Find the area of a regular pentagon with a side 8 centimeters.
Please provide an explanation!
center angle = 360/8 = 45
half center angle = 22.5
tan half center angle = 4/altitude
so altitude = 4/.414 = 9.657
area of half of section = (1/2)(4)(9.657)
= 19.31
times 16 = 309
by the way if you want to do this exactly tan 22.5 = sqrt 2 - 1
because
tan (45/2) = [ 1- cos(45) ] / sin (45)
and cos (45) = sin (45) = 1/sqrt 2
To find the area of a regular pentagon, you can use the formula:
Area = (1/4) * √(5(5+2√5)) * s^2
where 's' represents the length of one side of the pentagon.
In this case, the given length of one side of the pentagon is 8 centimeters. So we can substitute this value into the formula:
Area = (1/4) * √(5(5+2√5)) * (8)^2
Now let's solve this equation step by step:
Step 1: Calculate the expression inside the square root:
5(5+2√5) = 25 + 10√5
Step 2: Multiply the result from step 1 by 5:
5 * (25 + 10√5) = 125 + 50√5
Step 3: Square root the result from step 2:
√(125 + 50√5)
Step 4: Multiply the result from step 3 by (1/4):
(1/4) * √(125 + 50√5)
Step 5: Multiply the square of the side length (8^2 = 64) by the result from step 4:
(1/4) * √(125 + 50√5) * 64
After performing these calculations, you will find the area of the regular pentagon with a side length of 8 centimeters.