Pythagorean Theorem Glossary

This Pythagorean theorem glossary defines more than fifty terms you'll meet when working with right triangles, from leg and hypotenuse to primitive triple and law of cosines. Each definition is short and precise and links to the page that teaches the idea in depth. Use the letter index to jump straight to a term.

The parts of a right triangle

Most of the vocabulary below describes this one picture. The standard labels are a right angle at C, legs a and b, and hypotenuse c; each side is named with the lower-case letter of the vertex opposite it.

Numbers and A–C

3-4-5 triangle

The right triangle with sides 3, 4 and 5, the smallest one with whole-number sides, since 3² + 4² = 5². Its acute angles are about 36.87° and 53.13°. Builders use it to set out square corners. See the 3-4-5 triangle.

30-60-90 triangle

A right triangle with angles 30°, 60° and 90°. Its sides are always in the ratio 1 : √3 : 2, so the hypotenuse is twice the shortest leg. It is half of an equilateral triangle. See 30-60-90 triangle rules.

45-45-90 triangle

A right triangle with two 45° angles and two equal legs, also called an isosceles right triangle. Its sides are in the ratio 1 : 1 : √2. See the 45-45-90 triangle.

Acute triangle

A triangle whose three angles are all acute, that is, less than 90°. If c is its longest side, then a² + b² > c². See acute, right or obtuse.

Adjacent side

For an acute angle of a right triangle, the leg that forms the angle together with the hypotenuse. In the diagram above, b is adjacent to angle A. Used to define cosine and tangent. Compare opposite side.

Altitude

A segment from a vertex of a triangle that meets the opposite side (or its extension) at a right angle; its length is the triangle's height. In a right triangle each leg is the altitude to the other leg, and the altitude to the hypotenuse splits the triangle into two smaller similar triangles. See height of a triangle and the geometric mean theorem.

Congruent

Exactly the same shape and size; one figure can be moved, turned or flipped to fit exactly on the other. Many proofs of the theorem rearrange four congruent right triangles. See the rearrangement proof.

Converse

The statement made by swapping a theorem's "if" and "then" parts. The converse of the Pythagorean theorem says that if three sides satisfy a² + b² = c², the triangle has a right angle opposite c. It is also true (Euclid proves it in Elements I.48). See the converse of the Pythagorean theorem.

Corollary

A result that follows directly from a theorem with little extra work. For example, "the hypotenuse is the longest side of a right triangle" is a corollary of the Pythagorean theorem, and so is the distance formula. See the hypotenuse.

Cosine (cos)

For an acute angle in a right triangle, the ratio of the adjacent leg to the hypotenuse: cosA = b over c. See Pythagorean theorem or trigonometry?

D–H

De Gua's theorem

A three-dimensional version of the theorem for areas: in a tetrahedron with a corner where three edges meet at right angles, the square of the area of the face opposite that corner equals the sum of the squares of the other three faces' areas. See De Gua's theorem.

Diagonal

A segment joining two corners of a polygon that are not next to each other. A rectangle's diagonal is the hypotenuse of a right triangle whose legs are the length and width, so d = √ℓ² + w². See the diagonal calculator.

Difference of squares

An expression of the form x² − y², which always factors as (x − y)(x + y). It gives a mental-math shortcut for finding a leg: 25² − 24² = 1 × 49 = 49. See how to find a missing side.

Dissection proof

A proof that cuts shapes into pieces and rearranges them, showing that two regions have equal area. Several famous proofs of the theorem work this way. See Perigal's dissection.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) on a coordinate plane: d = √(x₂ − x₁)² + (y₂ − y₁)². It is the Pythagorean theorem applied to the horizontal and vertical gaps between the points. See the distance formula.

Euclidean distance

Ordinary straight-line distance, extended to any number of dimensions by adding the squares of the differences in every coordinate and taking the square root. It is used everywhere from physics to machine learning. See the theorem in n dimensions.

Euclid's formula

A recipe for Pythagorean triples: for whole numbers m > n > 0, the numbers m² − n², 2mn and m² + n² form a triple. With m = 2, n = 1 it gives 3, 4, 5. See Euclid's formula.

Exact answer

An answer with no rounding, often written with a radical, such as √13 or 5√2, as opposed to a decimal approximation such as 3.61. See square roots and radicals.

Fermat's Last Theorem

The statement that aⁿ + bⁿ = cⁿ has no solutions in positive whole numbers when n is greater than 2. Pierre de Fermat wrote it down in the 1600s; Andrew Wiles proved it in the 1990s. For n = 2 there are infinitely many solutions, the Pythagorean triples. See Fermat's Last Theorem.

Geometric mean

For two positive numbers x and y, the number √xy. In a right triangle, the altitude to the hypotenuse is the geometric mean of the two segments it creates: h² = pq. See the geometric mean theorem.

Gougu theorem

The traditional Chinese name for the Pythagorean theorem. Gou and gu are the names of the shorter and longer legs. It appears in ancient Chinese texts such as the Zhoubi Suanjing and The Nine Chapters on the Mathematical Art. See the Gougu theorem.

Hypotenuse

The side of a right triangle opposite the right angle, always its longest side, usually labeled c. The word comes from the Greek for "stretching under". See what the hypotenuse is and how to find it.

I–O

Irrational number

A number that can't be written as a fraction of two whole numbers; its decimal goes on forever without repeating. The square root of any whole number that is not a perfect square, such as √2 or √5, is irrational. See why √2 is irrational.

Isosceles right triangle

A right triangle with two equal legs, which makes both acute angles 45°. If each leg is s, the hypotenuse is s√2. It is half of a square, cut along a diagonal. See the 45-45-90 triangle.

