Evaluate the difference between the areas of a circle and a pentagon.
≈ 79.7 square units
Practice makes perfect
In our exercise we are asked to evaluate the area of the shaded region. To do this we need to find the difference between the areas of a circle and a pentagon.
To evaluate the area of the regular pentagon we need to find its apothem a. Since the central angle is 72^(∘) and the apothem bisects this angle and the corresponding side, we can use the trigonometric ratios to find a.
Let's write and solve an equation using the tangent ratio.
The apothem of the pentagon is approximately 8.26 units. Now let's recall the formula for the area of a regular polygon.
A=1/2ans
In this formula a is the apothem, n is the number of sides, and s is the side length. Let's substitute 8.26 for a, 5 for n, and 12 for s.
The area of the pentagon is approximately 247.75 square units. Next we will find the radius of the circle, as we need it to evaluate the area. Let's call the radius r.
The area of the circle is approximately 327.49 square units. Finally, we will find the area of the shaded region by subtracting the area of the pentagon from the area of the circle.
A_c-A_p=327.49-247.75≈ 79.7
The area of the shaded region is approximately 79.7 square units.
