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 $2\pi$.