We want to find the area of a regular nonagon with a radius that measures 10units and a side length of 6.84units. Recall that a regular nonagon is a regular polygon with nine congruent sides.
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 nonagon. Be aware that the radii divide a regular nonagon into nine 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 9 to obtain their measures.
360/9=40^(∘)
The vertex angles of the isosceles triangles measure 40^(∘) 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 hexagon. As a result, a right triangle is created. The length of its shorter leg is 6.84÷ 2=3.42units.
We can find the length of the apothem using the Pythagorean Theorem.
a^2+b^2=c^2
Let's substitute the corresponding values into the formula and solve the resulting equation.
Since a is an apothem it must be non-negative, which is why we only kept the principal root when solving the equation.
Perimeter
Since all nine sides are congruent, to find the perimeter we will multiply the side length 6.84 by 9.
P= 6.84* 9= 61.56
Area
We know that the apothem is sqrt(88.3036)units and that the perimeter is 61.56units. To find the area of the polygon we will substitute these values in the formula A= 12aP.
