The formula for the arc length of a quarter circle is πr/2, where 'r' represents the radius of the circle.
Understanding Quarter Circle Arc Length
The arc length of a quarter circle is precisely one-fourth of the total circumference of the full circle from which it originates. The circumference of a complete circle is the distance around its edge.
The Formulas for Arc Length
The arc length of a quarter circle can be expressed using either its radius (r) or its diameter (d).
-
Using the Radius (r):
The circumference of a full circle is given by the formula $C = 2πr$. Since a quarter circle's arc length is one-fourth of this, the formula becomes:
$L = \frac{1}{4} \times (2πr)$
$L = \frac{2πr}{4}$
$L = \frac{πr}{2}$ -
Using the Diameter (d):
Alternatively, the circumference of a full circle can be expressed as $C = πd$. Since the arc length is one-fourth of the circumference:
$L = \frac{1}{4} \times (πd)$
$L = \frac{πd}{4}$
It's important to remember that the diameter is twice the radius ($d = 2r$), so both formulas yield the same result.
Key Components of the Formula
- π (Pi): A mathematical constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter.
- r (Radius): The distance from the center of the circle to any point on its edge.
- d (Diameter): The distance across the circle passing through its center. It is equal to $2r$.
Practical Example
Let's calculate the arc length of a quarter circle with a radius of 7 units.
Given:
- Radius ($r$) = 7 units
Using the formula $L = \frac{πr}{2}$:
- Substitute the value of 'r' into the formula:
$L = \frac{π \times 7}{2}$ - Calculate the result:
$L = \frac{7π}{2}$
$L \approx \frac{7 \times 3.14159}{2}$
$L \approx \frac{21.99113}{2}$
$L \approx 10.995565$
The arc length of the quarter circle is approximately 11 units.
Arc Length Formulas at a Glance
For easy reference, here's a summary of arc length formulas for full and quarter circles:
| Circle Type | Radius (r) Formula | Diameter (d) Formula |
|---|---|---|
| Full Circle (Circumference) | $C = 2πr$ | $C = πd$ |
| Quarter Circle Arc Length | $L = \frac{πr}{2}$ | $L = \frac{πd}{4}$ |
Related Concepts
While the question focuses on arc length, it's helpful to distinguish it from the area of a quarter circle. The area of a quarter circle refers to the space enclosed by the two radii and the arc, given by $A = \frac{πr^2}{4}$. Arc length, on the other hand, measures only the curved boundary. Understanding circle geometry is fundamental to many mathematical and engineering applications. You can learn more about circle circumference and arc length concepts from educational resources like Khan Academy.
