Question

c the diagram shows a semicircle. show that
i the area of the semicircle is $72\pi$ m$^2$
ii the perimeter of the semicircle is $12\pi + 24$ m.

Explanation:

Part (i)

Step 1: Recall the formula for the area of a full circle

The area of a full circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.

Step 2: Derive the area of a semicircle

A semicircle is half of a full circle. So, to find the area of a semicircle, we take half of the area of a full circle.
The diameter of the semicircle is given as \( 24 \, \text{m} \), so the radius \( r \) of the semicircle is \( \frac{24}{2}=12 \, \text{m} \)? Wait, no, wait. Wait, maybe there is a mis - reading. Wait, the problem says "show that the area of the semicircle is \( 72\pi \, \text{m}^2 \)". Let's do it properly.

Wait, the formula for the area of a semicircle is \( A_{semicircle}=\frac{1}{2}\pi r^2 \). Wait, if the diameter \( d = 24 \, \text{m} \), then \( r=\frac{d}{2}=12 \, \text{m} \)? No, wait, if we substitute \( r = 12 \) into \( \frac{1}{2}\pi r^2 \), we get \( \frac{1}{2}\pi(12)^2=\frac{1}{2}\pi\times144 = 72\pi \, \text{m}^2 \). Ah, that's it.

So step by step:

Step 1: Identify the radius (from diameter)

The diameter of the semicircle is \( 24 \, \text{m} \). The radius \( r \) of a circle (and thus the semicircle) is related to the diameter \( d \) by \( r=\frac{d}{2} \). So \( r=\frac{24}{2}=12 \, \text{m} \).

Step 2: Recall the formula for the area of a semicircle

The area of a semicircle \( A_{semicircle}=\frac{1}{2}\times \) (area of full circle). The area of a full circle is \( \pi r^2 \), so the area of a semicircle is \( A_{semicircle}=\frac{1}{2}\pi r^2 \).

Step 3: Substitute the value of \( r \)

Substitute \( r = 12 \, \text{m} \) into the formula for the area of the semicircle:

\( A_{semicircle}=\frac{1}{2}\pi(12)^2 \)

\(=\frac{1}{2}\pi\times144 \)

\( = 72\pi \, \text{m}^2 \)

Part (ii)

Step 1: Recall the formula for the length of a semicircular arc

The length of the arc of a semicircle (the curved part) is half of the circumference of a full circle. The circumference of a full circle is \( C = 2\pi r \), so the length of the semicircular arc \( l=\frac{1}{2}\times2\pi r=\pi r \). Wait, if \( r = 12 \, \text{m} \), the length of the arc is \( 12\pi \, \text{m} \).

Step 2: Identify the straight part of the perimeter

The perimeter of a semicircle consists of two parts: the length of the semicircular arc and the length of the diameter (the straight edge). The diameter is \( 24 \, \text{m} \).

Step 3: Calculate the perimeter of the semicircle

The perimeter \( P \) of the semicircle is the sum of the length of the semicircular arc and the length of the diameter.

The length of the semicircular arc: using \( l=\pi r \), and since \( r = 12 \, \text{m} \), \( l = 12\pi \, \text{m} \).

The length of the diameter \( d=24 \, \text{m} \).

So the perimeter \( P=12\pi + 24 \, \text{m} \). Wait, but the problem says "show that the perimeter of the semicircle is \( 12\pi+24 \, \text{m} \)". Wait, if we consider the general case (maybe there was a mis - interpretation earlier). Wait, maybe the radius is \( r \), and the diameter is \( 24 \, \text{m} \), so \( 2r = 24 \)? No, that would make \( r = 12 \). Wait, let's do it with radius \( r \).

Wait, the formula for the length of the semicircular arc is \( \frac{1}{2}\times2\pi r=\pi r \). Wait, no, \( C = 2\pi r \) (circumference of full circle), so semicircular arc length is \( \pi r \). Wait, but if the diameter is \( 24 \, \text{m} \), then \( 2r=24\Rightarrow r = 12 \). Then the arc length is \( \pi\times12 = 12\pi \), and the straight side is \( 24 \, \text{m} \), so perimeter is \( 12\pi+24 \, \text{m} \).

