Answer :
To solve the problem, let's walk through the steps to find the area of sector [tex]\(AOB\)[/tex] when points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] lie on a circle centered at point [tex]\(O\)[/tex]. Here's how to do it:
1. Understand the Given Information:
- The radius of the circle [tex]\(OA = 5\)[/tex].
- The arc length of [tex]\(\overarc{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference.
2. Calculate the Circumference of the Circle:
[tex]\[
\text{Circumference} = 2 \times \pi \times \text{radius} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Find the Length of Arc [tex]\(\overarc{AB}\)[/tex]:
[tex]\[
\text{Length of } \overarc{AB} = \frac{1}{4} \times \text{Circumference} = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
4. Determine the Angle of the Sector:
- The arc length is part of the circumference, so the angle [tex]\(\theta\)[/tex] in radians corresponding to [tex]\(\overarc{AB}\)[/tex] can be calculated as:
[tex]\[
\theta = \frac{\text{Arc Length}}{\text{Radius}} = \frac{7.85}{5} = 1.57 \text{ radians}
\][/tex]
5. Calculate the Area of the Sector:
- The formula for the area of a sector given in radians is:
[tex]\[
\text{Area of Sector } AOB = \frac{1}{2} \times \text{Radius}^2 \times \theta
\][/tex]
- Substituting the known values:
[tex]\[
\text{Area of Sector } AOB = \frac{1}{2} \times 5^2 \times 1.57 = \frac{1}{2} \times 25 \times 1.57 = 19.625
\][/tex]
Based on the calculation, the closest answer to the area of sector [tex]\(AOB\)[/tex] is [tex]\( \boxed{19.6} \)[/tex] square units.
1. Understand the Given Information:
- The radius of the circle [tex]\(OA = 5\)[/tex].
- The arc length of [tex]\(\overarc{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference.
2. Calculate the Circumference of the Circle:
[tex]\[
\text{Circumference} = 2 \times \pi \times \text{radius} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Find the Length of Arc [tex]\(\overarc{AB}\)[/tex]:
[tex]\[
\text{Length of } \overarc{AB} = \frac{1}{4} \times \text{Circumference} = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
4. Determine the Angle of the Sector:
- The arc length is part of the circumference, so the angle [tex]\(\theta\)[/tex] in radians corresponding to [tex]\(\overarc{AB}\)[/tex] can be calculated as:
[tex]\[
\theta = \frac{\text{Arc Length}}{\text{Radius}} = \frac{7.85}{5} = 1.57 \text{ radians}
\][/tex]
5. Calculate the Area of the Sector:
- The formula for the area of a sector given in radians is:
[tex]\[
\text{Area of Sector } AOB = \frac{1}{2} \times \text{Radius}^2 \times \theta
\][/tex]
- Substituting the known values:
[tex]\[
\text{Area of Sector } AOB = \frac{1}{2} \times 5^2 \times 1.57 = \frac{1}{2} \times 25 \times 1.57 = 19.625
\][/tex]
Based on the calculation, the closest answer to the area of sector [tex]\(AOB\)[/tex] is [tex]\( \boxed{19.6} \)[/tex] square units.