Pi times diameter over 4, or (πd)/4, is equivalent to πr²/4, where r is the radius, because the diameter (d) is twice the radius (r). This also represents the area of a circle with a diameter of 'd' multiplied by 1/4.
Here's a breakdown:
- π (pi): A mathematical constant approximately equal to 3.14159.
- d (diameter): The distance across a circle through its center.
- (πd): This gives you a value related to the circumference of a circle. The circumference is actually πd.
- (πd)/4: This is the circumference divided by 4, or equivalently, pi times the radius squared divided by 4.
- r (radius): Half of the diameter of the circle.
Derivation:
Since d = 2r (diameter is twice the radius):
(πd)/4 = (π * 2r) / 4 = (2πr) / 4 = πr/2
(πd)/4 = π(d/2)² = πr²/4
Significance and Applications:
The expression (πd)/4 can be useful in various contexts:
- Area related calculations: Since the area of a circle is πr2, (πd)/4 isn't directly the area, but it is related by a factor of 4 to the area.
Example:
Let's say the diameter (d) of a circle is 4.
- (πd)/4 = (π * 4) / 4 = π
Therefore, if d = 4, (πd)/4 is simply π.
