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Angles, areas, diagonals, inscribed and circumscribed circles of regular polygons

Polygons are a part of a plane enclosed by lines. They are divided into groups based on the numbers of their sides. We already know triangles – three sided polygons and quadrilaterals – four sided polygons. Polygons are also divided into two special group, into regular and irregular polygons.

Regular polygons have all sides of equal length and all angles of equal measure.

In this lesson we’ll be learning there properties: diagonals of polygons,angles, areas and perimeters and inscribed and circumscribed circle.

First, we’ll remember things we learned about quadrilaterals, and expand things that will be needed in order to conclude some usefull things.

Quadrilaterals are poylgons with four sides.

four side polygon

Regular polygon is a square, because only square of quadrilaterals has all sides of equal length and all angles of equal measure. Area of a square is equal to the square of lenth of one side, and perimeter to four lengths of any side. All inner angles are 90˚, their sum is equal to 360˚.It has two diagonals. These diagonals intersect in one point which is a center of inscribed and circumscribed circle. Inscribed circle is a circle which touches all sides of a polygon, and circumscribed goes through al verices of a polygon.

To draw circumscribed circle of a square you simply put the needle of a compass into the intersection of diagonals and spread it to one vertex and draw. This circle must go through all verices.

 

circumscribed circle of a square

To draw inscribed circle, you must find the radius first. To find the radius you have to draw a perpendicular line from center to any side. When you draw that circle, it must touch all sides of a square.

Pentagons are polygons with five sides.

pentagon

Regular pentagon has five equal sides and five equal angles. It has five diagonals.

How would you know how many diagonals does a polygon have without having to draw it and count?

regular pentagon

Let’s try to logically come to a formula for number of diagonals of any polygon. Let’s say that polygon has n vertices. From any vertex you can draw n – 3 diagonals (you can’t draw from that vertex and two adjacent), and do that n times (from any vertex). And because every two diagonals overlap you have to divide that number with two. Final formula is:

d = \frac{n * (n - 3)}{2}

For pentagon that would be: d = \frac{5 * (5 - 3)}{2} = \frac{10}{2}

For quadrilateral that would be: d = \frac{4 * (4 - 3)}{2} = 2

How do we determine what is the sum of all interior angles, and what is the measure of a single angle in a regular polygon?

We’ll use a pentagon for example, but everything else goes the same.

diagonals of pentagon

We’ll draw diagonals of a pentagon from only one vertex. This divides it into three triangles. Since we know that the sum of all inner angles of a triangle is equal to 180˚, that means that the sum of all inner angles of a pentagon is equal to 180^{\circ} * 3 = 540^{\circ}.

The formula for a number of diagonals in a polygon of n vertexes is:

(n - 2) * 180^{\circ}
Let’s try it out:
For triangles: (3 - 2) * 180^{\circ} = 180^{\circ}
For quadrilaterals (4 - 2) * 180^{\circ} = 360^{\circ}
For pentagons (5 - 2) * 180^{\circ}= 540^{\circ} and so on…

Since all regular polygons have all angles of equal measure, to get to the measure of one angle in a polygon with n vertices you simply divide the sum of all inner angles by n.

For pentagon that will be: 540 : 5 =108^{\circ}. That means that every angle in a pentagon will be 108^{\circ}.

angles in pentagon

As you already noticed, diagonals in a regular polygon do not intersect in one point. That means that we have no candidates for center for inscribed and circumscribed circle. We’ll try to find it by bisecting angles. By doing this we got five congruent triangles.

bisection triangles give five congruent triangles

That means that i = k = j = n = m. and that means that point F is a center for inscribed and circumscribed circles. To draw circumscribed circle you simply put the needle of a compass in point F and spread it to any point of a pentagon. To get to the radius of a inscribed circle, you have to draw a perpendicular line to any side from center. This will be your radius. Again, circumscribed circle must go through all vertices and inscribed must touch all sides.

center for inscribed and circumscribed circles

These triangles we divided our pentagon will also be useful for finding area of our pentagon.

Since we know how to calculate area of a triangle, we simply multiply that area by 5 to get our whole area. All these triangles are isosceles triangles, whose angles you know. Than is fairly simple to calculate area.

For example:
Find the area of a pentagon whose one side is equal to 4 cm.

We know that all these triangles that we divided pentagon on are congruent. That also means that their areas are equal.

congruent triangles and same areas

The area of a triangle is equal to Pt = \frac{a * h_a}{2}.

That means that area of a pentagon will be:

Pp = 5· Pt

We know that one angle in a pentagram is equal to 108^{\circ}. And that sides of those triangles bisect our angle. That means that angles in our triangle are equal to the half of 108^{\circ} which is equal to 54^{\circ}. And we get third angle by subtracting the sum of these two from 180^{\circ}, which is equal to 180^{\circ} - 108^{\circ} = 72^{\circ}. That triangle can be divided into two right triangles with one side 2. By knowing that, we can use trigonometry:

right triangle in pentagon

Tan (54^{\circ}) = \frac{altitude}{2}
altitude = 2.75 c
Pt= 4 * 2.75 = 11 cm^2
Pp = 5 * 11 = 55 cm^2

Hexagon is a polygon with six sides.

centar of inscribed and circumscribed circle in intersection of opposite vertices

Regular hexagon has six equal sides, and six equal angles.

regular hexagon

Let’s use what we know to determine other properties.

Number of diagonals:
d = \frac{n (n - 3)}{2} = \frac{6 (6 - 3)}{2} = 9

Sum of all interior angles:
(n - 2) * 180^{\circ}= 4 * 180^{\circ}= 720^{\circ}

Measure of one angle:
720^{\circ} : 6 = 120^{\circ}

Center of inscribed and circumscribed circle is in intersection of opposite vertices. If you are unsure which point to use as center for inscribed and circumscribed circles, the safest way is to bisect the angles and their intersection will be the point you are looking for.

These diagonals divide a hexagon into six congruent equilateral triangles, which means that their sides are equal and all their angles are 60^{\circ}. For the area, you again calculate the area of one triangle and multiply it with 6.

diagonals in hexagon

The same rules and formulas apply on other polygons, so we won’t go so much in detail.

Heptagon – polygon with seven sides.

heptagon

Octagon – a polygon with eight sides.

octagon

Nonagon – a polygon with 9 sides.

nonagon

 

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