including angles for pitch specification.

TV screen diagonals use the same formula. A 48-inch wide and 27-inch tall screen has diagonal = √(2304 + 729) = √3033 = 55.07 inches — sold as a 55-inch TV. The aspect ratio 16:9 means width = diagonal × cos(arctan(9/16)) and height = diagonal × sin(arctan(9/16)). The Area Calculator can then compute the actual screen area: 48 × 27 = 1,296 square inches.

  • Foundation squaring: 3-4-5 rule or 6-8-10 for larger structures
  • Roof rafters: √(run² + rise²) = rafter length before overhang
  • Stair stringers: √(total_rise² + total_run²) = stringer length
  • Navigation: straight-line distance from coordinate offsets
  • Screen size: diagonal = √(width² + height²) for any aspect ratio

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Last Updated: Mar 26, 2026

This calculator is provided for informational and educational purposes only. Results are estimates and should not be considered professional financial, medical, legal, or other advice. Always consult a qualified professional before making important decisions. UseCalcPro is not responsible for any actions taken based on calculator results.

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