SOLUTION: If the area of a circle(A) has an area of a sector of the circle(a),an arc length(I) and circumference(C), deduce a formula for the arc length(I),in terms of the area of the sector

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Question 1167256: If the area of a circle(A) has an area of a sector of the circle(a),an arc length(I) and circumference(C), deduce a formula for the arc length(I),in terms of the area of the sector and the radius of the circle(r). Hence calculate arc length of the sector of a circle with radius 5cm and area 25cm^2.
Answer by CPhill(1987) (Show Source):
You can put this solution on YOUR website!
The formula for the arc length ($l$) in terms of the area of the sector ($a$) and the radius of the circle ($r$) is:
$$l = \frac{2a}{r}$$
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## Deduction of the Formula
This derivation uses the ratio between the sector's properties and the circle's total properties, which are proportional to the central angle ($\theta$).
### 1. Sector Area and Circle Area Relationship
The ratio of the sector's area ($a$) to the total circle's area ($A$) is equal to the ratio of the sector's central angle ($\theta$) to the total angle ($2\pi$ radians):
$$\frac{\text{Sector Area}}{\text{Circle Area}} = \frac{\text{Arc Length}}{\text{Circumference}} = \frac{\text{Sector Angle}}{\text{Full Angle}}$$
Using the area relationship:
$$\frac{a}{A} = \frac{\theta}{2\pi}$$
Since $A = \pi r^2$, we have:
$$\frac{a}{\pi r^2} = \frac{\theta}{2\pi}$$
Solving for the angle $\theta$:
$$\theta = \frac{2\pi a}{\pi r^2} = \frac{2a}{r^2}$$
(Note: This formula gives $\theta$ in radians.)
### 2. Arc Length Formula
The arc length ($l$) is defined by the formula:
$$l = r\theta$$
Substitute the expression for $\theta$ into the arc length formula:
$$l = r \left(\frac{2a}{r^2}\right)$$
Simplify by canceling one $r$:
$$l = \frac{2a}{r}$$
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## Calculation of Arc Length
Using the deduced formula, we can calculate the arc length with the given values:
* Radius ($r$): $5 \text{ cm}$
* Area of the sector ($a$): $25 \text{ cm}^2$
$$l = \frac{2a}{r}$$
$$l = \frac{2 \times 25 \text{ cm}^2}{5 \text{ cm}}$$
$$l = \frac{50 \text{ cm}^2}{5 \text{ cm}}$$
$$l = 10 \text{ cm}$$
The arc length of the sector is **$10 \text{ cm}$**.