The diameter of a circle is 20 meters. What is the area of a sector bounded by a 135° arc?
Give the exact answer in simplest form.
To find the area of a sector, we use the formula:
Area of sector = (θ/360°) × πr²
Where θ is the angle formed by the sector, and r is the radius of the circle.
In this case, the diameter of the circle is given as 20 meters, so the radius is half of that, which is 20/2 = 10 meters.
The angle formed by the sector is 135°, so substituting these values into the formula:
Area of sector = (135°/360°) × π(10 m)²
= (135/360) × π(100 m²)
= (3/8) × π(100 m²)
= (3/8) × 100π
= 300π/8
= 37.5π
Therefore, the area of the sector bounded by the 135° arc is 37.5π square meters in exact form.
To find the area of a sector, you can use the formula:
Area of Sector = (θ/360) * π * r^2
where:
- θ is the central angle in degrees,
- π is a mathematical constant approximately equal to 3.14159, and
- r is the radius of the circle.
Given that the diameter of the circle is 20 meters, the radius will be half of that, so r = 10 meters.
The central angle of the sector is 135 degrees.
Plugging these values into the formula, we get:
Area of Sector = (135/360) * π * 10^2
Simplifying further:
Area of Sector = (3/8) * π * 100
Finally, calculating the area:
Area of Sector = 3/8 * 100π
= 300/8π
= 37.5π
Therefore, the exact area of the sector bounded by a 135° arc is 37.5π square meters.