The area of a regular polygon is half the product of the apothem and the perimeter. We will use the Pythagorean Theorem to find the apothem.
Apothem
Let's now find the apothem. To do so, we will start by drawing the radii of the octagon. Be aware that the radii divide a regular octagon into eight 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 8 to obtain their measures.
360/8=45^(∘)
The vertex angles of the isosceles triangles measure 45^(∘) 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 octagon. As a result, a right triangle is created. One of its angles has the measure 45^(∘)2=22.5^(∘) and its hypotenuse measures 11 units.
We can find the length of the apothem using the cosine ratio.
cos(θ)=Adj./Hyp.
Let's substitute the corresponding values into the formula and solve the resulting equation.
The shorter leg in this triangle is also half the side length of the octagon. We find this length, b, using the sine ratio.
sin(θ)=Opp./Hyp.
Let's substitute the corresponding values into the formula and find b.
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 hexagon is 2* 11sin(22.5)units. Since this polygon has eight congruent sides, to find its perimeter we will multiply the side length by 8.
Perimeter
8* 2* 11sin(22.5)=176sin(22.5)units
Area
Now that we know that the apothem of the figure is 11cos(22.5)units and that the perimeter is 176sin(22.5)units. To find its area, we will substitute these values in the formula A= 12aP. Let's do it!
