Final Answer:
The area of a sector of a circle of diameter 8 whose central angle is 45 degrees is 2π square units.
Step-by-step explanation:
To solve this problem, we need to find the area of a sector of a circle. First, we need to identify some key pieces of information:
- The diameter of the circle is 8.
- The central angle of the sector is 45 degrees.
Let's go through the calculations step by step:
Step 1: Calculate the radius of the circle
Since the diameter (d) is 8, we divide it by 2 to find the radius (r).
r = d/2 = 8/2 = 4
Step 2: Calculate the area of the whole circle
The area (A) of a circle is calculated using the formula:
A = πr²
Plugging in our radius:
A = π(4)² = π*16
A = 16π
Step 3: Calculate the area of the sector
The area of the sector depends on the fraction of the circle it represents. Since the angle is 45 degrees, it represents 45/360 = 1/8 of the circle, since there are 360 degrees in a full circle.
We can calculate the area of the sector (A_sector) by multiplying the area of the whole circle by the fraction represented by the sector:
A_sector = A * (central angle/360)
Plugging in the values:
A_sector = 16π*1/8
A_sector = 2π
Hence, the area of the sector with a central angle of 45 degrees in a circle with a diameter of 8 is 2π square units.