A sector of a circle is a region bound by an arc and the two radii that share the arc's endpoints. Suppose you have a dartboard that has a diameter of 20 in and it is divided into 20 congruent sectors. Find the area of one sector.
area of sector $= \frac{m \overline{A B}}{360^{\circ}} \bullet \pi r^2$
Part I: Find the central angle. (Hint: A circle has $360^{\circ}$.)
Part II: Use your answer from Part I to find the fraction of the circle that one sector will take up.
Part III: Use the fractional part from Part II with the area formula to find the area of one sector of the circle to the nearest tenth.
Answer (2)
Find the radius of the dartboard: r = 2 20 = 10 inches.
Calculate the central angle of one sector: θ = 20 36 0 ∘ = 1 8 ∘ .
Determine the fraction of the circle that one sector occupies: Fraction = 36 0 ∘ 1 8 ∘ = 0.05 .
Calculate the area of one sector: Area of sector = 0.05 × π ( 10 in ) 2 ≈ 15.7 in 2 .
15.7 in 2
Explanation
Find the radius First, we need to find the radius of the dartboard. The diameter is given as 20 inches, so the radius r is half of that: r = 2 d = 2 20 = 10 inches
Calculate the central angle Next, we calculate the central angle of one sector. Since the dartboard is divided into 20 congruent sectors, the central angle θ of each sector is: θ = 20 36 0 ∘ = 1 8 ∘
Determine the fraction of the circle Now, we determine the fraction of the circle that one sector occupies. This is the central angle divided by the total degrees in a circle: Fraction = 36 0 ∘ θ = 36 0 ∘ 1 8 ∘ = 0.05
Calculate the area of the circle We calculate the area of the entire circle using the formula A = π r 2 :
A = π ( 10 in ) 2 = 100 π in 2
Calculate the area of the sector To find the area of one sector, we multiply the area of the circle by the fraction of the circle that one sector occupies: Area of sector = Fraction × A = 0.05 × 100 π in 2 = 5 π in 2
Round to the nearest tenth Finally, we round the area of one sector to the nearest tenth: Area of sector ≈ 5 × 3.14159 ≈ 15.70795 ≈ 15.7 in 2
Examples
Imagine you're designing a sprinkler system for a circular garden divided into equal sections. Knowing how to calculate the area of a sector helps you determine the coverage area of each sprinkler, ensuring even watering and optimal plant growth. This is also applicable when calculating areas of slices of a pie or pizza, ensuring everyone gets a fair share!
The area of one sector of the dartboard is approximately 15.7 square inches. This is derived by calculating the radius, central angle, and applying the area formula for a circle. Each sector occupies a fraction of the circle, which is then used to find the sector's area.
;
