Plane Geometry

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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Enlarging and reducing figures

When you zoom in on a photo, it becomes larger, but the face is not stretched or flattened. The photo was expanded: everything grew at the same rate. Enlarge without deforming Propriedade · The magnification rule To enlarge or reduce a figure without deforming: multiply all sides by same number; keep the angles (the corners) the same. If the number is greater than 1, the figure is enlarged; if it is less than 1, such as ½, it is reduced.

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Perimeter and area in the checkered mesh

To fence a vegetable garden, you need to know how much wire goes around it. To plant, you need to know how much floor is inside. There are two different measurements: the perimeter and the area. Perimeter: the path around Definição · Perimeter The perimeter is the measurement of the contour of a figure: it is the sum of the lengths of all sides. An ant that goes around a rectangular table measuring 2 m by 1 m walks 2 + 1 + 2 + 1 = 6 m. This is the perimeter of the table.

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Symmetry

Fold a sheet in half, cut out a design on the fold and open it: the two halves are equal, one “reflecting” the other. This figure has symmetry. The axis of symmetry Definição · Axis of symmetry A figure has axis of symmetry when there is a line that divides it into two halves that meet fit perfectly when folding the figure along this line. Each half is the reflection of the other in a mirror.

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Polygons: sides and vertices

Draw a figure using only straight line pieces, without taking the pencil off the paper, until you return to the beginning without crossing the path. You drew a polygon! What is a polygon Definição · Polygon A polygon is a flat closed figure, formed only by pieces of straight lines (the sides) that do not intersect. The points where two sides meet are the vertices. The circle is not a polygon: its contour is curved. An open figure is also not a polygon.

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