a circle of radius, r, with a sector of 45° shaded.
(a) Find the length of the arc of the colored sector in the figure when r = 7 cm
(b) Find the area A of the sector when r = 7 cm
45º / 360º = 1/8
(a) 1/8 of the circumference ... 1/8 * 2 * π * r
(b) 1/8 of the circle area ... 1/8 * π * r^2
To find the length of the arc of the colored sector in the figure, we can use the formula:
Arc Length = (Angle/360) * 2 * π * r
(a) Given that the sector has an angle of 45° and a radius of 7 cm, we can substitute these values into the formula:
Arc Length = (45/360) * 2 * π * 7
Simplifying, we get:
Arc Length = (1/8) * 2 * π * 7
Arc Length = 1/4 * π * 7
Arc Length = (π/4) * 7
Arc Length = 7π/4
Therefore, the length of the arc of the colored sector is 7π/4 cm.
To find the area of the sector, we can use the formula:
Area = (Angle/360) * π * r^2
(b) Given that the sector has an angle of 45° and a radius of 7 cm, we can substitute these values into the formula:
Area = (45/360) * π * (7)^2
Simplifying, we get:
Area = (1/8) * π * 49
Area = (π/8) * 49
Therefore, the area of the sector is (49π)/8 cm².
To find the length of the arc of the colored sector, you need to calculate the circumference of the circle and then multiply it by the ratio of the angle formed by the sector to a full circle (360°).
(a) The circumference of a circle can be found using the formula C = 2πr, where r is the radius of the circle. So, in this case, when r = 7 cm, C = 2π(7) ≈ 14π cm.
The angle of the sector is given as 45°. To calculate the length of the arc, you need to find the ratio of the angle of the sector to a full circle (360°). This can be calculated as 45°/360° = 1/8.
Now, multiply the circumference by this ratio to find the length of the arc of the colored sector: Arc Length = (1/8) * 14π cm ≈ π/2 cm.
Therefore, when r = 7 cm, the length of the arc of the colored sector is approximately π/2 cm.
(b) The area of a sector can be calculated using the formula A = (θ/360°) * π * r^2, where θ is the angle of the sector in degrees and r is the radius of the circle.
Using the same values as in part (a), we can calculate the area of the sector when r = 7 cm: A = (45°/360°) * π * (7 cm)^2 = (1/8) * π * 49 cm^2 ≈ 6.875π cm^2.
Therefore, when r = 7 cm, the area of the sector is approximately 6.875π cm^2.