Volume and surface area
Cross-sections and projections
A common AMC 12 move is to slice a 3D solid with a plane and reduce the problem to 2D geometry on that cross-section. Some cross-sections are worth recognizing immediately: a plane through a sphere's center gives a great circle; a plane through a cube parallel to a face gives a square; a plane cutting a cube diagonally can give a hexagon.
Similarly, an orthogonal projection of a 3D figure onto a plane (imagine shining a light straight through it) is often easier to reason about than the solid itself, especially for volume-comparison or shadow-area problems.
Worked example
Problem: Find the radius of the largest sphere that fits inside a cube with side length 6.
Solution: The largest inscribed sphere is tangent to all six faces, so its diameter equals the cube's side length:
Practice problems
1.Find the volume of a cylinder with radius 3 and height 10.
2.Find the surface area of a sphere with radius 5.
3.A cone has radius 6 and height 8. Find its volume and its slant height.
4.A rectangular prism has dimensions 4, 5, 6. Find its total surface area.