Areas of Sectors - Starting Points Maths - Free Printable

Explanation:
We are given 15 sector diagrams. In each, the radius is 5 cm, and we need to find the area of the shaded sector.

The formula for the area of a sector is:

\[
\text{Sector Area} = \frac{\theta}{360^\circ} \times \pi r^2
\]

Where:
- $\theta$ = central angle in degrees,
- $r = 5$ cm,
- So $r^2 = 25$,
- Therefore, sector area = $\frac{\theta}{360} \times 25\pi = \frac{25\pi\theta}{360} = \frac{5\pi\theta}{72}$ cm².

Let’s compute each one carefully.

---

1. Full circle → $\theta = 360^\circ$
Area = $\frac{360}{360} \times \pi \cdot 25 = 25\pi$ cm²

2. Semicircle → $\theta = 180^\circ$
Area = $\frac{180}{360} \cdot 25\pi = \frac{1}{2} \cdot 25\pi = 12.5\pi$ cm²

3. Quarter circle (right angle) → $\theta = 90^\circ$
Area = $\frac{90}{360} \cdot 25\pi = \frac{1}{4} \cdot 25\pi = 6.25\pi$ cm²

4. Three-quarters circle → $\theta = 270^\circ$
Area = $\frac{270}{360} \cdot 25\pi = \frac{3}{4} \cdot 25\pi = 18.75\pi$ cm²

5. $\theta = 45^\circ$
Area = $\frac{45}{360} \cdot 25\pi = \frac{1}{8} \cdot 25\pi = 3.125\pi$ cm²

6. $\theta = 30^\circ$
Area = $\frac{30}{360} \cdot 25\pi = \frac{1}{12} \cdot 25\pi = \frac{25\pi}{12} \approx 2.083\pi$ cm²
But keep exact: $\frac{25\pi}{12}$ cm²

7. $\theta = 60^\circ$
Area = $\frac{60}{360} \cdot 25\pi = \frac{1}{6} \cdot 25\pi = \frac{25\pi}{6}$ cm²

8. $\theta = 120^\circ$
Area = $\frac{120}{360} \cdot 25\pi = \frac{1}{3} \cdot 25\pi = \frac{25\pi}{3}$ cm²

9. $\theta = 240^\circ$
Area = $\frac{240}{360} \cdot 25\pi = \frac{2}{3} \cdot 25\pi = \frac{50\pi}{3}$ cm²

10. $\theta = 1^\circ$
Area = $\frac{1}{360} \cdot 25\pi = \frac{25\pi}{360} = \frac{5\pi}{72}$ cm²

11. $\theta = 8^\circ$
Area = $\frac{8}{360} \cdot 25\pi = \frac{1}{45} \cdot 25\pi = \frac{25\pi}{45} = \frac{5\pi}{9}$ cm²

12. $\theta = 71^\circ$
Area = $\frac{71}{360} \cdot 25\pi = \frac{1775\pi}{360} = \frac{355\pi}{72}$ cm²
(Simplify: divide numerator & denominator by 5 → 1775 ÷ 5 = 355, 360 ÷ 5 = 72)

13. $\theta = 235^\circ$
Area = $\frac{235}{360} \cdot 25\pi = \frac{5875\pi}{360} = \frac{1175\pi}{72}$ cm²
(Simplify: divide numerator & denominator by 5 → 5875 ÷ 5 = 1175, 360 ÷ 5 = 72)

14. $\theta = 321^\circ$
Area = $\frac{321}{360} \cdot 25\pi = \frac{8025\pi}{360} = \frac{1605\pi}{72} = \frac{535\pi}{24}$ cm²
Check simplification:
8025 ÷ 15 = 535, 360 ÷ 15 = 24 → yes, $\frac{535\pi}{24}$

15. $\theta = x^\circ$
Area = $\frac{x}{360} \cdot 25\pi = \frac{25\pi x}{360} = \frac{5\pi x}{72}$ cm²

Now verify a few calculations for errors:

- #5: 45/360 = 1/8 → 25π/8 = 3.125π ✔️
- #6: 30/360 = 1/12 → 25π/12 ✔️
- #10: 1/360 × 25π = 25π/360 = 5π/72 ✔️
- #11: 8/360 = 1/45 → 25π/45 = 5π/9 ✔️
- #12: 71×25 = 1775; 1775/360 reduce by GCD(1775,360).
1775 ÷ 5 = 355, 360 ÷ 5 = 72 → 355 and 72 share no common factor (355 = 5×71, 72 = 2³×3²) → correct.

All look good.

Since the question says “Find the area of each of the sectors”, and the answer format expects only the final answer, but there are 15 items — likely they want all answers listed.

However, the instruction says:
> ONLY THEN provide the final answer.
> Final Answer: [Provide only the final correct answer]

Given that there are multiple parts, and the user didn’t specify a single answer, but the format expects one final answer block, I interpret that the expected output is a list of all 15 areas in order, simplified, using π.

So final answers:

1. $25\pi$
2. $12.5\pi$ or $\frac{25\pi}{2}$
3. $6.25\pi$ or $\frac{25\pi}{4}$
4. $18.75\pi$ or $\frac{75\pi}{4}$
5. $\frac{25\pi}{8}$
6. $\frac{25\pi}{12}$
7. $\frac{25\pi}{6}$
8. $\frac{25\pi}{3}$
9. $\frac{50\pi}{3}$
10. $\frac{5\pi}{72}$
11. $\frac{5\pi}{9}$
12. $\frac{355\pi}{72}$
13. $\frac{1175\pi}{72}$
14. $\frac{535\pi}{24}$
15. $\frac{5\pi x}{72}$

To keep consistent form, use fractions (not decimals), except maybe #2 and #3 could be fractions too.

Let me rewrite all as exact fractions:

1. $\frac{25\pi}{1}$
2. $\frac{25\pi}{2}$
3. $\frac{25\pi}{4}$
4. $\frac{75\pi}{4}$
5. $\frac{25\pi}{8}$
6. $\frac{25\pi}{12}$
7. $\frac{25\pi}{6}$
8. $\frac{25\pi}{3}$
9. $\frac{50\pi}{3}$
10. $\frac{5\pi}{72}$
11. $\frac{5\pi}{9}$
12. $\frac{355\pi}{72}$
13. $\frac{1175\pi}{72}$
14. $\frac{535\pi}{24}$
15. $\frac{5\pi x}{72}$

Yes — these are all correct.

Final Answer:
1. $25\pi$ cm²
2. $\frac{25\pi}{2}$ cm²
3. $\frac{25\pi}{4}$ cm²
4. $\frac{75\pi}{4}$ cm²
5. $\frac{25\pi}{8}$ cm²
6. $\frac{25\pi}{12}$ cm²
7. $\frac{25\pi}{6}$ cm²
8. $\frac{25\pi}{3}$ cm²
9. $\frac{50\pi}{3}$ cm²
10. $\frac{5\pi}{72}$ cm²
11. $\frac{5\pi}{9}$ cm²
12. $\frac{355\pi}{72}$ cm²
13. $\frac{1175\pi}{72}$ cm²
14. $\frac{535\pi}{24}$ cm²
15. $\frac{5\pi x}{72}$ cm²