in a circle with center ooo, central angle aobaoba, o, b has a measure of \dfrac{5 \pi}{4} 4 5π start fraction, 5, pi, divided by, 4, end fraction radians. the area of the sector formed by central angle aobaoba, o, b is what fraction of the area of the circle?
The central angle of $aoba$ is $\frac{5\pi}{4}$ out of a full circle of $2\pi$, which means it covers $\frac{5\pi}{4}\div 2\pi=\frac{5}{8}$ of the circle. Therefore, the area of the sector formed by the central angle is $\frac{5}{8}$ of the total area of the circle.
So the fraction of the area of the sector formed by the central angle $aoba$ to the area of the circle is $\boxed{\frac{5}{8}}$.
To find the area of the sector, we need to know the radius of the circle. Let's assume it is "r".
The formula for the area of a sector is given by:
Area of sector = (θ / 2π) * π * r²
In this case, the central angle, aobaoba, is given as (5π/4) radians.
Area of sector = ((5π/4) / (2π)) * π * r²
Simplifying the expression, we get:
Area of sector = (5/8) * π * r²
Now, let's find the area of the entire circle:
Area of circle = π * r²
To find the fraction of the area of the sector relative to the area of the circle, we divide the area of the sector by the area of the circle:
Fraction of area = (Area of sector) / (Area of circle)
Fraction of area = [(5/8) * π * r²] / [π * r²]
We can simplify the expression by canceling out the terms:
Fraction of area = (5/8)
Therefore, the area of the sector formed by the central angle aobaoba, o, b is 5/8 of the area of the circle.