Geometry Cross Sections and Solids of Revolution Worksheet Solutions (PDF)

Every point of the cut is the same distance from the axis.

A circle

The cross section is a rectangle with width \(2 \cdot 5 = 10\) and height 12.

\(10 \cdot 12 = 120\)

Rectangle; area 120

A slanted plane section of a cylinder is an ellipse.

An ellipse

A square with side 6.

\(6^{2} = 36\)

Square; area 36

The rectangle has one side 6 and the other side is a face diagonal: \(6\sqrt{2}\).

\(6 \cdot 6\sqrt{2} = 36\sqrt{2} = 50.9\)

Rectangle; area 36√2, about 50.9

Each side is a face diagonal: \(6\sqrt{2}\). The triangle is equilateral.

\(\frac{\sqrt{3}}{4} \cdot (6\sqrt{2})^{2} = \frac{\sqrt{3}}{4} \cdot 72 = 18\sqrt{3} = 31.2\)

Equilateral triangle; area 18√3, about 31.2

Halfway up, the radius is half: \(2\) (similar triangles).

\(\pi \cdot 2^{2} = 4\pi = 12.6\)

Radius 2; area 4π, about 12.6

The cross section is a triangle with base \(2 \cdot 4 = 8\) and height 10.

\(\frac{1}{2} \cdot 8 \cdot 10 = 40\)

Triangle; area 40

\(r = \sqrt{13^{2} – 5^{2}} = \sqrt{144} = 12\).

\(\pi \cdot 12^{2} = 144\pi = 452.4\)

Circle of radius 12; area 144π, about 452.4

The cross section is a great circle with radius 13.

\(2\pi \cdot 13 = 26\pi\).

26π

The cross section is congruent to the 4 by 6 face.

\(4 \cdot 6 = 24\)

Rectangle; area 24

The cross section is a square similar to the base with scale factor \(\frac{1}{2}\): side 4.

\(4^{2} = 16\)

Square; area 16

For example: a square (cut parallel to a face), a rectangle (cut through two opposite edges), and a triangle (cut through three vertices). A hexagon is also possible.

Square, rectangle, triangle (also pentagon, hexagon)

The solid is a cylinder with radius 3 and height 8.

\(V = \pi \cdot 3^{2} \cdot 8 = 72\pi = 226.2\)

Cylinder; 72π, about 226.2

The solid is a cone with radius 6 and height 8.

\(V = \frac{1}{3}\pi \cdot 6^{2} \cdot 8 = 96\pi = 301.6\)

Cone; 96π, about 301.6

Now the radius is 8 and the height is 6.

\(V = \frac{1}{3}\pi \cdot 8^{2} \cdot 6 = 128\pi = 402.1\)

128π, about 402.1

The solid is a sphere of radius 5.

\(V = \frac{4}{3}\pi \cdot 5^{3} = \frac{500}{3}\pi = 523.6\)

Sphere; 500π/3, about 523.6

The rotation makes a hollow cylinder with inner radius 3, outer radius 5 and height 5.

\(V = \pi(5^{2} – 3^{2}) \cdot 5 = 80\pi = 251.3\)

Hollow cylinder; 80π, about 251.3

The face is a circle with radius 5.

\(\pi \cdot 5^{2} = 25\pi = 78.5\)

Circle; 78.5 square centimeters

Radius 9, height 12.

\(V = \frac{1}{3}\pi \cdot 81 \cdot 12 = 324\pi = 1017.9\)

1017.9 cubic units