Question

Determine the area of a segment of a circle with a central angle of $120$ degrees and a radius of $4$. Express answer to nearest integer.

• A. $11$
• B. $12$
• C. $10$
• D. $14$

Answer 1

The correct option is C $10$

Area of segment, i.e. area of shaded region = area of sector $OAPB-$ ar$\Delta ABO$

Area of sector OAPB = $\frac{\theta }{{360}^0}\times \pi r^2=\frac{{120}^0}{{360}^0}\times \frac{22}{7}\times 4\times 4$

= $\frac{352}{21}{cm}^2=16.76{cm}^2$

Now, area of $\Delta OAB$.

Draw $OM1AB$. This will results in forming OAB triangle in two ${30}^0-{60}^0-{90}^0$ triangles.

Using ar$\Delta OAB=2\times ar\Delta OAM=2\times \frac{1}{2}\times OM\times AM=2\times 2\sqrt{3}=4\sqrt{3}$

$=4$$\times 1.73=6.92$${cm}^2$

$\therefore$ area of segment $=16.76-6.92=9.84$$\approx$10${cm}^2$