https://file.aiquickdraw.com/mathbot/file/1726595882786wsu0pjf1.jpg

To calculate the perimeter of the quarter circle, we need to consider two components:

1. The curved part of the quarter circle, which is a quarter of the circumference of the full circle.
2. The straight parts, which are the radius lines \( AB \) and \( AC \).

### Given:
- Radius \( r = 21 \) cm.

### Step 1: Calculate the curved part (arc length)
The formula for the circumference of a full circle is:
\[
C = 2\pi r
\]
The arc length for a quarter circle is:
\[
L_{\text{arc}} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2}
\]
Substitute \( r = 21 \) cm:
\[
L_{\text{arc}} = \frac{\pi \times 21}{2} = 10.5\pi \text{ cm}
\]

### Step 2: Calculate the total perimeter
The total perimeter of the quarter circle is the sum of the arc length and the two straight radius parts:
\[
P = L_{\text{arc}} + 2r
\]
Substitute \( L_{\text{arc}} = 10.5\pi \) and \( r = 21 \):
\[
P = 10.5\pi + 2 \times 21 = 10.5\pi + 42
\]

Now, let's calculate the final perimeter rounded to two decimal places.The perimeter of the quarter circle is approximately \( 74.99 \) cm, rounded to two decimal places.

Would you like more details on any step? Here are some related questions to further explore:

1. What is the formula for the area of a quarter circle?
2. How would the perimeter change if the radius were doubled?
3. Can you explain how to find the arc length for different fractions of a circle?
4. How does the perimeter of a semi-circle compare to that of a quarter circle?
5. What are some real-world applications of calculating perimeters of quarter circles?

**Tip:** Remember, the perimeter of a quarter circle includes both the arc and the two radii, so always account for the straight sides as well.

Calculate the perimeter of the quarter circle shown above. Leave your answer to 2 decimal places.

Learn how to calculate the perimeter of a quarter circle with a radius of 21 cm. The solution covers key geometry concepts such as arc length and perimeter formulas. Includes step-by-step calculations and explanations suitable for students in Grades 7-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation