b To find the area of the polygon divide it into triangles by drawing diagonals.
A
a 20
B
b About 126.3 units^2
Practice makes perfect
a The exterior angle and its corresponding interior angle of a polygon are supplementary angles. This means they sum to 180^(∘). With this information we can determine the polygon's interior angle, θ.
θ+18^(∘) =180^(∘) ⇔ θ =162^(∘)
In a polygon, the sum of the interior angles is 180^(∘)(n-2), where n is the number of sides. Since this is a regular polygon where all interior angles are congruent we can equate this with 162^(∘) n and solve for n.
b To find the area of the 20-gon, we will first draw it including the diagonals between opposite vertices which creates 20 congruent and isosceles triangles. We know they are congruent by the SSS (Side-Side-Side) Congruence Theorem.
If we find one triangle's area we can calculate the area of the polygon. Since we have 20 congruent isosceles triangles, their vertex angle will be 360^(∘)20=18^(∘). We also know that the base of this triangle is 2 units. Let's draw one of these triangles, including its height.
With this information we can find the height, h, with the tangent ratio.
