Understanding the Problem: Finding Circle Radius from Sector Area

The question asks us to find the radius of a circle given the area of a sector and its central angle. We are provided with the area of the sector, the central angle, and the value of \(\pi\) to use for the calculation. This is a common problem in geometry that involves applying the formula for the area of a sector.

Formula for Area of a Sector

The area of a sector of a circle is a fraction of the total area of the circle. The fraction is determined by the ratio of the central angle of the sector to the total angle in a circle (360°). The formula for the area of a sector with a central angle \(\theta\) (in degrees) and radius \(r\) is:

\(\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2\)

Here, \(\theta\) is the central angle in degrees, \(r\) is the radius of the circle, and \(\pi\) is a mathematical constant approximately equal to 3.14159.

Given Information

From the question, we have the following information:

• Area of the sector = 154 cm²
• Central angle of the sector (\(\theta\)) = 45°
• Value of \(\pi\) to use = 22/7

We need to find the radius, \(r\), of the circle.

Step-by-Step Calculation

We will use the formula for the area of a sector and plug in the given values to solve for the radius, \(r\).

1. Write down the formula:

\(\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2\)

2. Substitute the given values into the formula:

\(154 = \frac{45^\circ}{360^\circ} \times \frac{22}{7} \times r^2\)

3. Simplify the fraction involving the angle:

\(\frac{45^\circ}{360^\circ} = \frac{45}{360}\)

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor. 45 goes into 360 exactly 8 times (45 * 8 = 360).

\(\frac{45}{360} = \frac{1}{8}\)

Now substitute this back into the equation:

\(154 = \frac{1}{8} \times \frac{22}{7} \times r^2\)

4. Multiply the fractions on the right side:

\(\frac{1}{8} \times \frac{22}{7} = \frac{1 \times 22}{8 \times 7} = \frac{22}{56}\)

The equation becomes:

\(154 = \frac{22}{56} \times r^2\)

5. Isolate \(r^2\) by multiplying both sides by the reciprocal of 22/56 (which is 56/22):

\(r^2 = 154 \times \frac{56}{22}\)

6. Simplify the expression. We can divide 154 by 22. \(154 \div 22 = 7\).

\(r^2 = 7 \times 56\)

7. Calculate the product \(7 \times 56\):

\(7 \times 56 = 7 \times (7 \times 8) = 49 \times 8 = 392\)

So, \(r^2 = 392\).

Alternatively, let's keep the factors to simplify finding the square root:

\(r^2 = 154 \times \frac{56}{22}\)

Divide 154 by 22:

\(r^2 = (154 \div 22) \times 56 = 7 \times 56\)

Factorize 56:

\(r^2 = 7 \times (8 \times 7) = 7^2 \times 8\)

Factorize 8:

\(r^2 = 7^2 \times (4 \times 2) = 7^2 \times 2^2 \times 2\)

\(r^2 = (7 \times 2)^2 \times 2 = 14^2 \times 2\)

8. Take the square root of both sides to find \(r\):

\(r = \sqrt{14^2 \times 2}\)

Using the property \(\sqrt{ab} = \sqrt{a}\sqrt{b}\):

\(r = \sqrt{14^2} \times \sqrt{2}\)

\(r = 14 \times \sqrt{2}\)

\(r = 14\sqrt{2}\)

The radius of the circle is \(14\sqrt{2}\) cm.

Final Answer

The radius of the circle is \(14\sqrt{2}\) cm.

Revision Table: Geometry Formulas

Additional Information: Understanding Sectors and Angles

A sector of a circle is like a slice of pizza or cake. It is bounded by two radii and the arc connecting their endpoints. The central angle is the angle formed by the two radii at the center of the circle.

Angles can be measured in degrees or radians. The formula used in this problem requires the angle in degrees (\(\theta\)). If the angle were given in radians (\(\alpha\)), the formula for the area of a sector would be:

\(\text{Area of Sector} = \frac{1}{2} r^2 \alpha\)

where \(\alpha\) is in radians.

To convert degrees to radians, you can use the relationship \(180^\circ = \pi\) radians. So, \(1^\circ = \pi/180\) radians and \(1 \text{ radian} = 180^\circ/\pi\). In our problem, 45° is equivalent to \(45 \times \frac{\pi}{180} = \frac{\pi}{4}\) radians.

Using the radian formula with \(r = 14\sqrt{2}\) and \(\alpha = \pi/4\):

\(\text{Area} = \frac{1}{2} (14\sqrt{2})^2 \times \frac{\pi}{4}\)

\(\text{Area} = \frac{1}{2} (14^2 \times (\sqrt{2})^2) \times \frac{\pi}{4}\)

\(\text{Area} = \frac{1}{2} (196 \times 2) \times \frac{\pi}{4}\)

\(\text{Area} = \frac{1}{2} \times 392 \times \frac{\pi}{4}\)

\(\text{Area} = 196 \times \frac{\pi}{4}\)

\(\text{Area} = 49\pi\)

If we use \(\pi = 22/7\):

\(\text{Area} = 49 \times \frac{22}{7} = 7 \times 22 = 154\)

This confirms our radius calculation is correct, as it yields the given sector area.