The area A of a circle can be expressed in different ways depending on whether we are using the radius, diameter, or circumference.
1. Area as a function of radius (r)
The area A of a circle in terms of its radius r is given by the formula:
A = πr²
In this formula, π (pi) is a constant approximately equal to 3.14159. This means that if you know the radius of the circle, you can easily find the area by squaring the radius and then multiplying by pi.
2. Area as a function of diameter (d)
The diameter d is twice the radius, which can be expressed as:
d = 2r
From this, we can express the radius in terms of the diameter:
r = d/2
Substituting this into the area formula gives:
A = π(d/2)² = (π/4)d²
So, the area A in terms of the diameter is calculated by squaring the diameter and multiplying by π/4.
3. Area as a function of circumference (C)
The circumference C of a circle is related to the radius by the formula:
C = 2πr
We can manipulate this equation to express the radius in terms of the circumference:
r = C/(2π)
Substituting this expression for r into the area formula A gives:
A = π(C/(2π))² = C²/(4π)
Thus, the area A can also be calculated using the circumference by squaring the circumference and dividing by 4π.
In summary, the area of a circle can be expressed in three forms:
- As a function of radius: A = πr²
- As a function of diameter: A = (π/4)d²
- As a function of circumference: A = C²/(4π)