Question

If the area of circle \(O\) is \(16 \pi,\) what is the length of an arc on the circle formed by a central angle measuring 45 degrees? $$\begin{array}{l} \pi \\ \frac{3 \pi}{2} \\ 2 \pi \\ \frac{5 \pi}{2} \\ 8 \pi \end{array}$$

Short answer

The length of the arc is \pi.

Step-by-step solution

- Find the Radius

Start by finding the radius of the circle. The area of a circle is given by the formula \(A = \pi r^2\). Given that the area \(A = 16 \pi\), we can set up the equation: \[\pi r^2 = 16 \pi \] Dividing both sides by \(\pi\), we get: \[r^2 = 16\] Now, take the square root of both sides: \[r = \sqrt{16} = 4\]

- Calculate the Circumference

The circumference of a circle can be found using the formula \(C = 2 \pi r\). Substituting the radius \(r = 4\), we get: \[C = 2 \pi \times 4 = 8 \pi\]

- Find the Fraction of the Circle

The central angle measures 45 degrees. To find the fraction of the circle that this angle represents, divide the angle by the total degrees in a circle. Since a full circle has 360 degrees: \[\frac{45}{360} = \frac{1}{8}\]

- Calculate the Arc Length

To find the length of the arc, multiply the circumference by the fraction of the circle. Using the circumference from Step 2 and the fraction from Step 3, we get: \[\text{Arc Length} = 8 \pi \times \frac{1}{8} = \pi\]

Key concepts

circle area calculation

To calculate the area of a circle, we use the formula: \( A = \pi r^2 \), where \( A \) is the area and \( r \) is the radius. In this problem, we know the area is \( 16 \pi \). To find the radius, we solve for \( r \) by setting up the equation: \[ \pi r^2 = 16 \pi \]
Divide both sides by \( \pi \): \[ r^2 = 16 \]
Now, take the square root of both sides: \[ r = \sqrt{16} = 4 \]
This shows that the radius of the circle is 4 units. Understanding how to manipulate the formula for the area helps us solve for the radius when the area is known.

• Area of Circle Formula: \( A = \pi r^2 \)
• Solve for radius: Isolate and square root \( r \)

radius determination

Determining the radius of a circle is crucial when solving various circle-related problems. From the previous section, we derived the radius as 4 units using the area formula. Knowing the radius, we can solve many more properties of the circle, like the circumference. Remember that the radius is the distance from the center of the circle to any point on its edge.

Key steps to determine the radius:

• Start with the area formula: \( A = \pi r^2 \)
• Plug in the given area and simplify
• Isolate \( r^2 \) and solve for \( r \)

This method is widely applicable and always reliable when given the area. 

circumference of a circle

The circumference of a circle refers to the perimeter or the distance around the circle. It can be calculated using the formula: \( C = 2 \pi r \). With our found radius of 4 units, we plug it into the formula: \[ C = 2 \pi \times 4 = 8 \pi \]
This gives us the entire distance around the circle. Knowing the circumference, we can easily calculate the length of an arc formed by any central angle.

Key points to remember:

• Formula: \( C = 2 \pi r \)
• Substitute the known radius into the formula
• Multiply and simplify to find the circumference

Calculating the circumference is essential before finding the arc length.

central angle and arc length

The arc length of a circle can be found using the circumference and the measure of the central angle. First, find the fraction of the circle that the central angle represents. In this problem, the central angle is 45 degrees. To get the fraction: \[ \frac{45}{360} = \frac{1}{8} \]

We next find the arc length by multiplying this fraction by the full circumference: \[ \text{Arc Length} = 8 \pi \times \frac{1}{8} = \pi \]

This tells us that the length of the arc corresponding to a central angle of 45 degrees is \( \pi \) units.

• Determine central angle fraction: Divide by 360
• Multiply this fraction by the circumference
• The product is the length of the arc

This approach is consistent and useful for various central angle measures.