A regular pentagon is inscribed in a circle whose radius measures 7 cm. Find the area of the pentagon.
To find the area of a regular pentagon inscribed in a circle, we can use the following steps:
Step 1: Calculate the side length of the pentagon.
In a regular pentagon, all sides are of equal length. To find the side length, we can use the radius of the circle inscribed in the pentagon.
In a triangle formed by the center of the circle, one vertex of the pentagon, and two neighboring vertices, we have a right triangle. The hypotenuse of this right triangle is equal to the radius of the circle, which is 7 cm. The side adjacent to the right angle is half the side length of the pentagon.
Using the Pythagorean theorem, we can calculate the side length of the pentagon as follows:
(side length)^2 + (half side length)^2 = (radius)^2
Let's say the side length of the pentagon is 's'. Then we have:
s^2 + (s/2)^2 = 7^2
Simplifying the equation:
s^2 + (1/4)s^2 = 49
(5/4)s^2 = 49
s^2 = (4/5) * 49
s^2 = 39.2
s ≈ 6.259
Therefore, the side length of the pentagon is approximately 6.259 cm.
Step 2: Find the apothem of the pentagon.
The apothem is a line segment that connects the center of the circle to the midpoint of any side of the pentagon. Since the pentagon is regular, the apothem is also the perpendicular bisector of any side.
In a right triangle formed by the apothem, the side length, and half the side length, we have the following relation:
(apothem)^2 + (half side length)^2 = (side length)^2
Let's say the apothem of the pentagon is 'a'. Then we have:
a^2 + (s/2)^2 = s^2
a^2 + (1/4)s^2 = s^2
(3/4)s^2 = a^2
Substituting the value of s from Step 1, we can calculate the apothem as follows:
(3/4)(6.259)^2 = a^2
a^2 ≈ 29.325
Therefore, the apothem of the pentagon is approximately √29.325 ≈ 5.415 cm.
Step 3: Calculate the area of the pentagon.
The area of a regular pentagon can be calculated using the formula: Area = (1/2) * (perimeter) * (apothem)
Since the pentagon is regular, the perimeter is equal to 5 times the side length. Therefore, the perimeter is: 5 * 6.259 = 31.295 cm.
Now, using the formula:
Area = (1/2) * (31.295) * (5.415)
Area ≈ 84.971 cm²
Therefore, the area of the pentagon is approximately 84.971 square cm.