Find the perimeter and the apothem of the figure. Then, substitute their values into the formula A= 12ap to find the area of the regular polygon.
≈ 769.5 square units
Practice makes perfect
The area of a regular polygon is half the product of the apothem and the perimeter of the figure. Here we are only given that the perimeter is 100 units and that the polygon is a decagon.
We need to find the apothem of the polygon. After that, we can use the formula for the area of a regular polygon to find the area of the figure.
A=1/2ap
Let's find the apothem.
Finding the Apothem
Before we can find the apothem, we will first find the side length of the decagon. Since the polygon is a regular decagon, the perimeter p is 10 times greater than the side length s.
10s = p
We know that the perimeter of the decagon is 100 units, so let's find the length of each side.
10s=100 ⇒ s=10
Each side of the decagon is 10 units long. Now let's find the apothem. To do so, we will start by drawing the radii of the decagon, which will divide the decagon into ten congruent isosceles triangles.
Since corresponding angles of congruent figures are congruent, the vertex angles of the isosceles triangles formed by the radii are congruent. Moreover, since a full turn measures 360^(∘), we can divide 360 by 10 to find their measure.
360/10=36^(∘)
The vertex angles of the isosceles triangles measure 36^(∘) each.
The apothem bisects both the vertex angle of the isosceles triangle and the opposite side of the vertex, which is the side of the decagon. As a result, a right triangle with one angle measuring 36^(∘) ÷ 2 = 18^(∘) is created. The length of its shorter leg is 10÷ 2= 5units.
Notice that the apothem is the leg adjacent to the 18^(∘) angle, while the leg measuring 5 units is opposite this angle. Therefore, we can use the cotangent ratio to find a.
cot( θ ) = adjacent/opposite ⇒
cot( 18^(∘) ) = a/5
Let's solve the equation above for a.
