Area of sector of a circle with radius \(18 \text{ cm}\) is \(198 \text{ cm}^2\). The measure of central angle is

Question

A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio a:b. If the radius of the circle is r, then the area of the triangle is

Answer 1

To find the measure of the central angle of a sector of a circle, we can use the formula for the area of a sector. The area \( A \) of a sector with a central angle \( \theta \) (in radians) in a circle of radius \( r \) is given by:

\[ A = \frac{1}{2} r^2 \theta \]

Given in the problem:

Area of the sector, \( A = 198 \text{ cm}^2 \)
Radius of the circle, \( r = 18 \text{ cm} \)

We need to find the central angle \( \theta \) in degrees. First, we will work with radians:

\[ 198 = \frac{1}{2} \times (18)^2 \times \theta \]

Simplifying:

\[ 198 = 162 \theta \]

\[ \theta = \frac{198}{162} = \frac{11}{9} \text{ radians} \]

To convert this angle from radians to degrees, we use the conversion factor \(\frac{180}{\pi}\):

\[ \theta (\text{in degrees}) = \frac{11}{9} \times \frac{180}{\pi} \]

Substituting the value of \(\pi \approx 3.14\), we get:

\[ \theta (\text{in degrees}) \approx \frac{11}{9} \times \frac{180}{3.14} \approx 70^{\circ} \]

Therefore, the measure of the central angle is \( 70^{\circ} \).

Hence, the correct answer is \( 70^{\circ} \).

Answer 2