Let's do it properly:

Step 1: Length of the semicircular arc

The formula for the circumference of a circle is \( C = 2\pi r \). A semicircular arc is half of the circumference of the circle. So the length of the semicircular arc \( l=\frac{1}{2}\times2\pi r=\pi r \). Wait, no, \( \frac{1}{2}\times2\pi r=\pi r \). Wait, if the diameter is \( 24 \, \text{m} \), then \( r = 12 \, \text{m} \), so the arc length is \( \pi\times12=12\pi \, \text{m} \).

Step 2: Length of the straight edge

The straight edge of the semicircle is the diameter, which is given as \( 24 \, \text{m} \).

Step 3: Calculate the perimeter

The perimeter of the semicircle \( P \) is the sum of the length of the semicircular arc and the length of the diameter. So \( P = 12\pi+24 \, \text{m} \).

Final Answers

(i) The area of the semicircle is shown to be \( 72\pi \, \text{m}^2 \) as derived above.

(ii) The perimeter of the semicircle is shown to be \( 12\pi + 24 \, \text{m} \) as derived above.

For part (i) answer: The area of the semicircle is \( \boldsymbol{72\pi \, \text{m}^2} \)

For part (ii) answer: The perimeter of the semicircle is \( \boldsymbol{12\pi + 24 \, \text{m}} \)

Answer:

Step 1: Recall the formula for the length of a semicircular arc

The length of the arc of a semicircle (the curved part) is half of the circumference of a full circle. The circumference of a full circle is \( C = 2\pi r \), so the length of the semicircular arc \( l=\frac{1}{2}\times2\pi r=\pi r \). Wait, if \( r = 12 \, \text{m} \), the length of the arc is \( 12\pi \, \text{m} \).

Step 2: Identify the straight part of the perimeter

The perimeter of a semicircle consists of two parts: the length of the semicircular arc and the length of the diameter (the straight edge). The diameter is \( 24 \, \text{m} \).

Step 3: Calculate the perimeter of the semicircle

The perimeter \( P \) of the semicircle is the sum of the length of the semicircular arc and the length of the diameter.

The length of the semicircular arc: using \( l=\pi r \), and since \( r = 12 \, \text{m} \), \( l = 12\pi \, \text{m} \).

The length of the diameter \( d=24 \, \text{m} \).

So the perimeter \( P=12\pi + 24 \, \text{m} \). Wait, but the problem says "show that the perimeter of the semicircle is \( 12\pi+24 \, \text{m} \)". Wait, if we consider the general case (maybe there was a mis - interpretation earlier). Wait, maybe the radius is \( r \), and the diameter is \( 24 \, \text{m} \), so \( 2r = 24 \)? No, that would make \( r = 12 \). Wait, let's do it with radius \( r \).

Wait, the formula for the length of the semicircular arc is \( \frac{1}{2}\times2\pi r=\pi r \). Wait, no, \( C = 2\pi r \) (circumference of full circle), so semicircular arc length is \( \pi r \). Wait, but if the diameter is \( 24 \, \text{m} \), then \( 2r=24\Rightarrow r = 12 \). Then the arc length is \( \pi\times12 = 12\pi \), and the straight side is \( 24 \, \text{m} \), so perimeter is \( 12\pi+24 \, \text{m} \).

Let's do it properly:

Step 1: Length of the semicircular arc

The formula for the circumference of a circle is \( C = 2\pi r \). A semicircular arc is half of the circumference of the circle. So the length of the semicircular arc \( l=\frac{1}{2}\times2\pi r=\pi r \). Wait, no, \( \frac{1}{2}\times2\pi r=\pi r \). Wait, if the diameter is \( 24 \, \text{m} \), then \( r = 12 \, \text{m} \), so the arc length is \( \pi\times12=12\pi \, \text{m} \).

Step 2: Length of the straight edge

The straight edge of the semicircle is the diameter, which is given as \( 24 \, \text{m} \).

Step 3: Calculate the perimeter

The perimeter of the semicircle \( P \) is the sum of the length of the semicircular arc and the length of the diameter. So \( P = 12\pi+24 \, \text{m} \).

Final Answers

(i) The area of the semicircle is shown to be \( 72\pi \, \text{m}^2 \) as derived above.

(ii) The perimeter of the semicircle is shown to be \( 12\pi + 24 \, \text{m} \) as derived above.

For part (i) answer: The area of the semicircle is \( \boldsymbol{72\pi \, \text{m}^2} \)

For part (ii) answer: The perimeter of the semicircle is \( \boldsymbol{12\pi + 24 \, \text{m}} \)