What is the size of each interior angle of a 12-sided regular polygon?

1. Analyze the question

You are asked to find the size of each interior angle of a regular polygon with 12 sides (a dodecagon).

2. Explain the concept

To solve this, we use the properties of regular polygons:

• Sum of interior angles: The sum of all interior angles of a polygon with $n$ sides is given by the formula: $(n - 2) \times 180^\circ$.
• Size of one interior angle: Since it is a regular polygon (all angles are equal), we divide the sum by the number of sides ($n$): $$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$$
• Alternative method: You can also find the exterior angle first using the formula $\frac{360^\circ}{n}$, then subtract that from $180^\circ$ (since interior and exterior angles lie on a straight line and sum to $180^\circ$).

3. Show the solution

Using the first method where $n = 12$:

• Step 1: Calculate the sum of interior angles. $$\text{Sum} = (12 - 2) \times 180^\circ$$ $$\text{Sum} = 10 \times 180^\circ = 1800^\circ$$

• Step 2: Divide by the number of sides to get the size of one angle. $$\text{Interior Angle} = \frac{1800^\circ}{12}$$ $$\text{Interior Angle} = 150^\circ$$

• Alternative method check: $$\text{Exterior Angle} = \frac{360^\circ}{12} = 30^\circ$$ $$\text{Interior Angle} = 180^\circ - 30^\circ = 150^\circ$$

Both methods confirm the result.

4. Identify the answer

The calculated size of each interior angle is $150^\circ$, which corresponds to option B.

5. Tip

When dealing with regular polygons in JAMB, it is often faster to calculate the exterior angle first ($\frac{360}{n}$) and subtract it from $180^\circ$. It involves smaller numbers and reduces the risk of calculation errors compared to multiplying by $(n-2)$ first!