Prisms and Cylinders

The same face all the way along

A prism is a solid with two identical flat ends joined by flat faces, so that it keeps the same shape all the way along its length. Cut straight across it anywhere, at right angles to its length, and the face you uncover is the same shape and size as the ends. That face is called the cross section.

A cuboid is a prism whose cross section is a rectangle. A prism whose ends are triangles is a triangular prism.

Wherever this prism is cut straight across, the face uncovered is the same rectangle, 4 by 3.

Cross section times length

The cross section of this prism is a rectangle 4 units by 3 units, so its area is 4 × 3 = 12 square units. Cut the prism into slices 1 unit thick. Each slice is a layer 1 unit deep on a face of 12 squares, so it holds 12 unit cubes.

The prism is 5 units long, so it is 5 of those slices: 12 × 5 = 60 cubic units. The volume of a prism is the area of its cross section times its length.

The cross section is 12 and the length is 5: 12 × 5 = 60.

The same prism stood on one end, so its cross section is the base. Every unit of its length adds another layer of the same 12 cubes, and 5 layers make 12 × 5 = 60.

Any cross section

The cross section does not have to be a rectangle. When it is a triangle, or any other flat shape, a slice 1 unit thick is not a neat set of whole cubes, but it still holds as many cubic units as its face holds square units: each square and each part of a square on the face, 1 unit deep. So the rule is the same.

A triangular prism has ends that are triangles with a base of 4 cm and a height of 3 cm, and it is 10 cm long. Its cross section is ½ × 4 × 3 = 6 cm², so its volume is 6 × 10 = 60 cm³.

The volume of a cuboid, length × width × height, is the same rule: width × height is the area of the end, and the length counts the slices.

The cylinder

A cylinder has two identical circular ends, and every cut straight across it uncovers the same circle. So a cylinder works like a prism. Its cross section is a circle of radius r, with area $\pi r^2$πr², and its length is its height, h. The volume of a cylinder is $\pi r^2\times h$πr² × h, written $\pi r^{2}h$πr²h.

A can has a radius of 3 cm and a height of 10 cm. Its cross section is $\pi \times 3^2=9\pi$π × 3² = 9π cm², so its volume is $9\pi \times 10=90\pi$9π × 10 = 90π cm³, which is about 282.7 cm³.

Every cut across the cylinder uncovers the same circle, of radius r and area $\pi r^2$πr², and the cylinder is h long: its volume is $\pi r^2\times h$πr² × h.

Find the face that repeats

A prism is not always standing on its end, and its length is not always the longest edge. Before multiplying, find the face that stays the same all the way through the solid. That is the cross section, and the length is measured at right angles to it, whichever way the solid lies.

A swimming pool whose floor slopes from the shallow end to the deep end is a prism lying on its side. Every cut parallel to its side wall gives the same trapezium, so the side wall is the cross section and the width of the pool is the length.

Cubic meters and liters

Water is measured in liters. A liter is the volume of a cube 10 cm along each edge, which is 10 × 10 × 10 = 1000 cm³. A cubic meter is a cube 100 cm along each edge, which is 100 × 100 × 100 = 1,000,000 cm³, so it holds 1,000,000 ÷ 1000 = 1000 liters.

Two slips

Adding the cross section to the length, 12 + 5 = 17, counts no cubes at all. Each of the 5 slices holds 12 cubes, so multiply: 12 × 5 = 60.

Doubling the sum, 2 × (12 + 5) = 34, treats the two numbers like the sides of a rectangle whose perimeter is being found. A volume multiplies an area by a length, and its unit is a cubed one, such as cm³.