Math Core

Slice a loaf of bread and every slice is roughly the same shape. Slice an onion and the rings shrink as you move toward the end. Spin a paper triangle quickly on a pencil and you see a cone. This lesson connects flat figures and solids in both directions: cutting solids to get 2D shapes, and spinning 2D shapes to get solids.

Cross sections

Definition

Cross section

A cross section is the flat shape you get where a plane cuts through a solid.

The same solid can have very different cross sections depending on how the plane is tilted.

Solid	Parallel to the base	Perpendicular to the base	Tilted
Cylinder	circle	rectangle	ellipse (oval)
Cone	circle	triangle (through the apex)	ellipse, or a curve open at one end
Prism	copy of the base	rectangle	other polygons
Pyramid	smaller copy of the base	triangle (through the apex)	other polygons
Sphere	always a circle	always a circle	always a circle

A cube alone can produce a surprising variety. Planes parallel to a face give squares. A plane through two opposite edges gives a rectangle that is longer than it is wide. A plane that cuts off one corner gives a triangle. A plane through the center, perpendicular to a long diagonal of the cube, cuts all six faces and gives a regular hexagon. Since each side of a cross section lies in a different face, a cube's cross section can never have more than six sides.

A plane through three corners next to one vertex slices a triangle off the cube. All three sides of the triangle are face diagonals, so it is equilateral.

Worked example: A triangle sliced from a cube

A cube has edges of 4 cm. A plane passes through the three corners next to one vertex, as in the figure. Describe the cross section and find its perimeter.

Each side of the cross section runs diagonally across one face of the cube. A face diagonal of a square with side 4 is $4\sqrt{2}$ , by the $45^\circ$ - $45^\circ$ - $90^\circ$ triangle rule. All three sides are equal, so the cross section is an equilateral triangle with side $4\sqrt{2}$ .

\text{Perimeter} = 3 \cdot 4\sqrt{2} = 12\sqrt{2} \approx 17.0 \text{ cm}

Slices parallel to the base

A plane parallel to the base of a pyramid or cone cuts off a smaller pyramid or cone that is similar to the original. If the cut is a distance $d$ from the apex and the whole height is $h$ , every length in the cross section is scaled by $\frac{d}{h}$ , so its area is scaled by $\left(\frac{d}{h}\right)^2$ .

A plane parallel to the base cuts a square pyramid in a smaller square.

Worked example: Area of a slice

A square pyramid has base edges of 10 in and a height of 15 in. A plane parallel to the base cuts the pyramid 9 in above the base. Find the area of the cross section.

The cut is $15 - 9 = 6$ in below the apex, so the scale factor is $\tfrac{6}{15} = \tfrac{2}{5}$ . The cross section is a square with side $10 \cdot \tfrac{2}{5} = 4$ in:

A = 4^2 = 16 \text{ in}^2

As a check, the base area is 100, and $100 \cdot \left(\tfrac{2}{5}\right)^2 = 100 \cdot \tfrac{4}{25} = 16$ . ✓

Common mistake

Measure $d$ from the apex, not from the base. In the last example, using $\tfrac{9}{15}$ gives a side of 6 and an area of 36, which is the slice 9 in below the apex, a different cut entirely.

Solids of revolution

A solid of revolution is formed by rotating a flat region all the way around a line, called the axis of rotation. Each point of the region traces a circle around the axis, so every cross section perpendicular to the axis is a circle (or a ring).

A right triangle spun around its vertical leg (the dashed axis) sweeps out a cone with radius 3 and height 5.

Common solids of revolution

A rectangle rotated around one of its sides makes a cylinder. The side on the axis becomes the height; the adjacent side becomes the radius.
A right triangle rotated around one of its legs makes a cone. The leg on the axis is the height; the other leg is the radius.
A semicircle rotated around its diameter makes a sphere.
A region that doesn't touch the axis leaves a hole down the middle.

A rectangle spun around one side sweeps out a cylinder.

To find the volume of a solid of revolution, first identify the solid, then read off its radius and height from the flat figure. The radius is always a distance from the axis.

Worked example: Same triangle, two axes

A right triangle has legs of 3 and 5. Find the volume of the solid formed by rotating it around (a) the leg of length 5 and (b) the leg of length 3.

a) The 5 leg lies on the axis, so it's the height. The 3 leg sweeps around as the radius. This is the cone in the figure:

V = \tfrac{1}{3}\pi(3)^2(5) = 15\pi

b) Now the height is 3 and the radius is 5:

V = \tfrac{1}{3}\pi(5)^2(3) = 25\pi

Same triangle, different volumes. The radius is squared, so the longer leg adds more volume when it's the radius.

Worked example: A solid with a hole

The rectangle with vertices $(1, 0)$ , $(3, 0)$ , $(3, 4)$ and $(1, 4)$ is rotated around the $y$ -axis. Describe the solid and find its volume.

The rectangle doesn't touch the axis. Its right side, 3 units away, sweeps out a cylinder of radius 3. Its left side, 1 unit away, sweeps out the edge of a hole of radius 1. The solid is a thick tube, 4 units tall.

V = \pi(3)^2(4) - \pi(1)^2(4) = 36\pi - 4\pi = 32\pi

Tip

Sketch the flat region and its mirror image across the axis. Together they show a cross section of the solid through the axis, which makes the radius and height easy to see.

Practice

Practice 1

A cone is cut by a plane parallel to its base (not through the apex). What shape is the cross section?

Practice 2

A 3 cm by 7 cm rectangle is rotated around its 7 cm side. Find the volume of the solid formed, in cubic centimeters, in terms of $\pi$ .