The surface area of a cube and a cuboid

Here is a drawing of an aquarium. We want to figure out how much glass it would take to make this aquarium. From the front, the aquarium looks like a quadrilateral, with four corners and four right angles: a rectangle. Looking from the side, it also looks like a quadrilateral, but with four equally long sides and four right angles: a square. And from above, the aquarium looks like a rectangle.

The aquarium is like a box, a block where all angles are right angles: it's a cuboid. In a cuboid, the front and back sides are equal in size.. ..the right and left sides are equal in size.. ..and the top and bottom sides are the same size. So all three pairs of opposite sides have the same area. To calculate the total area of a cuboid therefore, we only need to calculate the areas of three of the sides, and then multiply them by two. Let’s start with the glass for the front side.

The width should be 1 metre, And the height 0,6 metres. The area is equal to the base times the height. 1 metre times 0,6 metres is 0,6 square metres. That's this rectangle’s area. Since the front and back sides are the same size, we need 0,6 square metres times two.

That’s 1,2 square metres of glass to make the front and back. Now let’s calculate the area of the side of the aquarium. The height is 0,6 metres and the width is also 0,6 metres. The area is 0,6 meters by 0,6 metres, and that’s 0,36 square metres. For the right and left sides therefore, we need 0,36 times 2 square metres of glass: that’s 0,72 square metres.

The top side of the aquarium has the dimensions 1 metre times 0,6 metres. So this area is 0,6 square metres. The upper and lower sides are equal, and the total area for those sides is 0,6 square metres times 2: that is 1,2 square metres. The total area of the sides, that is, the total surface area of the cuboid, is 1,2 plus 0,72 plus 1,2 square metres. It is 3,12 square metres.

It takes at least 3,12 square metres of glass to make this aquarium. Here we have another drawing of an aquarium where the six sides are equally large squares. It's a cube. The width and height of the cube is 0,6 metres. In order to calculate this cube’s total surface area, we only need to calculate the area of one of the sides: 0,6 metres times 0,6 metres.

This is 0,36 square metres. All six sides have the same area, and therefore the cube's total surface area is 0,36 square metres times 6. That’s 2,16 square metres. It takes at least 2,16 square metres of glass to make this aquarium. Let’s start!

Oh. Guess I have to buy some more glass…