Question
How do you derive the formula for the area of a sector?
Divide the measurement of the sector’s angle by 180°
, then multiply the quotient by π
times the radius squared.
Divide the measurement of the sector’s angle by 180 degrees , then multiply the quotient by pi times the radius squared.
Divide the measurement of the sector’s angle by 360°
, then multiply the quotient by π
times the radius squared.
Divide the measurement of the sector’s angle by 360 degrees , then multiply the quotient by pi times the radius squared.
Divide the measurement of the sector’s angle by 360°
, then multiply the quotient by π
times the radius.
Divide the measurement of the sector’s angle by 360 degrees , then multiply the quotient by pi times the radius.
Divide the measurement of the sector’s angle by 360°
, then divide the quotient by π
times the radius squared.
Divide the measurement of the sector’s angle by 360 degrees , then divide the quotient by pi times the radius squared.
Divide the measurement of the sector’s angle by 180°
, then multiply the quotient by π
times the radius squared.
Divide the measurement of the sector’s angle by 180 degrees , then multiply the quotient by pi times the radius squared.
Divide the measurement of the sector’s angle by 360°
, then multiply the quotient by π
times the radius squared.
Divide the measurement of the sector’s angle by 360 degrees , then multiply the quotient by pi times the radius squared.
Divide the measurement of the sector’s angle by 360°
, then multiply the quotient by π
times the radius.
Divide the measurement of the sector’s angle by 360 degrees , then multiply the quotient by pi times the radius.
Divide the measurement of the sector’s angle by 360°
, then divide the quotient by π
times the radius squared.
Divide the measurement of the sector’s angle by 360 degrees , then divide the quotient by pi times the radius squared.
Answers
Answered by
GPT-4o mini
To derive the formula for the area of a sector of a circle, we start from the understanding that the area of a full circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
where \( r \) is the radius of the circle.
A sector is essentially a "slice" of the circle, defined by a central angle \( \theta \) (measured in degrees). The entire circle corresponds to an angle of \( 360^\circ \).
To find the area of the sector, we can express the area of the sector as a fraction of the area of the full circle. The fraction of the circle that the sector represents is given by the ratio of the angle of the sector to the total angle of the circle:
\[ \text{Fraction of Circle for Sector} = \frac{\theta}{360^\circ} \]
Now, to find the area of the sector, we multiply the area of the full circle by this fraction:
\[
\text{Area of Sector} = \text{Area of Circle} \times \text{Fraction of Circle for Sector}
\]
Substituting the area of the circle:
\[
\text{Area of Sector} = \pi r^2 \times \frac{\theta}{360^\circ}
\]
Thus, we arrive at the formula for the area of a sector:
\[
\text{Area of Sector} = \frac{\theta}{360^\circ} \cdot \pi r^2
\]
This shows that the correct approach to deriving the area of a sector is to divide the measurement of the sector’s angle by \( 360^\circ \), and then multiply that quotient by \( \pi r^2 \).
\[ \text{Area of Circle} = \pi r^2 \]
where \( r \) is the radius of the circle.
A sector is essentially a "slice" of the circle, defined by a central angle \( \theta \) (measured in degrees). The entire circle corresponds to an angle of \( 360^\circ \).
To find the area of the sector, we can express the area of the sector as a fraction of the area of the full circle. The fraction of the circle that the sector represents is given by the ratio of the angle of the sector to the total angle of the circle:
\[ \text{Fraction of Circle for Sector} = \frac{\theta}{360^\circ} \]
Now, to find the area of the sector, we multiply the area of the full circle by this fraction:
\[
\text{Area of Sector} = \text{Area of Circle} \times \text{Fraction of Circle for Sector}
\]
Substituting the area of the circle:
\[
\text{Area of Sector} = \pi r^2 \times \frac{\theta}{360^\circ}
\]
Thus, we arrive at the formula for the area of a sector:
\[
\text{Area of Sector} = \frac{\theta}{360^\circ} \cdot \pi r^2
\]
This shows that the correct approach to deriving the area of a sector is to divide the measurement of the sector’s angle by \( 360^\circ \), and then multiply that quotient by \( \pi r^2 \).