A 50% diameter reduction in a vessel is equivalent to what percentage area reduction?

A 50% diameter reduction in a vessel results in a significantly larger reduction in cross-sectional area due to the mathematical relationship between diameter and area. The area of a circle is calculated using the formula ( A = \pi r^2 ), where ( r ) is the radius of the vessel. When the diameter is reduced by 50%, the new diameter is half the original diameter, which means the new radius is also half the original radius.

The area of the vessel with the original diameter can be expressed as:

( A_{original} = \pi r^2 )

With a 50% reduction in diameter, the new radius is:

( r_{new} = \frac{1}{2} r )

Thus, the area of the vessel after the diameter reduction is:

( A_{new} = \pi (r_{new})^2 = \pi \left(\frac{1}{2} r\right)^2 = \pi \left(\frac{1}{4} r^2\right) = \frac{1}{4} A_{original} )

This indicates that the new area is 25% of the original area. Therefore, the area reduction can be calculated as follows