Cross-Sections of 3D Figures Math Example 1

Solution

1. 1
   Step 1: Visualize a right circular cone: a circular base tapering to a point (apex) at the top.
2. 2
   Step 2: A horizontal cut parallel to the base produces a cross-section that is a circle (since every horizontal level of a cone is circular by symmetry).
3. 3
   Step 3: A cut at the midpoint of the height means the cut is at half the total height. By similar triangles, the radius of the cross-section equals half the base radius.
4. 4
   Step 4: Therefore the cross-section is a circle with radius $= \frac{r}{2}$, and its area $= \pi\left(\frac{r}{2}\right)^2 = \frac{\pi r^2}{4}$, which is one-quarter of the base area.