How do you derive the formula for the area of a sector?
To derive the formula for the area of a sector, we need to consider the relationship between the area of a sector, the area of the entire circle and the angle measure of the sector.
The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.
A sector is a portion of a circle enclosed by two radii and an arc. The angle measure of the sector is represented by θ (in radians), where θ is a fraction of the total angle measure of a circle (2π radians).
Since a sector is a portion of a circle, the ratio of the area of the sector to the area of the entire circle is equal to the ratio of the angle measure of the sector to the total angle measure of the circle.
This can be expressed as:
Area of sector / Area of circle = θ / 2π
Substituting the formulas for the area of a sector and the area of a circle, we get:
(θ / 2π) = A_sector / (πr^2)
Cross multiplying, we get:
A_sector = (θ / 2π) * πr^2
Simplifying, we get:
A_sector = (θ / 2) * r^2
Therefore, the formula for the area of a sector is given by A_sector = (θ / 2) * r^2, where θ is the angle measure of the sector in radians and r is the radius of the circle.