Answer :
Sure! Let's solve this step by step.
### Step 1: Determine the Radius
The diameter of the circle is given as 20 inches. The radius (which is half the diameter) can be calculated as follows:
[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{20 \text{ inches}}{2} = 10 \text{ inches} \][/tex]
### Step 2: Calculate the Area of the Entire Circle
The area of a circle can be calculated using the formula:
[tex]\[ \text{Area} = \pi r^2 \][/tex]
Substituting the radius we found:
[tex]\[ \text{Area} = \pi \times (10 \text{ inches})^2 = \pi \times 100 = 100\pi \text{ square inches} \][/tex]
### Step 3: Determine the Area of the Sector
A sector is a fraction of the entire circle's area. The fraction is determined by the central angle of the sector.
The central angle of the sector is given as 36 degrees. There are 360 degrees in a full circle. Therefore, the fraction of the circle that the sector represents is:
[tex]\[ \text{Fraction of circle} = \frac{\text{Central Angle}}{360} = \frac{36}{360} = \frac{1}{10} \][/tex]
### Step 4: Calculate the Area of the Sector
The area of the sector is then this fraction of the total area of the circle:
[tex]\[ \text{Area of Sector} = \left(\frac{1}{10}\right) \times 100\pi \text{ square inches} = 10\pi \text{ square inches} \][/tex]
### Final Answer
The area of the sector with a 36° arc is:
[tex]\[ 10\pi \text{ square inches} \][/tex]
### Step 1: Determine the Radius
The diameter of the circle is given as 20 inches. The radius (which is half the diameter) can be calculated as follows:
[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{20 \text{ inches}}{2} = 10 \text{ inches} \][/tex]
### Step 2: Calculate the Area of the Entire Circle
The area of a circle can be calculated using the formula:
[tex]\[ \text{Area} = \pi r^2 \][/tex]
Substituting the radius we found:
[tex]\[ \text{Area} = \pi \times (10 \text{ inches})^2 = \pi \times 100 = 100\pi \text{ square inches} \][/tex]
### Step 3: Determine the Area of the Sector
A sector is a fraction of the entire circle's area. The fraction is determined by the central angle of the sector.
The central angle of the sector is given as 36 degrees. There are 360 degrees in a full circle. Therefore, the fraction of the circle that the sector represents is:
[tex]\[ \text{Fraction of circle} = \frac{\text{Central Angle}}{360} = \frac{36}{360} = \frac{1}{10} \][/tex]
### Step 4: Calculate the Area of the Sector
The area of the sector is then this fraction of the total area of the circle:
[tex]\[ \text{Area of Sector} = \left(\frac{1}{10}\right) \times 100\pi \text{ square inches} = 10\pi \text{ square inches} \][/tex]
### Final Answer
The area of the sector with a 36° arc is:
[tex]\[ 10\pi \text{ square inches} \][/tex]
