The area of a regular polygon is half the product of the apothem and the perimeter. We will first use the Pythagorean Theorem to find the side length. When the side length is known we can find the perimeter.
Side Length
Let's now find the side length. To do so, we will start by drawing the radii of the pentagon. Be aware that the radii divide a regular pentagon into five 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 5 to obtain their measures.
360/5=72^(∘)
The vertex angles of the isosceles triangles measure 72^(∘) each.
Let's look at one of the isosceles triangles with the apothem drawn.
The apothem bisects both the vertex angle of the isosceles triangle and the opposite side of the vertex, which is a side of the pentagon. As a result, a right triangle is created. One of its sides measures 5 units and one of its angles has the measure 72^(∘)2= 36^(∘).
We can find the length of the triangle's other side, b, using the tangent ratio.
tan(θ)=Opp./Adj.
Let's substitute the corresponding values into the formula and solve the resulting equation.
As previously mentioned, the apothem bisects the side of the regular octagon. Therefore, the length of the side of the given polygon is twice the length of the side of the above triangle.
Consequently, the side length of the regular pentagon is 2* 5tan(36)units.
Perimeter
Since this polygon has five congruent sides, to find its perimeter we will multiply the side length by 5.
Perimeter
5* 2* 5tan(36)=50tan(36)units
Area
Now that we know that the apothem of the figure is 5units and that the perimeter is 50tan(36)units. To find its area, we will substitute these values in the formula A= 12aP. Let's do it!
