Area of Sectors and Segments (With Clues) Worksheet | Cazoom Maths ... - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1) Two Shaded Sectors

Goal: Find the total area of the two orange sectors.
* Step 1: Find the angle. The angles around the center of a circle add up to $360^\circ$. We are given one angle of $158^\circ$. The remaining angle for the two shaded sectors combined is:
$$360^\circ - 158^\circ = 202^\circ$$
* Step 2: Calculate the area. Use the sector area formula with the combined angle ($202^\circ$) and radius ($4 \text{ cm}$).
$$\text{Area} = \frac{202}{360} \times \pi \times 4^2$$
$$\text{Area} = \frac{202}{360} \times \pi \times 16$$
$$\text{Area} \approx 28.204...$$
* Rounding: To 3 significant figures, this is 28.2.

2) Quarter Annulus (Ring Sector)

Goal: Find the area of the orange shape, which is a large quarter-circle minus a smaller white quarter-circle.
* Step 1: Identify dimensions.
* Outer radius ($R$) = $14 \text{ cm}$.
* Inner radius ($r$) = $14 - 10 = 4 \text{ cm}$.
* Angle = $90^\circ$ (quarter circle).
* Step 2: Calculate Area.
$$\text{Area} = \left( \frac{90}{360} \times \pi \times 14^2 \right) - \left( \frac{90}{360} \times \pi \times 4^2 \right)$$
$$\text{Area} = \frac{1}{4}\pi(196) - \frac{1}{4}\pi(16)$$
$$\text{Area} = 49\pi - 4\pi = 45\pi$$
$$\text{Area} \approx 141.37...$$
* Rounding: To 3 significant figures, this is 141.

3) Circular Segment

Goal: Find the area of the orange segment (Sector minus Triangle).
* Step 1: Sector Area. Radius $r=9$, Angle $\theta=90^\circ$.
$$\text{Area}_{\text{sector}} = \frac{90}{360} \times \pi \times 9^2 = \frac{1}{4} \times 81\pi \approx 63.617$$
* Step 2: Triangle Area. This is a right-angled triangle with base and height equal to the radius ($9$).
$$\text{Area}_{\text{triangle}} = \frac{1}{2} \times 9 \times 9 = 40.5$$
* Step 3: Subtract.
$$\text{Shaded Area} = 63.617 - 40.5 = 23.117...$$
* Rounding: To 3 significant figures, this is 23.1.

4) Circular Segment (Non-right angle)

Goal: Find the shaded area using the formula provided in the clue.
* Step 1: Sector Area. Radius $r=16$, Angle $\theta=55^\circ$.
$$\text{Area}_{\text{sector}} = \frac{55}{360} \times \pi \times 16^2 \approx 122.866$$
* Step 2: Triangle Area. Use the clue formula: $\text{Area} = \frac{1}{2}ab \sin C$. Here $a=16, b=16, C=55^\circ$.
$$\text{Area}_{\text{triangle}} = \frac{1}{2} \times 16 \times 16 \times \sin(55^\circ)$$
$$\text{Area}_{\text{triangle}} = 128 \times 0.81915... \approx 104.851$$
* Step 3: Subtract.
$$\text{Shaded Area} = 122.866 - 104.851 = 18.015...$$
* Rounding: To 3 significant figures, this is 18.0.

5) Major Sector

Goal: Find the area of the large orange sector.
* Step 1: Find the angle. The unshaded part is $169^\circ$. The shaded part is the rest of the circle.
$$\text{Angle} = 360^\circ - 169^\circ = 191^\circ$$
* Step 2: Calculate Area. Radius $r=7$.
$$\text{Area} = \frac{191}{360} \times \pi \times 7^2$$
$$\text{Area} = \frac{191}{360} \times \pi \times 49 \approx 81.656...$$
* Rounding: To 3 significant figures, this is 81.7.

6) Segment from Chord Distance

Goal: Find the area of the segment defined by a chord $2 \text{ cm}$ from the center.
* Step 1: Find the angle ($\theta$).
Imagine a triangle formed by the center and the chord endpoints. Split it in half to make a right-angled triangle.
* Hypotenuse (Radius) = $6$.
* Adjacent side (distance to chord) = $2$.
* Let $\alpha$ be half the central angle.
$$\cos(\alpha) = \frac{2}{6} = \frac{1}{3}$$
$$\alpha = \arccos\left(\frac{1}{3}\right) \approx 70.529^\circ$$
Total angle $\theta = 2 \times 70.529^\circ = 141.058^\circ$.
* Step 2: Sector Area.
$$\text{Area}_{\text{sector}} = \frac{141.058}{360} \times \pi \times 6^2 \approx 44.315$$
* Step 3: Triangle Area. Using $\frac{1}{2} r^2 \sin \theta$:
$$\text{Area}_{\text{triangle}} = \frac{1}{2} \times 6^2 \times \sin(141.058^\circ)$$
$$\text{Area}_{\text{triangle}} = 18 \times 0.6285... \approx 11.314$$
* Step 4: Subtract.
$$\text{Shaded Area} = 44.315 - 11.314 = 33.001...$$
* Rounding: To 3 significant figures, this is 33.0.

Final Answer:
1) 28.2 cm²
2) 141 cm²
3) 23.1 cm²
4) 18.0 cm²
5) 81.7 cm²
6) 33.0 cm²