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In circle \( L \) with \( m \angle KLM = 70^{\circ} \) and \( KL = 5 \), find the area of sector \( KLM \). Round to the nearest hundredth.

Answer :

Radius = 2.5. Circle area ≈ 19.63. Sector angle = 70°. Sector area ≈ (70/360) * 19.63 ≈ 3.82.

To find the area of sector KLM, we first need to find the area of the entire circle and then multiply it by the fraction of the circle that sector KLM represents.

The formula for the area of a circle is given by [tex]\( A = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle.

Given that [tex]\( KL = 5 \),[/tex] we can find the radius [tex](\( r \))[/tex] using the fact that the radius is half the diameter, so [tex]\( r = \frac{5}{2} = 2.5 \).[/tex]

Now, to find the area of the entire circle:

[tex]\[ A_{\text{circle}} = \pi \times (2.5)^2 \]\[ A_{\text{circle}} = \pi \times 6.25 \]\[ A_{\text{circle}} \approx 19.63 \][/tex]

Now, to find the area of sector KLM, we use the formula for the area of a sector, which is given by [tex]\( A_{\text{sector}} = \frac{\theta}{360} \[/tex] times A_[tex]{\text{circle}} \)[/tex], where [tex]\( \theta \)[/tex] is the angle of the sector in degrees.

Given that [tex]\( \theta = 70^\circ \)[/tex], we plug in the values:

[tex]\[ A_{\text{sector}} = \frac{70}{360} \times 19.63 \][/tex]

[tex]\[ A_{\text{sector}} \approx 3.82 \][/tex]

So, the area of sector KLM is approximately 3.82 square units, rounded to the nearest hundredth.

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