Think about slicing a radish with a knife. Lay it on its side and cut straight across, and you get a round circular cross-section; tilt the cut at an angle, and you get a long, stretched-out ellipse. When you cut a solid with a plane like this, the shape that appears where you cut is called a cross-section. The key idea here is that the same solid's cross-section shape changes completely depending on where and at what angle you cut it.
The most interesting example is a cone. Cut a cone parallel to its base and you always get a circle. Tilt the cut slightly and it becomes an ellipse; tilt it further, and the moment the cut becomes exactly parallel to the cone's slant side, the cross-section turns into an open-ended parabola. Cut it even steeper, almost vertically, and you get one piece each from the upper and lower halves of a double cone (two cones joined tip to tip), producing two separate hyperbola-shaped cross-sections. Circles, ellipses, parabolas, and hyperbolas together are called conic sections — true to the name, every one of these shapes can be explained entirely by the single angle you cut a cone at.
Solids made entirely of flat faces, like prisms and pyramids, behave a bit differently. Cut a cube parallel to its base and you always get a square, but cut it at an angle and you can get a rectangle, a parallelogram, or a trapezoid — there's even a surprising case where cutting exactly perpendicular to the cube's main diagonal gives a regular hexagon. The number of sides and the shape of the cross-section are determined by how many faces the cutting plane passes through, and in what order.
A cylinder combines the properties of both a cone and a cube. Cut it parallel to its base and you get a circle; cut it straight down, parallel to its side, and you get a rectangle; cut it at an angle in between, and you get an ellipse. Prisms and pyramids work similarly — a cross-section parallel to the base is always a shape similar to the base, and tilting the angle can produce more complex cross-sections, like a triangle or a pentagon.
This principle is used well beyond the textbook too. CT and MRI scans at a hospital work by continually slicing thin planes through the body, taking hundreds of cross-sectional images, and then stacking them back together to reconstruct the 3D structure. Drawing a "cross-sectional diagram" of an object in architecture or mechanical design is exactly the same idea — slicing a solid with a plane to check its internal structure.
On our activity page, you can rotate five solids yourself in 3D — a cone, cylinder, cube, triangular prism, and square pyramid — and move the cutting angle (θ), direction (φ), and position with sliders to see the cross-section change in real time. You can also press pre-set buttons to jump straight to special cross-sections, like "the spot that gives a circle" or "the spot that gives a parabola."