Draw segments from the remaining vertices of the pentagon to the pentagon's center.
About 103.2 cm^2
Practice makes perfect
To find the area of the shaded region, we will draw segments from the remaining vertices of the pentagon to the pentagon's center, which creates 5 congruent triangles. We know they are congruent by the SSS (Side-Side-Side) Congruence Theorem.
If we find one triangle's area, we can then calculate the area of the pentagon. In order to do this, we will first determine the sum of the pentagon's interior angles.
The sum of the interior angles is 540^(∘). Since this is a regular pentagon, each interior angle will measure 540^(∘)5=108^(∘). Additionally, the segment between each vertex and the center of the pentagon bisects the angles at each vertex. Therefore, if each of the pentagon's angles is 108^(∘) half of this is 54^(∘).
If we draw the height from the vertex angle, it cuts the opposite side in two equal halves.
By finding the height of this triangle we can determine its area and then the area of the pentagon. With the given information, we can find the height by using the tangent ratio.
Now we can calculate the area of one of these triangles.
Area triangle: 1/2(10)(6.88)=34.4 cm^2
From the diagram, we see that three of these triangles are shaded. With this information, we can calculate the total area by multiplying the area of one triangle by 3.
Area shaded region: 34.4(3) = 103.2 cm^2
