Answer :
To determine the area of a sector of a circle given its diameter and the measure of the central angle, let's follow a step-by-step procedure:
1. Understand the Problem:
- The diameter of the circle is given as 2 feet.
- The central angle of the sector is given as 144°.
2. Calculate the Radius:
- The radius (r) of the circle is half of the diameter.
- [tex]\[ r = \frac{\text{diameter}}{2} = \frac{2 \text{ feet}}{2} = 1 \text{ foot} \][/tex]
3. Formula for the Area of a Circle:
- The area (A) of a full circle is given by:
- [tex]\[ A = \pi r^2 \][/tex]
- Substituting the radius:
- [tex]\[ A = \pi (1 \text{ foot})^2 = \pi \text{ square feet} \][/tex]
4. Calculate the Proportion of the Circle Represented by the Sector:
- The full circle is 360°.
- The given sector represents 144°.
- Therefore, the proportion of the circle occupied by the sector is:
- [tex]\[ \frac{\theta}{360^{\circ}} = \frac{144^{\circ}}{360^{\circ}} = \frac{144}{360} = \frac{2}{5} \][/tex]
5. Calculate the Area of the Sector:
- The area of the sector is this fraction of the total area of the circle.
- So, the area of the sector (A_sector) is:
- [tex]\[ A_{\text{sector}} = \frac{2}{5} \times \pi \text{ square feet} = \frac{2\pi}{5} \text{ square feet} \][/tex]
6. Final Answer:
- The exact area of the sector in simplest form is:
- [tex]\[ \boxed{\frac{2\pi}{5} \text{ square feet}} \][/tex]
1. Understand the Problem:
- The diameter of the circle is given as 2 feet.
- The central angle of the sector is given as 144°.
2. Calculate the Radius:
- The radius (r) of the circle is half of the diameter.
- [tex]\[ r = \frac{\text{diameter}}{2} = \frac{2 \text{ feet}}{2} = 1 \text{ foot} \][/tex]
3. Formula for the Area of a Circle:
- The area (A) of a full circle is given by:
- [tex]\[ A = \pi r^2 \][/tex]
- Substituting the radius:
- [tex]\[ A = \pi (1 \text{ foot})^2 = \pi \text{ square feet} \][/tex]
4. Calculate the Proportion of the Circle Represented by the Sector:
- The full circle is 360°.
- The given sector represents 144°.
- Therefore, the proportion of the circle occupied by the sector is:
- [tex]\[ \frac{\theta}{360^{\circ}} = \frac{144^{\circ}}{360^{\circ}} = \frac{144}{360} = \frac{2}{5} \][/tex]
5. Calculate the Area of the Sector:
- The area of the sector is this fraction of the total area of the circle.
- So, the area of the sector (A_sector) is:
- [tex]\[ A_{\text{sector}} = \frac{2}{5} \times \pi \text{ square feet} = \frac{2\pi}{5} \text{ square feet} \][/tex]
6. Final Answer:
- The exact area of the sector in simplest form is:
- [tex]\[ \boxed{\frac{2\pi}{5} \text{ square feet}} \][/tex]
