find the area bounded by a regular pentagon with radius 6 cm. round to the nearest tenth.
To find the area bounded by a regular pentagon, we can use trigonometry.
First, let's find the apothem (distance from the center of the pentagon to the midpoint of its side). In a regular pentagon, the apothem is the perpendicular distance from the center of the pentagon to one of its sides.
Given that the radius of the pentagon is 6 cm, we can drop a perpendicular from the center of the pentagon to one of its sides. This forms a right triangle, with the apothem as the hypotenuse and half the length of one side as the adjacent side.
Since the regular pentagon has five sides, each side is the same length as the radius (6 cm). Hence, the length of half a side is 6/2 = 3 cm.
Using Pythagoras' theorem, we can find the apothem (a):
a^2 = r^2 - (s/2)^2,
where r is the radius and s is the length of half a side.
Substituting the known values into the equation, we get:
a^2 = 6^2 - (3)^2
a^2 = 36 - 9
a^2 = 27
Taking the square root of both sides, we find:
a ≈ √27
To find the area of the regular pentagon, we can use the formula:
Area = (a * s * 5)/2,
where a is the apothem and s is the length of one side.
Substituting the known values into the equation, we get:
Area ≈ (√27 * 6 * 5)/2
Evaluating this expression, we find:
Area ≈ (5.196 * 6 * 5)/2
Area ≈ (30.976)/2
Area ≈ 15.488
Rounding to the nearest tenth, the area bounded by the regular pentagon with a radius of 6 cm is approximately 15.5 square centimeters.