How do you calculate the cross-sectional area of a circle given its diameter?

To determine the cross-sectional area of a circle from its diameter, we can use the formula for the area of a circle, which is derived from the radius. The area ( A ) is calculated using the equation:

[

A = \pi r^2

]

where ( r ) is the radius of the circle. The radius is half of the diameter ( D ), which can be expressed as:

[

r = \frac{D}{2}

]

Substituting this expression for ( r ) into the area formula gives:

[

A = \pi \left(\frac{D}{2}\right)^2

]

This simplifies to:

[

A = \pi \left(\frac{D^2}{4}\right) = \frac{\pi D^2}{4}

]

Therefore, the correct formula for calculating the cross-sectional area of a circle when given its diameter is indeed ( A = \frac{\pi D^2}{4} ). This correctly captures the relationship between diameter and area for a circle.