The area of a sector of a circle is 30.8 square centimetres and the corresponding central angle is 72 degrees. What is the radius of the circle in centimetres?

Introduction / Context:
This problem connects the area of a sector of a circle with its central angle and radius. Given the area of the sector and the central angle in degrees, we are asked to find the radius of the circle. It tests the ability to apply the sector area formula and manipulate it to solve for the radius.

Given Data / Assumptions:

• Area of the sector A = 30.8 square centimetres.
• Central angle θ = 72 degrees.
• Radius of the circle is r cm.
• Use pi approximately equal to 22 / 7.

Concept / Approach:
The area of a sector of a circle when the central angle is given in degrees is A = (θ / 360) * pi * r^2. With A and θ known, we can rearrange this formula to solve for r^2 and then take the square root to obtain r. Careful substitution and algebra will yield a nice value when using the given approximation for pi.

Step-by-Step Solution:
Step 1: Write the sector area formula: A = (θ / 360) * pi * r^2.Step 2: Substitute A = 30.8 and θ = 72 degrees: 30.8 = (72 / 360) * pi * r^2.Step 3: Simplify the fraction 72 / 360 = 1 / 5.Step 4: The equation becomes 30.8 = (1 / 5) * pi * r^2.Step 5: Multiply both sides by 5: 154 = pi * r^2.Step 6: Use pi = 22 / 7 to get 154 = (22 / 7) * r^2.Step 7: Multiply both sides by 7 / 22: r^2 = 154 * 7 / 22.Step 8: Compute 154 / 22 = 7, so r^2 = 7 * 7 = 49.Step 9: Take the square root: r = sqrt(49) = 7 cm.

Verification / Alternative check:
Substitute r = 7 cm back into the formula to verify the area.Compute A = (72 / 360) * pi * 7^2 = (1 / 5) * pi * 49.Using pi = 22 / 7, A = (1 / 5) * (22 / 7) * 49 = (1 / 5) * 22 * 7 = 154 / 5 = 30.8 square centimetres.This matches the given sector area, confirming the calculation.

Why Other Options Are Wrong:
If r were 3.5 cm, then r^2 would be 12.25 and the sector area would be far smaller than 30.8 square centimetres.If r were 14 cm, r^2 would be 196 and the area would be much larger than 30.8 square centimetres.Values like 10.5 cm or 21 cm also produce sector areas that do not match 30.8 when substituted into the formula.Only r = 7 cm satisfies the equation.

Common Pitfalls:
A common error is to forget to convert the central angle from degrees to a fraction of a full circle by dividing by 360.Another mistake is to use the full circle area formula pi * r^2 directly without applying the fraction θ / 360.Misplacing the factor of 1 / 5 when rearranging the equation can also lead to incorrect values of r^2.

Final Answer:
The radius of the circle is 7 cm.