Before getting to the point of how to find the surface area of a rectangular prism, we will give a definition of the prism and learn how to build a prism, as well as study its basic properties. We’ll also learn what a straight prism is and what the height of the prism is. We will recall the concept of a perpendicular line and surface in order to formulate a definition of direct and oblique prism.

Prism is a geometric figure, a polyhedron with two equal and parallel faces that called bases and have the shape of a polygon. Other faces have a common base side and are called the sides.

Even in ancient times there were two ways of determining the geometric concepts. The first way led from figures of a higher order to the figures of a lower order. Euclid in particular agreed with this point of view. He defined a surface as a boundary of a body, a line as a boundary surface, and the ends of the line as points. The second way is, on the contrary, led from the figures of the lower dimensions to higher figures. The line is formed by the movement of a point similarly the surface is formed with the lines, etc.

Heron of Alexandria was one of the first, who connected these two points of view. He said that that the body is limited by the surface and at the same time it can be considered as something formed by the surface movement. In books on geometry that appeared later over the centuries, sometimes one point of view was accepted, sometimes – the other, and at times even two of the points were agreed with.

Just as a triangle in the sense of Euclid is not empty, i.e. triangle is a part of the surface bounded by the three non-competitive (i.e. not intersected at any point) segments, the polyhedron is not empty either. It is not hollow, but filled with part of the space. In ancient mathematics, however, the concept of abstract space did not exist. Euclid defines prism as a bodily shape, enclosed between two equal and parallel planes (bases) and with lateral faces – parallelograms. In order for this definition to be quite correct, it would be advisable to prove that the planes passing through the pair of non-parallel sides of the base intersect along parallel lines. Euclid uses the term «plane» both in the broad sense (considering it indefinitely extended in all directions) and in the sense of ultimate and limited part of it, in particular a face, which is similar to Euclid’s use of the term «direct» (in the broad sense – an infinite straight line and in a narrow sense – a cut).

In the 18th century, Taylor defined prism as a polyhedron whose all faces, except for two, are parallel to one single line.

The Babylonian and Egyptian monuments of architecture include such geometric figures as cube, parallelepiped, and prism. The most important task of the Egyptian and Babylonian geometry was to determine the volume of the various spatial figures and surface area of a rectangular prism. This task responded to the need to build houses, palaces, temples, and other buildings.

Part geometry, which studies the properties of a cube, prism, parallelepiped and other geometric shapes and three-dimensional shapes, have long called stereometry. The word is of Greek origin and it can be found even in the works of a famous ancient Greek philosopher Aristotle. Stereometry occurred later than planimetry.

Euclid defines the prism as a bodily (i.e. spatial) figure enclosed between the planes, of which two opposite are equal and parallel, while the rest are parallelograms. Here, as in many other places, Euclid uses the term «plane» not in the sense of an infinitely extended plane, but in the sense of a limited part of it, a face. The same as direct means a line segment. The term prism is of Greek origin and literally means the body that was intercepted.

Rectangular prism is the name used to refer to the hex object, resembling an ordinary box. Imagine a brick or a box from under shoes, and you’ll have a clear understanding of how the rectangular prism looks like.

The surface area of a rectangular prism is the total area of all its facets. The calculation of the surface area of a rectangular prism is similar to the question’s answer «how much paper is needed to wrap the box?»

In order to calculate the surface area of a rectangular prism, mark the length, width, and height of the prism. Each rectangular prism has length, width, and height. Draw the prism and mark its various edges with letters l, w, and h.

If you are not sure how to mark each rib, select any angle of a prism. Mark three edges with the corresponding letters coming out of this corner.

Let’s say, for example, the base of the prism is a rectangle 3 by 4 centimeters, and a prism height is 5 centimeters. Since the long side of the base equals to 4 centimeters, we get l = 4, w = 3, and h = 5.

Take a look at the six faces of the prism. To cover the entire surface of the figure, it is necessary to paint over all of its six faces. Imagine each face, or take a box from under the shoes and take a look at it:

- There is one face located above and one face located below, and they have the same size.
- There is one face in front and one face behind, and their areas are equal.
- There is also one face on the left and one face on the right, they are also the same.

If it is difficult for you to imagine this picture, cut around the edges of the box and expand it.

Now, let’s try to find the surface area of a rectangular prism. At first, let’s find out the area of one face, namely the bottom. This face, like the rest faces, is a rectangle. One side of this rectangle was marked by you as the length, and the second was marked as the width. To find the area of a rectangle, you need to multiply the length of the two sides of a rectangular. Thus, the area (of the lower face) = the length multiplied by the width = lw.

If we take the values we proposed above, for the area of the base of the prism we get 4 cm x 3 cm = 12 square centimeters.

The second step on the way to finding the surface area of a rectangular prism is calculating the area of the upper face. As you we said above, the top and bottom faces have the same area. Thus, the area of the upper face is also equal to lw. In our example, the upper face area is 12 square centimeters.

Next, define the area of the front and rear faces. The front face’s sides are the width and the height. Thus, the area of the front face = width multiplied by the height = wh. The area of the rear face is also wh.

In our example, w = 3 cm and h = 5 cm, so the area of the front face is 3 cm x 5 cm = 15 square centimeters. The area of the rear face is also equal to 15 square centimeters.

Now, let’s find the area of the left and right faces. Their areas are also the same that is why it is enough to find the area of the left side. It is limited in length and height of the prism. Thus, the area of the left face equals lh the area of the right face is also equal to lh.

In our example, l = 4 cm and h = 5 cm, so the area of the left side = 4 cm x 5 cm = 20 square centimeters. The area of the right side is also equal to 20 square centimeters.

Now we will summarize the areas we found to calculate surface area of a rectangular prism. We have found the area of each of the six faces of the prism. By summarizing them together, we can find the surface area of a rectangular prism: lw + lw + wh + wh + lh + lh. This formula can be used to calculate the area of the surface for any a rectangular prism.

Let’s go back to our example and find the surface area of a rectangular prism: 12 + 12 + 15 + 15 + 20 + 20 = 94 square centimeters.

We can simplify the formula we mentioned above in order to faster find surface area of a rectangular prism. We already know how to calculate the surface area of a rectangular prism. However, it can be done faster if we do simple algebraic transformations. Let's start with the equation we mentioned above: the surface area of a rectangular prism = lw + lw + wh + wh + lh + lh. Combining the similar summand, we obtain: the surface area of a rectangular prism = 2lw + 2wh + 2lh.

Impose a common factor 2 out of the brackets. If you can lay out factoring algebraic equation, this formula can be simplified as follows: area = 2lw + 2wh + 2lh = 2 (lw + wh + lh).

Now, you can verify this formula with our example. Go back to the values we used above: the length is 4, the width is 3, and the height is 5. Substitute these numbers into our formula: surface area of a rectangular prism = 2 (lw + wh + lh) = 2 x (lw + wh + lh) = 2 x (4x3 + 3x5 + 4x5) = 2 x (12 + 15 + 20) = 2 x (47) = 94 square centimeters. This response coincides with the one that we got earlier. Using this equation, the surface area of a rectangular prism can be computed much faster.

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