Find the area of a regular pentagon with radius 5in.
To find the area of a regular pentagon with a given radius, you can use the formula for the area of a regular polygon:
Area = (1/2) * Perimeter * Apothem
The perimeter of a regular pentagon is simply the sum of the lengths of all its sides. For a regular pentagon, all sides are equal in length, so the perimeter can be calculated by multiplying the length of one side by the number of sides.
Let's begin by finding the length of one side of the regular pentagon. In a regular pentagon, each interior angle is 108 degrees. By drawing a line that divides the pentagon into two congruent triangles from the center to one of the vertices, we create a right triangle with one angle measuring 54 degrees. Since we know the radius (5 inches), the opposite side of this right triangle is the apothem (distance from the center to the middle of one side).
Using the trigonometric ratio tangent (tan), we can set up the equation:
tan(54 degrees) = opposite side / adjacent side
tan(54 degrees) = apothem / radius
Since we know the radius (5 inches), we can calculate the apothem by rearranging the equation:
apothem = radius * tan(54 degrees)
apothem = 5 inches * tan(54 degrees)
Now that we have the apothem, we can calculate the perimeter by multiplying the length of one side by the number of sides:
perimeter = length of one side * 5
Finally, we can use the formula to find the area of the regular pentagon:
Area = (1/2) * perimeter * apothem
= (1/2) * perimeter * (5 inches * tan(54 degrees))
= (1/2) * (length of one side * 5) * (5 inches * tan(54 degrees))
By plugging in the calculated values, we can determine the area of the regular pentagon.