Answer:
1. Area = 113.1 square units
2. Area = 28.27 square units
3. Area of Shaded = 22.27 square units
Step-by-step explanation:
1.
Area of the circle is given by the formula [tex]A=\pi r^2[/tex]
where A is the area and r is the radius
In our case the radius is 6 units, thus we have:
[tex]A=\pi r^2\\A=\pi (6)^2\\A=\pi *36\\A=113.1[/tex]
Area = 113.1 square units
2.
When we have the angle given in radians, the area of sector is given by the formula [tex]A=\frac{1}{2}r^2\theta[/tex]
Where [tex]\theta[/tex] is the central angle ( in our case [tex]\theta = \frac{\pi}{2}[/tex] and r is the radius (it is 6)
Plugging in these info into the formula we have area of sector:
[tex]A=\frac{1}{2}r^2\theta\\A=\frac{1}{2}(6)^2(\frac{\pi}{2})\\A=\frac{1}{2}*36*\frac{\pi}{2}\\A=28.27[/tex]
Area = 28.27 square units
3.
Area of shaded region = Area of Sector - Area of Triangle
We know area of sector is 28.27
Since the angle is 90 degrees, we have a right triangle, we can use the pythagorean theorem to find the height of the triangle, CE.
Thus
[tex]DC^2+CE^2=DE^2\\3^2+CE^2=5^2\\9+CE^2=25\\CE^2=25-9\\CE^2=16\\CE=4[/tex]
The area of triangle is [tex]A=\frac{1}{2}bh[/tex]
where b is the base (in our case it is DC = 3 ) and h is the height (in our case it is CE, which is 4). Plugging into the formula we have the area of triangle as:
[tex]A=\frac{1}{2}bh\\A=\frac{1}{2}(3)(4)\\A=6[/tex]
Area of Shaded = 28.27 - 6 = 22.27 square units
