Question

What is the area of a semicircle with a 15 foot diameter?

Answer 1

Hint: To find the area of a semicircle, we have to use the formula $A=\dfrac{\pi {{r}^{2}}}{2}$ , where r is the radius of the semi-circle. From the given diameter, we have to find r by dividing the diameter by 2. Then, we have to substitute the value of r in the formula for the area of the semi-circle.

Complete step by step solution:
We have to find the area of a semicircle. We are given with the diameter of the semi-circle, that is,
$d=15\text{ ft}$

We know that the radius of a semicircle is half of its diameter. Mathematically, we can denote this as
$r=\dfrac{d}{2}$
Let us substitute the value of d in the above formula.
$\Rightarrow r=\dfrac{15}{2}=7.5\text{ ft}$
We know that area of a semicircle is given by
$A=\dfrac{\pi {{r}^{2}}}{2}$
Let us substitute the value of r in the above formula.
$\Rightarrow A=\dfrac{\pi {{\left( 7.5 \right)}^{2}}}{2}$
We know that $\pi =3.14$ . Therefore, the above equation becomes
$\Rightarrow A=\dfrac{3.14\times {{\left( 7.5 \right)}^{2}}}{2}$
Let us cancel the common factor 2.
$\Rightarrow A=\dfrac{{{\require{cancel}\cancel{3.14}}^{1.57}}\times {{\left( 7.5 \right)}^{2}}}{{{\require{cancel}\cancel{2}}^{1}}}$
We can write the result of above simplification as
$\Rightarrow A=1.57\times {{\left( 7.5 \right)}^{2}}$
Let us write the value of the square of 7.5.
$\Rightarrow A=1.57\times 56.25$
We have to perform the multiplication operation.
$\Rightarrow A=88.312\text{ f}{{\text{t}}^{2}}$
Hence, the area of the given semi-circle is 88.312 sq.ft.

Note: Students must be very thorough with the formulas of semi-circle. They should not get confused with the formulas of area and circumference of the semicircle. We can see that the area of the semicircle is half of the area of the circle. The circumference of a semicircle is given by $C=r\left( \pi +2 \right)$.