Answer:
[tex]Area = 216\pi[/tex]
Step-by-step explanation:
Given
See attachment for semicircles
Required
The area of the paper
First, calculate the area of each petal.
The petal is made up of:
1. Semicircle of diameter 12cm
2. Semicircle of diameter 18cm (i.e. 12cm + 6cm)
3. A hole semicircle of diameter 6cm
The area of each semicircle is calculated as follows:
[tex]Area = \frac{\pi r^2}{2}[/tex]
For (1):
[tex]d = 12cm[/tex] --- diameter
[tex]r = 0.5 * d = 0.5* 12cm = 6cm[/tex] --- radius
So:
[tex]A_1 = \frac{\pi * 6^2}{2}[/tex]
Evaluate the exponent
[tex]A_1 = \frac{\pi * 36}{2}[/tex]
Divide 36 by 2
[tex]A_1 = 18\pi[/tex]
For (2):
[tex]d = 18cm[/tex]
[tex]r = 0.5* 18cm = 9cm[/tex]
So:
[tex]A_2 = \frac{\pi * 9^2}{2}[/tex]
[tex]A_2 = \frac{\pi * 81}{2}[/tex]
[tex]A_2 = 40.5\pi[/tex]
For (3):
[tex]d = 6cm[/tex]
[tex]r = 0.5* 6cm = 3cm[/tex]
So:
[tex]A_3 = \frac{\pi * 3^2}{2}[/tex]
[tex]A_3 = \frac{\pi * 9}{2}[/tex]
[tex]A_3 = 4.5\pi[/tex]
Recall that (3) is a hole.
So, the area (A) of a petal is:
[tex]A = A_1 + A_2 - A_3[/tex]
[tex]A = 18\pi + 40.5\pi - 4.5\pi[/tex]
[tex]A = 54\pi[/tex]
The paper uses 4 petals.
So:
[tex]Area = 4 * A[/tex]
[tex]Area = 4 * 54\pi[/tex]
[tex]Area = 216\pi[/tex]
