A circular sector has an 6.26-inch radius and a 10.84-inch arc length. There is another sector that has the same area and the same perimeter. What are its measurements?
To find the measurements of the sector with the same area and perimeter, we need to determine its radius and arc length.
Let's start by finding the measure of the central angle of the given sector.
We know that the circumference of a full circle is given by 2πr, where r is the radius. Therefore, the circumference of the circle with a radius of 6.26 inches is:
C1 = 2π * 6.26
= 12.52π inches
The ratio of the arc length to the circumference of the full circle gives us the ratio of the central angle to 360°. Therefore, the central angle of the given sector can be found as:
Central angle 1 = (10.84 / C1) * 360°
= (10.84 / (12.52π)) * 360°
Now, let's find the area of the given sector.
The area of a sector is given by (θ/360°) * πr^2, where θ is the central angle and r is the radius. We can plug in the values to find the area of the given sector:
Area 1 = (Central angle 1 / 360°) * π * (6.26)^2
Now, since we want the other sector to have the same area, we can set the area of the other sector equal to the area of the given sector:
Area 1 = Area 2
(Central angle 1 / 360°) * π * (6.26)^2 = (Central angle 2 / 360°) * π * (r2)^2
We know that the perimeter of a sector is the sum of its arc length and the length of its two radii. Therefore, the perimeter of the given sector can be calculated as:
Perimeter 1 = 2 * 6.26 + 10.84
= 12.52 + 10.84
= 23.36 inches
Now, we want the other sector to have the same perimeter, so we can set the perimeter of the other sector equal to the perimeter of the given sector:
Perimeter 1 = Perimeter 2
23.36 = 2 * r2 + Arc length 2
We have two equations:
1. (Central angle 1 / 360°) * π * (6.26)^2 = (Central angle 2 / 360°) * π * (r2)^2
2. 23.36 = 2 * r2 + Arc length 2
Simplifying these equations will give us the measurements of the other sector (r2 and Arc length 2).
To find the measurements of the other circular sector with the same area and perimeter, we need to find the radius and arc length of the second sector.
Here's how you can calculate it:
1. Calculate the area of the first sector.
- The formula for the area of a circular sector is A = (1/2) * r^2 * θ, where A is the area, r is the radius, and θ is the central angle in radians.
- Let's assume the central angle of the first sector is θ1. We need to find θ1 using the given arc length.
- The formula for the arc length, s, in a circular sector is s = r * θ. Rearranging the formula, we get θ = s / r.
- Substitute the given values: θ1 = 10.84 inches / 6.26 inches ≈ 1.731 radians.
- Calculate the area of the first sector: A1 = (1/2) * (6.26 inches)^2 * 1.731 ≈ 32.193 square inches.
2. Calculate the radius of the second sector.
- Since the two sectors have the same area, the area of the second sector, A2, should also be 32.193 square inches.
- Use the formula for the area of a circular sector to find the radius of the second sector.
- Rearrange the area formula to solve for r: r = sqrt(2 * A / θ), where A is the area and θ is the central angle.
- Substitute the known values: r2 = sqrt(2 * 32.193 square inches / 1.731 radians) ≈ 7.149 inches.
3. Calculate the arc length of the second sector.
- Since the two sectors have the same perimeter, the arc length of the second sector, s2, should be equal to the arc length of the first sector, which is 10.84 inches.
- Use the formula for the arc length of a circular sector to find the central angle of the second sector.
- Rearrange the arc length formula to solve for θ: θ = s / r, where s is the arc length and r is the radius.
- Substitute the known values: θ2 = 10.84 inches / 7.149 inches ≈ 1.516 radians.
4. Summarize the measurements of the second sector.
- The radius of the second sector is approximately 7.149 inches.
- The arc length of the second sector is approximately 10.84 inches.
- The central angle of the second sector is approximately 1.516 radians.