Practice problems for finding the area of shaded sectors and segments in circles, including given angles and radii.

To find the area of a shaded segment, we follow these steps:
1. Find the Area of the Sector: This is the "pie slice" shape. The formula is $\frac{\theta}{360} \times \pi r^2$, where $\theta$ is the angle and $r$ is the radius.
2. Find the Area of the Triangle: This is the triangle formed by the two radii and the chord. The formula is $\frac{1}{2} r^2 \sin(\theta)$.
3. Subtract: Area of Segment = Area of Sector - Area of Triangle.

*Note: For diagrams where the shaded region is the larger part of the circle (reflex angle), we use the reflex angle for the sector calculation but keep the triangle calculation based on the inner angle (since $\sin(360^\circ - x) = -\sin(x)$, subtracting the negative effectively adds the triangle area back in).*

Here are the step-by-step solutions for each diagram, rounded to 1 decimal place.

a. Radius $r=5$, Angle $\theta=210^\circ$.
* Sector Area: $\frac{210}{360} \times \pi \times 5^2 \approx 45.81$
* Triangle Area: $\frac{1}{2} \times 5^2 \times \sin(210^\circ) = 12.5 \times (-0.5) = -6.25$
* Segment Area: $45.81 - (-6.25) = 52.06$
* Answer: 52.1 cm²

b. Radius $r=3$, Angle $\theta=125^\circ$.
* Sector Area: $\frac{125}{360} \times \pi \times 3^2 \approx 9.82$
* Triangle Area: $\frac{1}{2} \times 3^2 \times \sin(125^\circ) \approx 3.69$
* Segment Area: $9.82 - 3.69 = 6.13$
* Answer: 6.1 cm²

c. Radius $r=4.1$, Angle $\theta=52^\circ$.
* Sector Area: $\frac{52}{360} \times \pi \times 4.1^2 \approx 7.62$
* Triangle Area: $\frac{1}{2} \times 4.1^2 \times \sin(52^\circ) \approx 6.62$
* Segment Area: $7.62 - 6.62 = 1.00$
* Answer: 1.0 cm²

d. Radius $r=8.7$, Angle $\theta=305^\circ$.
* Sector Area: $\frac{305}{360} \times \pi \times 8.7^2 \approx 201.99$
* Triangle Area: $\frac{1}{2} \times 8.7^2 \times \sin(305^\circ) \approx -30.99$
* Segment Area: $201.99 - (-30.99) = 232.98$
* Answer: 233.0 cm²

e. Radius $r=7$, Angle $\theta=95^\circ$.
* Sector Area: $\frac{95}{360} \times \pi \times 7^2 \approx 40.58$
* Triangle Area: $\frac{1}{2} \times 7^2 \times \sin(95^\circ) \approx 24.39$
* Segment Area: $40.58 - 24.39 = 16.19$
* Answer: 16.2 cm²

f. Radius $r=4$, Angle $\theta=165^\circ$.
* Sector Area: $\frac{165}{360} \times \pi \times 4^2 \approx 23.04$
* Triangle Area: $\frac{1}{2} \times 4^2 \times \sin(165^\circ) \approx 2.07$
* Segment Area: $23.04 - 2.07 = 20.97$
* Answer: 21.0 cm²

g. Radius $r=15$, Angle $\theta=28^\circ$.
* Sector Area: $\frac{28}{360} \times \pi \times 15^2 \approx 54.98$
* Triangle Area: $\frac{1}{2} \times 15^2 \times \sin(28^\circ) \approx 52.82$
* Segment Area: $54.98 - 52.82 = 2.16$
* Answer: 2.2 cm²

h. Radius $r=3.9$, Angle $\theta=127^\circ$.
* Sector Area: $\frac{127}{360} \times \pi \times 3.9^2 \approx 16.86$
* Triangle Area: $\frac{1}{2} \times 3.9^2 \times \sin(127^\circ) \approx 6.07$
* Segment Area: $16.86 - 6.07 = 10.79$
* Answer: 10.8 cm²

i. Radius $r=5$, Angle $\theta=60^\circ$.
* Sector Area: $\frac{60}{360} \times \pi \times 5^2 \approx 13.09$
* Triangle Area: $\frac{1}{2} \times 5^2 \times \sin(60^\circ) \approx 10.83$
* Segment Area: $13.09 - 10.83 = 2.26$
* Answer: 2.3 cm²

j. Radius $r=9$, Angle $\theta=100^\circ$.
* Sector Area: $\frac{100}{360} \times \pi \times 9^2 \approx 70.69$
* Triangle Area: $\frac{1}{2} \times 9^2 \times \sin(100^\circ) \approx 39.92$
* Segment Area: $70.69 - 39.92 = 30.77$
* Answer: 30.8 cm²

k. Radius $r=11$, Angle $\theta=135^\circ$.
* Sector Area: $\frac{135}{360} \times \pi \times 11^2 \approx 142.55$
* Triangle Area: $\frac{1}{2} \times 11^2 \times \sin(135^\circ) \approx 42.78$
* Segment Area: $142.55 - 42.78 = 99.77$
* Answer: 99.8 cm²

l. Radius $r=4.7$, Angle $\theta=135^\circ$.
* Sector Area: $\frac{135}{360} \times \pi \times 4.7^2 \approx 26.06$
* Triangle Area: $\frac{1}{2} \times 4.7^2 \times \sin(135^\circ) \approx 7.81$
* Segment Area: $26.06 - 7.81 = 18.25$
* Answer: 18.3 cm²

Final Answer:
a. 52.1 cm²
b. 6.1 cm²
c. 1.0 cm²
d. 233.0 cm²
e. 16.2 cm²
f. 21.0 cm²
g. 2.2 cm²
h. 10.8 cm²
i. 2.3 cm²
j. 30.8 cm²
k. 99.8 cm²
l. 18.3 cm²