Start by finding the perimeter. Then, find the apothem. Finally, substitute the values of the perimeter and the apothem into the formula A= 12 to find the area of the polygon.
123.1yd^2
Practice makes perfect
We want to find the area of a decagon with a side length of 4yd. To do so we will start by finding its perimeter. 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.
Perimeter
Let's start by drawing our decagon.
Recall that in a decagon, all 10 sides are congruent, and we are told that the side length is 4yd. Therefore, we can find the perimeter by multiplying 4 by 10.
Perimeter: 4 * 10= 40yd
Therefore, the perimeter is 40yd.
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 10 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: 360/10= 36^(∘)
The measure of the vertex angle of each of the isosceles triangles formed by the radii is 36^(∘).
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 36^(∘) and the side whose length is 4yd. Therefore, the apothem divides the isosceles triangle into two right triangles with an acute angle of measure 362= 18^(∘) and opposite side length 42= 2yd. 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 6.2yd.
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 6.2yd and the perimeter is 40yd, we can substitute these values in the formula A= 12ap to find the area.
