The Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.
In Spanish
teorema de Pitágoras
Also called
a squared plus b squared, Pythagoras' theorem, theorem of Pythagoras
The two squares on the legs (9 and 16 unit squares) hold exactly as much area as the square on the hypotenuse (25). That is what a2 + b2 = c2 says.
Formal definition and notation
In any right triangle with legs of lengthsa and b and hypotenuse of length c, a2 + b2 = c2: the area of the square drawn on the hypotenuse equals the sum of the areas of the squares drawn on the legs. The converse is also true: if the side lengths of a triangle satisfy a2 + b2 = c2, the triangle has a right angle opposite the side c. Whole-number solutions, such as 3, 4, 5, are called Pythagorean triples.
Symbol
Read it as
Meaning
a, b
a and b
the legs: the two sides that meet at the right angle
c
c
the hypotenuse, opposite the right angle
c = √a2 + b2
c equals the square root of a squared plus b squared
the theorem solved for the hypotenuse
Key facts
The hypotenuse is always the longest side and always sits across from the right angle.
Common Pythagorean triples: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25. Multiples work too: 6, 8, 10 is 2 × (3, 4, 5).
The converse tests for a right angle: 9, 40, 41 is a right triangle because 81 + 1600 = 1681 = 412.
The distance between two points comes from the theorem: from (1, 2) to (4, 6) the legs are 3 and 4, so the distance is 5.
For a triangle that is not right, a2 + b2 ≠ c2: with c the longest side, a2 + b2 > c2 means acute and a2 + b2 < c2 means obtuse.
Pythagorean theorem
Pythagorean theorem
a2 + b2 = c2
a, b
the lengths of the legs
c
the length of the hypotenuse, the longest side
Hypotenuse missing: c = √a2 + b2. Leg missing: a = √c2 − b2. Square first, add or subtract, then take the square root last.
Try it: drag the legs
Interactive
Drag the blue or orange point, or select one and use the arrow keys, to change the legs. The readout squares them, adds, and takes the square root. Look for whole-number triples like 5, 12, 13.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Pythagorean Theorem examples, step by step
Example 1: Find the hypotenuse
Warm-up
A right triangle has legs 5 and 12. How long is the hypotenuse?
The hypotenuse is one side of a right triangle, c. The Pythagorean theorem is the equation that links it to the legs.
Pythagorean Theoremvs.Distance formula
The distance formula, d = √(x2 − x1)2 + (y2 − y1)2, is the Pythagorean theorem used on a coordinate plane, with the legs as the horizontal and vertical changes.
Pythagorean Theoremvs.Pythagorean triple
A Pythagorean triple is a set of three whole numbers that fit the theorem, like 5, 12, 13. The theorem itself works for any right triangle, including sides like 1, 1 and √2.
Quick check: 3 questions
Question 1 of 3
A right triangle has legs 9 and 12. How long is the hypotenuse?
Show answer and explanation
Answer: B.81 + 144 = 225 and √225 = 15. Choice A adds the legs, C forgets the square root, and D subtracts the squares as if 12 were the hypotenuse.
Question 2 of 3
Which set of side lengths makes a right triangle?
Show answer and explanation
Answer: D.72 + 242 = 49 + 576 = 625 = 252. For the others, the squares of the two shorter sides do not add up to the square of the longest: 16 + 25 ≠ 36, 25 + 100 ≠ 144, 4 + 9 ≠ 16.
Question 3 of 3
What is the distance between the points (1, 2) and (7, 10)?
Show answer and explanation
Answer: A. The legs are 7 − 1 = 6 and 10 − 2 = 8, so the distance is √36 + 64 = 10. Choice B adds the legs, C forgets the square root, and D is only the vertical leg.
Pythagorean Theorem: frequently asked questions
Does the Pythagorean theorem work for all triangles?
No. It works only for right triangles, those with a 90° angle. For other triangles, a2 + b2 is bigger or smaller than c2. You can use that as a test: if the side lengths satisfy a2 + b2 = c2, the triangle must be a right triangle. That is the converse of the theorem.
How do I know which side is c?
c is the hypotenuse: the side across from the right angle, which is also the longest side. The two sides that touch the right-angle mark are the legs, a and b, and it does not matter which leg you call a. If a problem gives you three lengths, the largest one is c.
What is a Pythagorean triple?
A set of three whole numbers that fit a2 + b2 = c2, such as 3, 4, 5 or 5, 12, 13. Multiplying a triple by any whole number gives another one: 3, 4, 5 times 3 is 9, 12, 15. Recognizing triples saves time on tests, because you can skip the square root.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.G.B.6, CCSS 8.G.B.7, CCSS 8.G.B.8
Cite this entry
Effortless Math Team. “Pythagorean Theorem.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/pythagorean-theorem/
Flashcard
TermPythagorean Theorema2 + b2 = c2
DefinitionIn a right triangle, the squares of the legs add up to the square of the hypotenuse: 32 + 42 = 52.
Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1FootA foot is a US customary unit of length equal to 12 inches, about the length of a standard school ruler. Its abbreviation is ft, and its plural is feet. There are 3 feet in a yard and 5,280 feet in a mile, and 1 foot is exactly 30.48 centimeters.Measurement · Grade 2, Grade 3, Grade 4HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6Law of CosinesThe law of cosines relates the three sides of any triangle to one of its angles: c2 = a2 + b2 − 2abcosC, where C is the angle across from side c. Use it to find the third side when you know two sides and the angle between them, or to find an angle when you know all three sides.Trigonometry · Geometry, Algebra 2, PrecalculusDistance FormulaThe distance formula finds the straight-line length between two points on the coordinate plane: d = √(x2 − x1)2 + (y2 − y1)2. It is the Pythagorean theorem in disguise: the horizontal and vertical differences are the legs of a right triangle, and the distance is the hypotenuse.Geometry · Grade 8, Algebra 1, GeometryCoordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8TrigonometryTrigonometry is the branch of math that studies how the angles and the side lengths of triangles are related. Its three main ratios, sine, cosine and tangent, compare two sides of a right triangle. Knowing one side and one acute angle, or two sides, you can find every other side and angle without measuring them.Trigonometry · Geometry, Algebra 2, PrecalculusSpecial Right TrianglesSpecial right triangles are the two right triangles whose angles are 45°-45°-90° or 30°-60°-90°. Their sides always come in fixed ratios: x : x : x√2 for the 45-45-90 triangle and x : x√3 : 2x for the 30-60-90 triangle. From one side you can find the other two without the Pythagorean theorem.Geometry · Grade 8, Geometry, Algebra 2TheoremA theorem is a math statement that has been proved true using definitions, postulates and statements already proved. Once proved, it is true in every case it covers, so you can use it to solve problems. The Pythagorean theorem and the Vertical Angles Theorem are famous examples.Geometry · Grade 8, Geometry