Law of cosines

The rule c² = a² + b² − 2abcosC, which works in every triangle. When C = 90°, cosC = 0 and it becomes the Pythagorean theorem. See the law of cosines.

Leg

Either of the two sides of a right triangle that form the right angle, usually labeled a and b. Each leg is shorter than the hypotenuse. See parts of a right triangle.

Obtuse triangle

A triangle with one obtuse angle, an angle greater than 90° and less than 180°. If c is the longest side, then a² + b² < c². See acute, right or obtuse.

Opposite side

For an angle in a triangle, the side that does not touch that angle. In a right triangle, the leg opposite angle A is a, and the side opposite the right angle is the hypotenuse. Used to define sine and tangent.

P

Perfect square

A whole number that is the square of a whole number: 1, 4, 9, 16, 25, 36… Its square root is a whole number, so a Pythagorean problem whose c² is a perfect square has a whole-number answer. See the perfect squares table.

Perpendicular

Meeting at a right angle, written ⊥. The two legs of a right triangle are perpendicular. See right triangles.

Plimpton 322

A Babylonian clay tablet from about 1800 BCE, now at Columbia University, with columns of numbers connected to Pythagorean triples. It shows the relationship was used more than a thousand years before Pythagoras, though what the tablet was for is still debated. See Plimpton 322.

Primitive Pythagorean triple

A Pythagorean triple whose three numbers have no common factor other than 1, such as 3-4-5 or 5-12-13. The triple 6-8-10 is not primitive, because it is 3-4-5 doubled. See primitive Pythagorean triples.

Proof

A chain of logical steps showing that a statement must be true, starting from definitions, accepted facts and earlier results. Hundreds of different proofs of the Pythagorean theorem are known, using algebra, rearranged areas, similar triangles and more. See proofs of the Pythagorean theorem.

Pythagoras

Greek thinker of the 6th century BCE (about 570 to 495 BCE), born on the island of Samos, who founded a community of followers in southern Italy. The theorem carries his name, but it was known earlier, and none of his own writings survive to show how, or whether, he proved it. See who was Pythagoras?

Pythagorean identity

The equation sin²θ + cos²θ = 1, true for every angle θ. It is the Pythagorean theorem applied to a right triangle with hypotenuse 1. See the Pythagorean identity.

Pythagorean inequality

The comparison that classifies any triangle with longest side c: if a² + b² > c² it is acute, if a² + b² = c² it is right, and if a² + b² < c² it is obtuse. See acute, right or obtuse.

Pythagorean quadruple

Four positive whole numbers with a² + b² + c² = d², such as 1, 2, 2, 3. They are the boxes whose edges and space diagonal are all whole numbers. See Pythagorean quadruples.

Pythagorean theorem

In any right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². Seen as areas, the square drawn on the hypotenuse has the same area as the squares on the two legs put together. See what is the Pythagorean theorem?

Pythagorean triple

Three positive whole numbers a, b, c with a² + b² = c², such as 3-4-5, 5-12-13 and 8-15-17. Every triple gives a right triangle with whole-number sides. See Pythagorean triples.

R–S

Radical

An expression with a root sign, such as √50 or 3√2. The √ symbol is the radical sign, and the number under it is the radicand. See square roots and radicals.

Right angle

An angle of exactly 90°, a quarter turn, marked in diagrams with a small square in the corner. See right triangles.

Right triangle

A triangle with one right angle. Its other two angles are acute and add up to 90°. British English calls it a right-angled triangle. See parts of a right triangle.

Similar triangles

Triangles with the same angles, so one is a scaled copy of the other and their corresponding sides are in the same ratio. One classic proof of the theorem uses the similar triangles made by the altitude to the hypotenuse. See the similar triangles proof.

Simplest radical form

A radical with no perfect-square factor (other than 1) left under the root: √72 in simplest form is 6√2. See the simplify square root calculator.

Sine (sin)

For an acute angle in a right triangle, the ratio of the opposite leg to the hypotenuse: sinA = a over c. See Pythagorean theorem or trigonometry?

Space diagonal

A segment through the inside of a box joining opposite corners. For a box ℓ × w × h, its length is √ℓ² + w² + h², found by using the theorem twice. See the Pythagorean theorem in 3D.

Spiral of Theodorus

A spiral built from right triangles, each with one leg 1 and the previous hypotenuse as its other leg. The hypotenuses are √2, √3, √4, √5 and so on. See the spiral of Theodorus.

Square (of a number)

A number multiplied by itself, written with a small 2: 7² = 7 × 7 = 49. It is also the area of a square with that side length, which is why the theorem is about squares. See the Pythagorean theorem formula.

Square root

The non-negative number that gives a number when multiplied by itself: √49 = 7. It is the last step in finding a side with the theorem, turning an area back into a length. See square roots.

T–V

Tangent (tan)

For an acute angle in a right triangle, the ratio of the opposite leg to the adjacent leg: tanA = a over b. (A tangent line to a circle is a different idea.) See Pythagorean theorem or trigonometry?

Theorem

A mathematical statement that has been proved true. The Pythagorean theorem is one of the oldest and most widely used. Compare corollary and converse. See how it is proved.

Vector magnitude

The length of a vector. For a vector with components x and y, the magnitude is √x² + y², used in physics for resultant forces and velocities. See the theorem in physics.

Vertex

A corner point of a shape, where two sides meet (plural vertices). A triangle's vertices are usually labeled with capital letters A, B, C, and each side with the lower-case letter of the opposite vertex. See parts of a right triangle.