Plane Geometry — interactive book

Chapter 16 — Regular Polygons

16 Capítulo 16 · Neste capítulo você vai Regular polygons recognize regular polygons and prove that they are inscriptible and circumscriptible; identify center, apothem and centric angle; calculate side and apothem of the square, hexagon and equilateral triangle as a function of the radius; obtain the side of the regular decagon and pentagon and recognize the golden segment. Hexagonal nuts, honeycombs, the STOP sign, the gears of a clock: regular polygons are where you need to repeat the same piece around a center. In this chapter, we prove that every regular polygon fits exactly between two circles with the same center and we calculate its sides and apothem from the radius — calculations that, almost two thousand years ago, were used to create the first trigonometric tables.

October 12, 2026 Read

Exercises from chapter 15 — Any triangles

Law of sines 624 Knowing that $\operatorname{sen}(180^{\circ} - x) = \operatorname{sen}x$ and $\cos(180^{\circ} - x) = -\cos x$, determine: a) $\operatorname{sen}120^{\circ}$; b) $\cos 150^{\circ}$; c) $\cos 135^{\circ}$; d) $\operatorname{sen}150^{\circ}$; e) $\operatorname{sen}135^{\circ}$; f) $\cos 120^{\circ}$. Ver respostaa) $\dfrac{\sqrt{3}}{2}$ · b) $-\dfrac{\sqrt{3}}{2}$ · c) $-\dfrac{\sqrt{2}}{2}$ · d) $\dfrac{1}{2}$ · e) $\dfrac{\sqrt{2}}{2}$ · f) $-\dfrac{1}{2}$ 625 Determine the value of $x$ in the following cases: Ver respostaa) $6\sqrt{2}$ · b) $6\sqrt{6}$

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Calculation of notable lines

Medians, heights and angle bisectors can only be calculated from the three sides. All formulas come out of metric relations — and a single relation, Stewart’s, solves them all at once. Who discovered The median formula is known as Apollonius’ theorem: Papo of Alexandria attributes it to Apollonius of Perga (3rd century BC). The height formula uses the area of the triangle as a function of the sides, the Heron’s formula, demonstrated by Heron of Alexandria in the 1st century, in the Metric (the Arabs attributed it to Archimedes). The relation that holds for any segment connecting a vertex to the opposite side was published by the Scotsman Matthew Stewart in 1746, in the book Some General Theorems of Considerable Use in the Higher Parts of Mathematics.

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Metric relations and cosine theorem

The Pythagorean theorem has a correction term when the angle is not right. This term is the law of cosines. Who discovered The metric relations of this topic are in The Elements of Euclid (c. 300 BC), in area language: the proposition II.12 deals with the side opposite to an obtuse angle and II.13, opposite an acute angle. The cosine form, $a^2 = b^2 + c^2 - 2bc\cos A$, appeared much later, along with the trigonometric tables. The Persian astronomer Jamshid al-Kashi presented it in 1427, in The Key to Arithmetic; in France, the law of cosines is still called théorème d’Al-Kashi. François Viète wrote it with cosines in 1593.

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Sines theorem

In any triangle, each side divided by the sine of the opposite angle always gives the same number: the diameter of the circumscribed circle. Who discovered The law of sines was born in astronomy, for triangles drawn on the celestial sphere. The spherical version was obtained in the 10th century by astronomers from the Islamic world — including Abu al-Wafa and Abu Nasr Mansur. The form for flat triangles appears systematically in the Treatise on the quadrilateral (1260) by the Persian Nasir al-Din al-Tusi, who made trigonometry a discipline independent of astronomy. In Europe, it spread with the book On triangles of all kinds, by Regiomontanus, written in 1464 and printed in 1533.

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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

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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.

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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.

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Exercises from chapter 13 — Similarity of triangles and point powers

Exercises and resolutions 445 The triangles $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$ in the figure are similar. If the similarity ratio of the 1st to the 2nd is $\dfrac{3}{2}$, determine: a) $a$, $b$ and $c$; b) the ratio between their perimeters. Ver respostaa) $a = 21$, $b = 18$ e $c = 15$ · b) $\dfrac{3}{2}$ 446 Triangles $ABC$ and $PQR$ are similar. Determine $x$ and $y$. Ver resposta$x = 16$ e $y = 14$

October 11, 2026 Read

Point power

Two chords that intersect within a circle: the product of the parts of one is equal to the product of the parts of the other. With this idea, the distance to the horizon can be calculated. Who discovered The two classical forms are in The Elements of Euclid (c. 300 BC): the proposition III.35, about cords that cut each other, and the III.36, on a secant and a tangent drawn from the same exterior point. The name power of a point was created much later, in 1826, by the Swiss Jakob Steiner, who realized that the product is always the same, whatever the straight line drawn by the point.

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Cases (criteria) of similarity

To ensure that two triangles are similar, it is not necessary to check everything: two angles, or three ratios between sides, are enough. Who discovered The three criteria are in book VI of The Elements by Euclid (c. 300 BC): triangles equiangular have proportional sides (VI.4, the case AA), triangles with three proportional sides are similar (VI.5, LLL), and two proportional sides with the angle between them equal are also sufficient (VI.6, LAL). Much earlier, tradition attributes to Tales the use of triangles of the same shape to measure inaccessible distances.

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Similarity of triangles

Two figures are similar when they have the same shape, even if they are different sizes. It’s the mathematics of photos, maps and models. Who discovered The idea is in the stories about Tales of Miletus, who would have measured the height of a pyramid by comparing its shadow with that of a stick. The complete theory is in book VI of The Elements by Euclid (c. 300 BC), which defines similar figures as those with equal angles and proportional sides. It depends on the theory of proportions by Eudoxus of Cnidus (4th century BC), from book V, which also allowed the treatment of ratios between incommensurable segments.

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