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Answer :
To find the area of sector [tex]\( AOB \)[/tex] in the circle, let's go through the steps to solve the problem:
1. Identify the Given Information:
- The circle is centered at point [tex]\( O \)[/tex].
- Radius [tex]\( OA = 5 \)[/tex] units.
- Arc [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference of the circle.
2. Calculate the Total Circumference of the Circle:
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi r
\][/tex]
Substituting the given radius [tex]\( r = 5 \)[/tex]:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Determine the Fraction of the Circle:
We are given that arc [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference. Therefore, the arc length also indicates the fraction of the circle that sector [tex]\( AOB \)[/tex] covers.
4. Calculate the Area of the Sector:
The formula for the area of a sector is:
[tex]\[
\text{Area of sector} = \left( \frac{\text{Arc Length}}{\text{Circumference}} \right) \times \pi r^2
\][/tex]
Given that [tex]\(\frac{\text{Arc Length}}{\text{Circumference}} = \frac{1}{4}\)[/tex], substitute the values into the formula:
[tex]\[
\text{Area of sector} = \frac{1}{4} \times 3.14 \times 5^2
\][/tex]
[tex]\[
\text{Area of sector} = \frac{1}{4} \times 3.14 \times 25
\][/tex]
[tex]\[
\text{Area of sector} = \frac{1}{4} \times 78.5
\][/tex]
[tex]\[
\text{Area of sector} = 19.625
\][/tex]
5. Choose the Closest Option:
From the provided choices, 19.6 square units is the closest to our calculated area of 19.625. Therefore, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units.
Thus, the correct answer is:
A. 19.6 square units
1. Identify the Given Information:
- The circle is centered at point [tex]\( O \)[/tex].
- Radius [tex]\( OA = 5 \)[/tex] units.
- Arc [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference of the circle.
2. Calculate the Total Circumference of the Circle:
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi r
\][/tex]
Substituting the given radius [tex]\( r = 5 \)[/tex]:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Determine the Fraction of the Circle:
We are given that arc [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference. Therefore, the arc length also indicates the fraction of the circle that sector [tex]\( AOB \)[/tex] covers.
4. Calculate the Area of the Sector:
The formula for the area of a sector is:
[tex]\[
\text{Area of sector} = \left( \frac{\text{Arc Length}}{\text{Circumference}} \right) \times \pi r^2
\][/tex]
Given that [tex]\(\frac{\text{Arc Length}}{\text{Circumference}} = \frac{1}{4}\)[/tex], substitute the values into the formula:
[tex]\[
\text{Area of sector} = \frac{1}{4} \times 3.14 \times 5^2
\][/tex]
[tex]\[
\text{Area of sector} = \frac{1}{4} \times 3.14 \times 25
\][/tex]
[tex]\[
\text{Area of sector} = \frac{1}{4} \times 78.5
\][/tex]
[tex]\[
\text{Area of sector} = 19.625
\][/tex]
5. Choose the Closest Option:
From the provided choices, 19.6 square units is the closest to our calculated area of 19.625. Therefore, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units.
Thus, the correct answer is:
A. 19.6 square units
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