How do I find the area and perimeter of 75% of a circle when the radius is 4?

To find the area and perimeter of 75% of a circle with a radius of 4, we can use the following steps:

### 1. **Area of 75% of the Circle:**

The formula for the area of a circle is:

\[
A = \pi r^2
\]

For a full circle with radius \( r = 4 \):

\[
A_{\text{circle}} = \pi (4)^2 = 16\pi
\]

Now, since you're looking for 75% of the area:

\[
A_{\text{75%}} = 0.75 \times 16\pi = 12\pi \approx 37.6991 \, \text{square units}
\]

### 2. **Perimeter of 75% of the Circle:**

The perimeter consists of two parts:
1. The arc length of 75% of the circle's circumference.
2. The two radii that connect the ends of the arc to the center.

#### (i) **Arc Length:**

The full circumference of a circle is:

\[
C = 2\pi r = 2\pi \times 4 = 8\pi
\]

Now, for 75% of the circle, the arc length is:

\[
L_{\text{arc}} = 0.75 \times 8\pi = 6\pi \approx 18.8496 \, \text{units}
\]

#### (ii) **Radii:**

You also need to account for the two radii that connect the arc to the center. Each radius is 4 units long, so the total contribution from the two radii is:

\[
\text{Radii Length} = 2 \times 4 = 8 \, \text{units}
\]

#### (iii) **Total Perimeter:**

The total perimeter of 75% of the circle is the sum of the arc length and the two radii:

\[
P_{\text{75%}} = L_{\text{arc}} + 2 \times r = 6\pi + 8 \approx 18.8496 + 8 = 26.8496 \, \text{units}
\]

### Final Answers:
- **Area of 75% of the circle**: \( 12\pi \approx 37.6991 \, \text{square units} \)
- **Perimeter of 75% of the circle**: \( 26.8496 \, \text{units} \)

Would you like further details or have any questions?

### 5 Related Questions:
1. How do you calculate the area of a sector with any given angle?
2. What is the difference between an arc and a chord in a circle?
3. How does the formula for the circumference of a circle change with different percentages?
4. How can you find the area of a circle if given only the diameter?
5. How do radians relate to calculating arc lengths?

**Tip:** Remember that the arc length formula can also be derived from the fraction of the circle's angle (in radians) over \(2\pi\).

Find out how to calculate the area and perimeter of 75% of a circle with a radius of 4. This solution includes detailed steps, key formulas for area, arc length, and perimeter, and is perfect for students in Grades 9-11.

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