We are given three figures: a circle with a diameter of 13 inches, a regularpentagon with a side length of 9 inches, and a triangle with a base of 18 inches and a height of 15 inches. First let's focus on the pentagon, as it seems to have the greatest area.
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 6.2 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 6.2 for a, 5 for n, and 9 for s.
The area of the pentagon is approximately 139.5 square inches. Next, let's move to the circle. It seems to have the least area, as it looks like it would fit in the other figures.
Since the diameter of the circle is 13 we can evaluate the radius by dividing 13 by 2.
r=13/2=6.5
The radius of this circle is 6.5 inches. Let's use this value to evaluate the approximate area of the circle.
The area of the circle is approximately 132.7 square inches. Finally, let's find the area of the triangle.
Recall that the area of a triangle is half of the product of its height and the corresponding side.
A_t=1/2( 15)( 18)=135
The area of the triangle is 135 square inches. Let's summarize the areas of the figures.
Figure
Area
Pentagon
139.5in.^2
Circle
132.7in.^2
Triangle
135in.^2
As we can see, the pentagon has the greatest area and the circle has the least area.
