Calculating Circle Area from Arc Length and Angle

This problem requires us to find the area of a circle when we are given the length of an arc and the angle it subtends at the centre. We are provided with the arc length, the central angle, and the value of \(\pi\).

Here's how we can solve it:

First, we need to find the radius of the circle using the given arc length and central angle. The formula for the length of an arc is:

\[ \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r \]where:

• \(\theta\) is the central angle in degrees.
• \(r\) is the radius of the circle.

We are given:

• Arc Length = 23.1 cm
• \(\theta = 18^\circ\)
• \(\pi = \frac{22}{7}\)

Substitute these values into the arc length formula:

\[ 23.1 = \frac{18}{360} \times 2 \times \frac{22}{7} \times r \]

Simplify the fraction \(\frac{18}{360}\):

\[ \frac{18}{360} = \frac{18 \div 18}{360 \div 18} = \frac{1}{20} \]

Substitute this back into the equation:

\[ 23.1 = \frac{1}{20} \times 2 \times \frac{22}{7} \times r \]

Simplify the multiplication on the right side:

\[ 23.1 = \frac{2}{20} \times \frac{22}{7} \times r \]

\[ 23.1 = \frac{1}{10} \times \frac{22}{7} \times r \]

\[ 23.1 = \frac{22}{70} \times r \]

Now, solve for \(r\):

\[ r = \frac{23.1 \times 70}{22} \]

To make calculation easier, convert 23.1 to a fraction \(\frac{231}{10}\):

\[ r = \frac{231}{10} \times \frac{70}{22} \]

Cancel out common factors:

\[ r = \frac{231}{\cancel{10}} \times \frac{\cancel{70}^7}{22} \]

\[ r = \frac{231 \times 7}{22} \]

Now, divide 231 and 22 by 11:

\[ r = \frac{\cancel{231}^{21} \times 7}{\cancel{22}^2} \]

\[ r = \frac{21 \times 7}{2} = \frac{147}{2} = 73.5 \text{ cm} \]

So, the radius of the circle is 73.5 cm.

Next, we need to find the area of the circle using the formula:

\[ \text{Area} = \pi r^2 \]

Substitute the values of \(\pi\) and \(r\):

\[ \text{Area} = \frac{22}{7} \times (73.5)^2 \]

Substitute \(73.5 = \frac{147}{2}\):

\[ \text{Area} = \frac{22}{7} \times \left(\frac{147}{2}\right)^2 \]

\[ \text{Area} = \frac{22}{7} \times \frac{147 \times 147}{2 \times 2} \]

\[ \text{Area} = \frac{22}{7} \times \frac{147 \times 147}{4} \]

Cancel out common factors. Divide 147 by 7:

\[ \text{Area} = \frac{22}{\cancel{7}^1} \times \frac{\cancel{147}^{21} \times 147}{4} \]

\[ \text{Area} = \frac{22 \times 21 \times 147}{4} \]

Divide 22 and 4 by 2:

\[ \text{Area} = \frac{\cancel{22}^{11} \times 21 \times 147}{\cancel{4}^2} \]

\[ \text{Area} = \frac{11 \times 21 \times 147}{2} \]

Multiply the numbers in the numerator:

\[ 11 \times 21 = 231 \]

\[ 231 \times 147 = 33957 \]

So, the area is:

\[ \text{Area} = \frac{33957}{2} = 16978.5 \text{ cm}^2 \]

The area of the circle is 16978.50 cm².

Revision Table: Circle Geometry Formulas

Additional Information on Circle Properties

Circles are fundamental shapes in geometry with many interesting properties. Understanding the relationship between the radius, diameter, circumference, area, arc length, and sector area is crucial for solving problems.

• The radius (\(r\)) is the distance from the center to any point on the circle.
• The diameter (\(d\)) is the distance across the circle through the center (\(d=2r\)).
• \(\pi\) (pi) is a mathematical constant approximately equal to 3.14159 or \(\frac{22}{7}\), representing the ratio of a circle's circumference to its diameter.
• A central angle is an angle whose vertex is the center of the circle and whose sides are radii.
• The arc subtended by a central angle is the portion of the circle's circumference between the two radii.
• A sector is the region of a circle bounded by two radii and their intercepted arc. Its area is a fraction of the total circle area, proportional to the central angle.

Remember that angles in formulas are typically given in degrees or radians. The formula used here requires the angle in degrees. If an angle is given in radians, you would convert it (\(180^\circ = \pi \text{ radians}\)) or use the formula for arc length with radians: \(L = r\theta\) (where \(\theta\) is in radians).