Surface Area Of A Rectangular Prism Calculator

What this calculator does

The surface area of a rectangular prism, a box, is the total area of its six rectangular faces, the whole of its outside. The wrapping paper for a parcel, the material to line a chest, the paint for the walls, floor and ceiling of a room. This works it out from the box's width, length and height.

Type the three in and you have the area of the outside.

Using the calculator

1. Type the width, the length and the height.
2. Press Calculate.

All three values have to be positive. There is no unit setting, so the result is a plain number in square units of whatever unit you used.

The formula | surface area = 2 × (wl + lh + wh)

The surface area of a rectangular prism is:

surface area = 2 × (width × length + length × height + width × height)

A box has six faces, but they come in three matching pairs, so there are really only three different rectangles to work out. Add those three, then double, because each one appears twice.

Three matching pairs of faces

The clean way to keep this straight is to think in pairs. The top and bottom are identical, each width × length. The front and back are identical, each length × height. The two sides are identical, each width × height. Work out one of each, the three distinct rectangles, add them, and double. That doubling is what turns three faces into all six. Unfold the box flat into a net of six rectangles and you would add the same six areas, arriving at the same total.

The wrapping-paper number

This is the "how much to cover it" figure. Wrapping a present, lining a drawer, cladding a crate, or painting the inside faces of a room all come down to the surface area: enough material to cover every face once. If you can cover all six faces, you have it.

Surface area or volume?

A box has both, and they measure different things. The surface area, 2 × (wl + lh + wh), is the skin around the outside, in square units. The volume, length × width × height, is the space inside, in cubic units. The skin covers the box; the space fills it. For the space inside, see the volume of a rectangular prism calculator.

When the box is a cube

If the width, length and height are all equal, the box is a cube, and 2 × (wl + lh + wh) becomes 2 × (3 × edge²), which is 6 × edge². So the cube's tidy formula is just this one with all three sides the same. For that case, see the surface area of a cube calculator, and for cylinders, cones and spheres, the surface area calculator.

A worked example | 3 by 4 by 5

Say the box is 3 wide, 4 long and 5 tall.

1. The three distinct faces: 3 × 4 = 12, 4 × 5 = 20, and 3 × 5 = 15.
2. Add them: 12 + 20 + 15 = 47.
3. Double it, for the matching pairs: 2 × 47 = 94.

So the surface area is 94 square units. The same box's volume, by contrast, is 3 × 4 × 5 = 60 cubic units, a different measurement of a different thing.

Questions people ask

What is the surface area of a 3 by 4 by 5 box?

It is 94 square units. Add the three distinct faces, 12 + 20 + 15 = 47, and double.

What is the formula?

Surface area = 2 × (width × length + length × height + width × height), since the six faces form three matching pairs.

Why three pairs?

Because opposite faces of a box are identical: top and bottom, front and back, and the two sides. You compute three faces and double.

What is the difference between surface area and volume?

Surface area is the outside skin, 2 × (wl + lh + wh), in square units. Volume is the inside space, length × width × height, in cubic units.

What if it is a cube?

Then all three sides are equal, and the formula collapses to 6 × edge². The surface area of a cube calculator does that from one number.

References

A note on the idea behind it. The surface area of a rectangular prism is the sum of its six rectangular faces, which fall into three identical pairs, so the area is 2 × (wl + lh + wh). It can equally be found by unfolding the box into a flat net of six rectangles and adding their areas. For further reading, see Cuboid.

1. The six faces of a rectangular prism in three matching pairs, giving a surface area of 2 × (wl + lh + wh).
2. The net of a rectangular prism, six rectangles whose total area is the surface area.