Answer :
- Calculate the circumference of the circle using $C = 2\pi r$, which equals $31.4$.
- Determine the arc length $AB$ as $\frac{1}{4}$ of the circumference, resulting in $7.85$.
- Compute the area of the entire circle using $A = \pi r^2$, which equals $78.5$.
- Find the area of sector $AOB$ as $\frac{1}{4}$ of the circle's area, giving $\boxed{19.6}$ square units.
### Explanation
1. Problem Analysis
We are given a circle with center $O$, and points $A$ and $B$ on the circle. The radius $OA$ is 5 units, and the ratio of the length of arc $AB$ to the circumference of the circle is $\frac{1}{4}$. We need to find the area of sector $AOB$.
2. Calculate Circumference and Arc Length
The circumference of the circle is given by $C = 2\pi r$, where $r$ is the radius. In our case, $r = 5$ and $\pi = 3.14$, so the circumference is:
$$C = 2 \times 3.14 \times 5 = 31.4$$
The length of arc $AB$ is $\frac{1}{4}$ of the circumference:
$$\text{length of } \widehat{AB} = \frac{1}{4} \times 31.4 = 7.85$$
3. Calculate Circle and Sector Area
The area of the entire circle is given by $A = \pi r^2$. In our case, $r = 5$ and $\pi = 3.14$, so the area is:
$$A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5$$
The area of sector $AOB$ is $\frac{1}{4}$ of the area of the entire circle:
$$\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625$$
4. Final Answer
The area of sector $AOB$ is 19.625 square units. Looking at the options, the closest answer is 19.6 square units.
### Examples
Understanding the area of a sector is useful in many real-world applications. For example, if you're designing a sprinkler system for a circular lawn, you need to know the area that each sprinkler head will cover. Sectors also come into play when calculating slices of a pie or pizza, determining coverage areas in surveillance systems, or even in the design of certain types of gears and mechanical components.
- Determine the arc length $AB$ as $\frac{1}{4}$ of the circumference, resulting in $7.85$.
- Compute the area of the entire circle using $A = \pi r^2$, which equals $78.5$.
- Find the area of sector $AOB$ as $\frac{1}{4}$ of the circle's area, giving $\boxed{19.6}$ square units.
### Explanation
1. Problem Analysis
We are given a circle with center $O$, and points $A$ and $B$ on the circle. The radius $OA$ is 5 units, and the ratio of the length of arc $AB$ to the circumference of the circle is $\frac{1}{4}$. We need to find the area of sector $AOB$.
2. Calculate Circumference and Arc Length
The circumference of the circle is given by $C = 2\pi r$, where $r$ is the radius. In our case, $r = 5$ and $\pi = 3.14$, so the circumference is:
$$C = 2 \times 3.14 \times 5 = 31.4$$
The length of arc $AB$ is $\frac{1}{4}$ of the circumference:
$$\text{length of } \widehat{AB} = \frac{1}{4} \times 31.4 = 7.85$$
3. Calculate Circle and Sector Area
The area of the entire circle is given by $A = \pi r^2$. In our case, $r = 5$ and $\pi = 3.14$, so the area is:
$$A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5$$
The area of sector $AOB$ is $\frac{1}{4}$ of the area of the entire circle:
$$\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625$$
4. Final Answer
The area of sector $AOB$ is 19.625 square units. Looking at the options, the closest answer is 19.6 square units.
### Examples
Understanding the area of a sector is useful in many real-world applications. For example, if you're designing a sprinkler system for a circular lawn, you need to know the area that each sprinkler head will cover. Sectors also come into play when calculating slices of a pie or pizza, determining coverage areas in surveillance systems, or even in the design of certain types of gears and mechanical components.
