We want to find the area of a regular 15-sided polygon with a perimeter of 180cm. To do so we will start by finding its side length and the measure of the angles formed by two radii. Then, we will use a right triangle to find the apothem. Finally, we will use the formula A= 12ap to find the area of the polygon.
Length of a Side
Let's start by drawing our 15-sided regular polygon.
All 15 sides are congruent, and we are told that the perimeter is 180cm. Therefore, we can find the length of a side by dividing 180 by 15.
Side Length: 180/15= 12cm
The length of each side is 12cm.
Measure of the Angles Formed by Two Radii
Now let's draw the radii of the polygon. Since all of the radii are congruent, they form 15 congruent isosceles triangles. Moreover, since corresponding angles of congruent figures are congruent, all the vertex angles of the isosceles triangles formed by the radii are congruent.
Recall that a full turn of a circle measures 360^(∘). Therefore, we can find the measure of the central angles by dividing 360 by the number of angles.
Central Angle Measure: 360/15= 24^(∘)
The measure of the vertex angle of each of the isosceles triangles formed by the radii is 24^(∘).
Apothem
Let's consider one of the isosceles triangles formed by two radii and a side of the polygon.
The apothem bisects both the angle whose measure is 24^(∘) and the side whose length is 12cm. Therefore, the apothem divides the isosceles triangle into two right triangles with an acute angle of measure 242= 12^(∘) and opposite side length 122= 6cm. Let's look at just one of these right triangles.
In the above right triangle we know the measure of an angle and the length of its opposite side, so we can use the tangent ratio to find the length of its adjacent side.
tan θ = Length of opposite side toθ/Length of adjacent side toθ
We found that the apothem of the regular polygon is approximately 28.227781cm.
Area of the Regular Polygon
The area of a regular polygon is half the product of the apothem and the perimeter. Since we already know that the apothem is approximately 28.227781cm and the perimeter is 180cm, we can substitute these values in the formula A= 12ap to find the area.
