Math worksheet for calculating the area of semi-circles using radius measurements.

To solve the problem of finding the area of a semicircle, we need to use the formula for the area of a semicircle. The formula is:

\[
\text{Area of a semicircle} = \frac{1}{2} \pi r^2
\]

where \( r \) is the radius of the semicircle.

Let's go through each part of the task step by step.

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Part 1: Finding the Area of a Semicircle with Radius \( r \)

#### Step 1: Understand the formula
The area of a full circle is given by:
\[
\text{Area of a circle} = \pi r^2
\]
Since a semicircle is half of a full circle, its area is:
\[
\text{Area of a semicircle} = \frac{1}{2} \pi r^2
\]

#### Step 2: Apply the formula
For a semicircle with radius \( r \):
\[
\text{Area} = \frac{1}{2} \pi r^2
\]

#### Step 3: Match the answer
The correct answer is:
\[
\boxed{\frac{1}{2} \pi r^2}
\]

---

Part 2: Finding the Area of a Semicircle with Diameter \( d \)

#### Step 1: Relate diameter to radius
The diameter \( d \) is twice the radius \( r \):
\[
d = 2r \quad \Rightarrow \quad r = \frac{d}{2}
\]

#### Step 2: Substitute \( r \) in the formula
Using the formula for the area of a semicircle:
\[
\text{Area} = \frac{1}{2} \pi r^2
\]
Substitute \( r = \frac{d}{2} \):
\[
\text{Area} = \frac{1}{2} \pi \left( \frac{d}{2} \right)^2
\]

#### Step 3: Simplify the expression
\[
\left( \frac{d}{2} \right)^2 = \frac{d^2}{4}
\]
So:
\[
\text{Area} = \frac{1}{2} \pi \cdot \frac{d^2}{4} = \frac{\pi d^2}{8}
\]

#### Step 4: Match the answer
The correct answer is:
\[
\boxed{\frac{\pi d^2}{8}}
\]

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Part 3: Finding the Area of a Semicircle with Given Dimensions

#### Step 1: Identify the given dimensions
We are given specific dimensions for each semicircle. We will calculate the area for each case using the appropriate formula.

##### Case (i): Radius \( r = 7 \) cm
Using the formula:
\[
\text{Area} = \frac{1}{2} \pi r^2
\]
Substitute \( r = 7 \):
\[
\text{Area} = \frac{1}{2} \pi (7)^2 = \frac{1}{2} \pi (49) = \frac{49\pi}{2} \text{ cm}^2
\]

##### Case (ii): Diameter \( d = 28 \) cm
First, find the radius:
\[
r = \frac{d}{2} = \frac{28}{2} = 14 \text{ cm}
\]
Using the formula:
\[
\text{Area} = \frac{1}{2} \pi r^2
\]
Substitute \( r = 14 \):
\[
\text{Area} = \frac{1}{2} \pi (14)^2 = \frac{1}{2} \pi (196) = 98\pi \text{ cm}^2
\]

##### Case (iii): Radius \( r = 3.5 \) cm
Using the formula:
\[
\text{Area} = \frac{1}{2} \pi r^2
\]
Substitute \( r = 3.5 \):
\[
\text{Area} = \frac{1}{2} \pi (3.5)^2 = \frac{1}{2} \pi (12.25) = \frac{12.25\pi}{2} = 6.125\pi \text{ cm}^2
\]

##### Case (iv): Diameter \( d = 4 \) cm
First, find the radius:
\[
r = \frac{d}{2} = \frac{4}{2} = 2 \text{ cm}
\]
Using the formula:
\[
\text{Area} = \frac{1}{2} \pi r^2
\]
Substitute \( r = 2 \):
\[
\text{Area} = \frac{1}{2} \pi (2)^2 = \frac{1}{2} \pi (4) = 2\pi \text{ cm}^2
\]

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Final Answers

1. For radius \( r \):
\[
\boxed{\frac{1}{2} \pi r^2}
\]

2. For diameter \( d \):
\[
\boxed{\frac{\pi d^2}{8}}
\]

3. For specific cases:
- Case (i): \( \frac{49\pi}{2} \text{ cm}^2 \)
- Case (ii): \( 98\pi \text{ cm}^2 \)
- Case (iii): \( 6.125\pi \text{ cm}^2 \)
- Case (iv): \( 2\pi \text{ cm}^2 \)

Thus, the final boxed answers are:
\[
\boxed{\frac{1}{2} \pi r^2, \frac{\pi d^2}{8}, \frac{49\pi}{2}, 98\pi, 6.125\pi, 2\pi}
\]