To find the area of the sector [tex]\( AOB \)[/tex], where points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] lie on a circle centered at point [tex]\( O \)[/tex], we need to follow several steps given the information:
1. Radius of the circle:
The radius [tex]\( OA = 5 \)[/tex] units.
2. Circumference of the circle:
The circumference of a circle is given by the formula:
[tex]\[
\text{Circumference} = 2 \pi r
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( \pi = 3.14 \)[/tex]. Substituting the given radius:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Arc ratio:
The ratio of the length of the arc [tex]\( \widehat{AB} \)[/tex] to the circumference is given by:
[tex]\[
\frac{\text{length of} \; \widehat{AB}}{\text{circumference}} = \frac{1}{4}
\][/tex]
4. Arc length:
The length of the arc [tex]\( \widehat{AB} \)[/tex] can be calculated using the above ratio and the circumference:
[tex]\[
\text{Arc length} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
5. Area of the circle:
The area of the entire circle is computed using the formula:
[tex]\[
\text{Area of Circle} = \pi r^2
\][/tex]
Substituting the radius:
[tex]\[
\text{Area of Circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
6. Area of the sector:
The sector [tex]\( AOB \)[/tex] is proportional to [tex]\( \frac{1}{4} \)[/tex] of the circle because the arc length is [tex]\( \frac{1}{4} \)[/tex] of the circumference. Therefore, the area of the sector is also [tex]\( \frac{1}{4} \)[/tex] of the total area of the circle:
[tex]\[
\text{Area of Sector} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
Thus, the closest answer to the area of sector [tex]\( AOB \)[/tex] is:
[tex]\[
\boxed{19.6} \text{ square units}
\][/tex]
