In circle $L$ with $ m\angle KLM = 46^\circ $ and $ KL = 13 $, find the area of sector $ KLM $.

Round to the nearest hundredth.

Question

High School

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Answer :

souma00ira souma00ira · Answer 1 · 2024-01-05T12:43:46-05:00

The area of sector KLM in Circle L is approximately 76.81 square units.

Let's find the area of sector KLM in Circle L step by step:

1. Calculate the Area of Circle L:

- The area of a circle is given by the formula: [tex]$$A_{\text{circle}} = \pi r^2$$[/tex]

- Given that the radius of Circle L, KL, is 13 units, we can calculate the area:

[tex]$$A_{\text{circle}} = \pi (13)^2 = 169\pi$$[/tex]

2. Find the Fraction of Circle L's Area Represented by Sector KLM:

- The central angle ∠KLM is given as 46 degrees.

- To find the fraction of the circle's area represented by sector KLM, divide the central angle by the total angle of a circle (360 degrees):

[tex]\text{Fraction} = \frac{\angle KLM}{360} = \frac{46}{360}[/tex]

3. Calculate the Area of Sector KLM:

- The area of a sector is given by the formula: [tex]$A_{\text{sector}} = \text{Fraction} \cdot A_{\text{circle}}$$[/tex]

- Substitute the known values:

[tex]$$A_{\text{sector}} = \left(\frac{46}{360}\right) \cdot 169\pi$$[/tex]

- Calculate the expression:

[tex]$$A_{\text{sector}} = \frac{23}{180} \cdot 169\pi \approx 76.81$$[/tex]

4. Round to the Nearest Hundredth:

- Rounding to two decimal places, the area of sector KLM is approximately 76.81 square units.

Therefore, the area of sector KLM in Circle L is approximately 76.81 square units.

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