Question

given: ab is the height of the cone.
a plane, parallel to the base of a cone intersects the cone at the midpoint between points a and b. determine the area of the cross - section formed by the intersection of the plane and the cone. leave your answer in terms of pi.

Explanation:

Step1: Identify total cone height

The total height $AB = 20$ in.

Step2: Find cross-section height

The plane cuts at the midpoint, so the distance from the apex $A$ to the plane is $\frac{20}{2}=10$ in.

Step3: Find cross-section radius

By similar cones, the ratio of radii equals the ratio of heights. Let $r$ be the cross-section radius, base radius $R=6$ in.
$\frac{r}{R}=\frac{10}{20}$
$r = 6 \times \frac{10}{20}=3$ in

Step4: Calculate cross-section area

The cross-section is a circle, area formula $A=\pi r^2$.
$A=\pi \times 3^2$

Answer:

$9\pi$ square inches