Pythagoras Theorem

Metric elements and relationships

We know that a triangle is rectangle when one of its internal angles is right. In this chapter we will develop a geometric-trigonometric theory that allows us to calculate the measurements of segments and angles of a right-angled triangle based on a minimum number of data. The story The metric relationships of the right triangle are among the oldest in mathematics. Babylonian tablets from around 1800 BC, such as the famous Plimpton 322, list suits of numbers that satisfy $a^2 = b^2 + c^2$, and the YBC 7289 tablet gives an excellent approximation of $\sqrt{2}$, the diagonal of the square on its side 1. The first known proof of the theorem, however, is Greek: it appears as the proposition 47 of book I of Euclid’s The Elements (around 300 BC), and tradition attributes it to the school of Pythagoras.

October 24, 2026 Read

The sine of π/n and notable arcs

We will verify that the circular functions of the reals $x = \dfrac{\pi}{n}$, with $n \in \mathbb{N}$ and $n \ge 3$, can be calculated from $\ell_n$, the side of the regular polygon with $n$ sides inscribed in the cycle. The story In book I of Almagest, Ptolemy constructed his table of chords exactly like this: he started from the sides of inscribed regular polygons — triangle, square, pentagon, hexagon and decagon —, which Euclid’s geometry knew how to calculate, and obtained the other arcs through formulas for sum, difference and half of arcs, which we will see in chapter VI. The chord of an arc $2\alpha$ in a circle of radius $R$ is $2R\operatorname{sen}\alpha$: the theorem in this chapter is the modern translation of this idea.

October 23, 2026 Read

Fundamental relationships

All relations in this topic come from two plane geometry tools: the Pythagorean theorem and the triangle similarity. The figure below shows the three triangles we will use. Drag $P$ and choose the relationship. The panel numerically checks each equality. Ative o JavaScript para ver esta figura interativa. The story In Almagest, Ptolemy already used the equivalent of $\operatorname{sen}^2 x + \cos^2 x = 1$: if he knew the chord of a bow, he obtained the additional bowstring through Pythagoras’ theorem in the triangle inscribed in the semicircle. The relations with tangent and secant appear in the works of Abu al-Wafa (10th century) and Nasir al-Din al-Tusi (13th century), whose Treatise on the quadrilateral made trigonometry a discipline independent of astronomy.

October 22, 2026 Read

Chapter exercises 14 — Right triangles

Metric relationships 506 Determine the value of $x$ in the following cases: Ver respostaa) 5 · b) 12 · c) $\sqrt{7}$ · d) $3\sqrt{3}$ 507 Determine $x$ as a function of $a$ in the following cases: Ver respostaa) $a\sqrt{3}$ · b) $\dfrac{a}{2}$ 508 Determine $x$ in cases: Ver respostaa) 12 · b) 24 · c) 20 · d) 8

October 9, 2026 Read

Applications of the Pythagorean theorem

The diagonal of the square, the height of equilateral triangle and the trigonometric ratios of 30°, 45° and 60° all come from the same theorem. Who discovered The diagonal of the square has already been calculated by Babylonians: tablet YBC 7289 (c. 1800–1600 B.C.) gives $\sqrt{2}$ with six correct decimal places. The ratios between the sides of a right triangle began as astronomical tables: Hipparchus of Nicaea (2nd century BC) set up a string table, the Indian Aryabhata (499) started to use the half-rope — our sine — and Islamic astronomers like Abu al-Wafa (10th century) organized shadow tables, our tangents. The formula that generates all Pythagorean triangles, with sides $x^2 - y^2$, $2xy$ and $x^2 + y^2$, appears in a motto from book Elements of Euclid.

October 9, 2026 Read

Metric relationships

The square of the hypotenuse is the sum of the squares of the legs. But this is just one of several relationships that height reveals within the right triangle. Who discovered The relationship between the sides of the right triangle is much older than Pythagoras of Samos (6th century BC). Tablets from Babylon from around 1800 BC use terms like 3, 4, 5 and calculate the diagonal of the square; the Sulba Sutras of India (c. 800–500 B.C.) enunciate the rule for building altars; and the Chinese text Zhoubi Suanjing provides visual proof. Greek tradition associated the theorem with Pythagoras, but there is no documentary proof that he demonstrated it. The oldest demonstration that has come down to us is in the The Elements of Euclid (c. 300 BC): the proposition I.47 proves the theorem by areas, and the I.48 proves the reciprocal. The relations with height — each leg and height as proportional averages — are in the proposition VI.8 and its corollary.

October 9, 2026 Read

Pythagoras Theorem

The Pythagorean Theorem is one of the best-known and fundamental propositions in mathematics, establishing an elegant relationship between the sides of a right-angled triangle. The history of this theorem is as fascinating as its mathematical application, as it spans cultures and eras, revealing how mathematical knowledge is accumulated and shared across generations. Traditionally attributed to the Greek philosopher and mathematician Pythagoras, who lived between approximately 570 BC and 495 BC, the theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, known as legs.

July 11, 2024 Read