Answer:
Explanation:
a. To find the area of the sector, we can use the formula:
A = (θ/360)πr^2
where A is the area of the sector, θ is the central angle in degrees, r is the radius of the circle, and π is the constant pi.
In this case, the radius is 4 cm and the central angle is 45 degrees. Substituting these values into the formula, we get:
A = (45/360)π(4^2)
A = (1/8)π(16)
A = 2π
Therefore, the area of the sector is 2π square cm.
b. To find the area of the segment of a circle, we need to subtract the area of the triangle formed by the two radii and the chord from the area of the sector.
The central angle of the sector is 45 degrees, so the angle between the chord and one of the radii is 22.5 degrees. We can use trigonometry to find the length of the chord:
cos(22.5) = adjacent/hypotenuse
cos(22.5) = x/4
x = 4cos(22.5)
So the length of the chord is approximately 3.54 cm (rounded to two decimal places).
The area of the triangle can be found using the formula:
A = (1/2)bh
where b is the length of the base (which is the chord) and h is the height (which is the distance from the midpoint of the chord to the center of the circle). The height is equal to the radius minus half the length of the chord:
h = 4 - (3.54/2)
h = 1.23 (rounded to two decimal places)
Substituting the values of b and h, we get:
A = (1/2)(3.54)(1.23)
A = 2.17 (rounded to two decimal places)
So the area of the triangle is approximately 2.17 square cm.
Finally, we can find the area of the segment by subtracting the area of the triangle from the area of the sector:
Area of segment = Area of sector - Area of triangle
Area of segment = 2π - 2.17
Area of segment = 0.85 (rounded to two decimal places)
Therefore, the area of the segment of the circle is approximately 0.85 square cm.