Question

Q. If the radius of the sector in the image is 12 cm, what is the area of the violet segment (in cm²)?
a) 36π-48√3
b) 36π-16√3
c) 48π-36√3
d) 48π-18√3

Answer 1

5994646 45..474442. it's not correct answer

Answer 2

Area of violet segment is $\bf 48\pi - 36 \sqrt{3}\:{cm}^{2}$

Option (c) is correct.

Given:

• Figure
• Radius of sector (r)= 12 cm

To find:

• what is the area of the violet segment (in cm²)?
• a) 36π-48√3
• b) 36π-16√3
• c) 48π-36√3
• d) 48π-18√3

Solution:

Formula to be used:

Area of segment: Area of sector-Area of triangle

Area of segment $\bf = \frac{ \theta}{360} \pi {r}^{2} - \frac{1}{2} {r}^{2} sin\theta$

Step 1:

Find area of segment.

r= 12 cm

$\theta = 120^{ \circ}$

Area of sector $= \frac{120}{360}\pi \times 12 \times 12$

or

on simplify

Area of sector$\bf = 48\pi \: {cm}^{2}$

Step 2:

Find area of ∆OAB.

Area of ∆OAB $= \frac{1}{2} \times 12 \times 12 \times sin 120^{\circ}$

As

$sin 120^{\circ}=\frac{ \sqrt{3} }{2}$

so,

Area of ∆OAB $= \frac{1}{2} \times 12 \times 12 \times \frac{ \sqrt{3} }{2}$

or

Area of ∆OAB$\bf = 36 \sqrt{3} \: {cm}^{2}$

Step 3:

Area of segment $\bf = 48\pi - 36 \sqrt{3} \: {cm}^{2}$

Thus,

Area of violet segment is $48\pi - 36 \sqrt{3} \: {cm}^{2}$

Option (c) is correct.

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