Answer :
To find the area of a [tex]\(60^\circ\)[/tex] sector of a circle with an area of [tex]\(30 \text{ in}^2\)[/tex], we can follow these steps:
1. Understand the Problem:
- We need the area of a sector of a circle, given that the total area of the circle is [tex]\(30 \text{ in}^2\)[/tex] and the central angle of the sector is [tex]\(60^\circ\)[/tex].
2. Formula for the Area of a Sector:
- The area of a sector of a circle can be found using the formula:
[tex]\[ \text{Sector Area} = \left( \frac{\text{Angle of Sector}}{360^\circ} \right) \times \text{Total Area of Circle} \][/tex]
3. Substitute the Values:
- The given angle of the sector is [tex]\(60^\circ\)[/tex].
- The total area of the circle is [tex]\(30 \text{ in}^2\)[/tex].
Substituting these values into the formula, we get:
[tex]\[ \text{Sector Area} = \left( \frac{60^\circ}{360^\circ} \right) \times 30 \text{ in}^2 \][/tex]
4. Simplify the Fraction:
- Simplify [tex]\(\frac{60^\circ}{360^\circ}\)[/tex]:
[tex]\[ \frac{60^\circ}{360^\circ} = \frac{1}{6} \][/tex]
5. Calculate the Sector Area:
- Now, multiply the simplified fraction by the total area of the circle:
[tex]\[ \text{Sector Area} = \frac{1}{6} \times 30 \text{ in}^2 = 5 \text{ in}^2 \][/tex]
6. Conclusion:
- The area of the [tex]\(60^\circ\)[/tex] sector of the circle is [tex]\(5 \text{ in}^2\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{5 \text{ in}^2} \][/tex]
1. Understand the Problem:
- We need the area of a sector of a circle, given that the total area of the circle is [tex]\(30 \text{ in}^2\)[/tex] and the central angle of the sector is [tex]\(60^\circ\)[/tex].
2. Formula for the Area of a Sector:
- The area of a sector of a circle can be found using the formula:
[tex]\[ \text{Sector Area} = \left( \frac{\text{Angle of Sector}}{360^\circ} \right) \times \text{Total Area of Circle} \][/tex]
3. Substitute the Values:
- The given angle of the sector is [tex]\(60^\circ\)[/tex].
- The total area of the circle is [tex]\(30 \text{ in}^2\)[/tex].
Substituting these values into the formula, we get:
[tex]\[ \text{Sector Area} = \left( \frac{60^\circ}{360^\circ} \right) \times 30 \text{ in}^2 \][/tex]
4. Simplify the Fraction:
- Simplify [tex]\(\frac{60^\circ}{360^\circ}\)[/tex]:
[tex]\[ \frac{60^\circ}{360^\circ} = \frac{1}{6} \][/tex]
5. Calculate the Sector Area:
- Now, multiply the simplified fraction by the total area of the circle:
[tex]\[ \text{Sector Area} = \frac{1}{6} \times 30 \text{ in}^2 = 5 \text{ in}^2 \][/tex]
6. Conclusion:
- The area of the [tex]\(60^\circ\)[/tex] sector of the circle is [tex]\(5 \text{ in}^2\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{5 \text{ in}^2} \][/tex]
