Have you ever tried to wrap a box and ended up with too little wrapping paper? Or built a paper model of a building that did not quite fold up right? In both cases, you are really working with nets of three-dimensional figures. When you understand nets, you can plan exactly how much material you need and how the pieces must fit together. That is a powerful geometry skill.
In this lesson, you learn how to:
You will work with shapes like rectangular prisms (box shapes), triangular prisms (like some tents), and square pyramids (like certain roofs or monuments). You will not need to memorize surface area formulas; instead, you will focus on the areas of each individual part of the net.
Three-dimensional (3D) shapes, like boxes and pyramids, take up space. Their sides are called faces. Each face is a flat, two-dimensional (2D) shape, like a rectangle or a triangle.
A net is what you get if you could carefully cut along some edges of a 3D shape and unfold it so it lies flat on a table. The net is made up of the faces of the solid, all connected along edges. When you fold a net back up, it makes the original 3D shape again.
We care about nets because they let us see all the faces at once, without overlapping. That makes it easier to:
In this lesson, we focus on nets made of rectangles and triangles. These are the most common kinds of faces for prisms and pyramids you see in everyday life.
Rectangular prisms are box-shaped solids, like cereal boxes, shipping boxes, or bricks. Every face of a rectangular prism is a rectangle. There are six faces in total.
It is easier to understand how the faces connect when you look at the solid and its net side by side, as shown in [Figure 1]. Each rectangle in the net matches exactly one face on the prism.
Key facts about rectangular prisms:
A typical net for a rectangular prism might look like a row of four rectangles in a line (front, right, back, left), with one rectangle attached above one of them (top) and one attached below (bottom). The important thing is that when you fold the rectangles up, the right edges touch.
To connect this to area, remember:
Even if two faces are the same size, you can think of them separately and then add their areas together.
Triangular prisms look like a “stretched-out” triangle. Imagine a long triangular piece of cheese or the main shape of some camping tents, as shown in [Figure 2]. Each triangular prism has:
When you unfold a triangular prism, you get a net made of two triangles and three rectangles.
In the net, the three rectangles are usually connected in a strip, and each triangle is attached to one of the long sides of the strip. The sides that touched in the 3D prism must touch in the net as well.
To connect this to area:
Sometimes, in real life, you might not need all the faces. For example, if a tent is open on the bottom, you only find the area of the sides and top, not the floor. You can choose which faces to include by looking carefully at the net.
Pyramids have a flat base and triangular faces that meet at a single point called the vertex. A square pyramid has a square base. A rectangular pyramid has a rectangular base. Seeing the pyramid and its net together (see [Figure 3]) makes the relationship between the base and the triangular faces very clear.
For a square pyramid:
When you unfold the pyramid into a net, you get one square in the middle with four triangles attached to its sides.
The base side length of the pyramid matches the base of each triangle in the net. The triangles usually have the same “slant height,” which is the height of each triangle measured straight up from its base to its tip in the net.
To connect this to area:
For a rectangular pyramid, the base is a rectangle, and the four triangles may not all be congruent, but the idea of using a net and adding the triangles’ and base’s areas is the same.
Surface area is the total area of all the outside faces of a 3D shape. A net shows all these faces laid out flat, which makes it easier to calculate. Our goal is to find surface area by adding the areas of each rectangle and triangle in the net. We do not rely on memorizing special surface area formulas.
Here is a simple step-by-step method:
As long as you correctly identify all the faces and their sizes, this method works for any 3D shape whose faces are rectangles and triangles.
Suppose you have a small shipping box with:
You want to know how much cardboard is used to make the outside of the box. That is the surface area.
Step 1: Think about the net. The net will have six rectangles:
Step 2: Find the area of each kind of face.
Front (and back) face area:
\[A_{\textrm{front}} = 10 \times 4 = 40 \textrm{ cm}^2.\]
Since the back is the same size:
\[A_{\textrm{back}} = 40 \textrm{ cm}^2.\]
Left (and right) face area:
\[A_{\textrm{left}} = 6 \times 4 = 24 \textrm{ cm}^2.\]
\[A_{\textrm{right}} = 24 \textrm{ cm}^2.\]
Top (and bottom) face area:
\[A_{\textrm{top}} = 10 \times 6 = 60 \textrm{ cm}^2.\]
\[A_{\textrm{bottom}} = 60 \textrm{ cm}^2.\]
Step 3: Add all the areas to get the surface area.
\[\begin{aligned} A_{\textrm{total}} &= A_{\textrm{front}} + A_{\textrm{back}} + A_{\textrm{left}} + A_{\textrm{right}} + A_{\textrm{top}} + A_{\textrm{bottom}} \ &= 40 + 40 + 24 + 24 + 60 + 60 \ &= 248 \textrm{ cm}^2. \end{aligned}\]
The surface area of the box is \(248 \textrm{ cm}^2\). That tells you how much cardboard is on the outside of the box.
The length of the tent (the distance between the two triangular ends) is \(L = 5 \textrm{ m}\).
We want the surface area of the tent cloth if it covers all faces, including the floor. (Later, you could adjust this if there were no floor.)
Step 1: Picture or draw the net. The net has:
Step 2: Find the area of the triangular faces.
Each triangle area is:
\[A_{\textrm{triangle}} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 3 \times 2 = 3 \textrm{ m}^2.\]
There are two triangles, so:
\[A_{\textrm{two triangles}} = 2 \times 3 = 6 \textrm{ m}^2.\]
Step 3: Find the dimensions of the rectangles.
To keep things simple, suppose the triangle is a right triangle with legs \(3 \textrm{ m}\) (base) and \(2 \textrm{ m}\) (height). Then its hypotenuse can be found using the Pythagorean theorem, but here we will just say the hypotenuse length is \(\sqrt{3^2 + 2^2} = \sqrt{13}\) meters. We will use this distance as a side of one rectangle.
Step 4: Add all the areas for total surface area.
\[\begin{aligned} A_{\textrm{total}} &= A_{\textrm{two triangles}} + A_{1} + A_{2} + A_{3} \ &= 6 + 15 + 10 + 5\sqrt{13}. \end{aligned}\]
We can group the whole-number parts:
\[6 + 15 + 10 = 31,\]
so
\[A_{\textrm{total}} = 31 + 5\sqrt{13} \textrm{ m}^2.\]
If you want an approximate value, you can estimate \(\sqrt{13}\) as about \(3.6\), so
\[5\sqrt{13} \approx 5 \times 3.6 = 18.0,\]
and
\[A_{\textrm{total}} \approx 31 + 18 = 49 \textrm{ m}^2.\]
So it takes about \(49 \textrm{ m}^2\) of fabric to cover this tent completely.
Now imagine a small model of a roof that is a square pyramid. The base is a square with side length \(s = 4 \textrm{ m}\). Each triangular face has:
You want to find how much roofing material is needed for all four triangular sides, plus maybe the base if it is covered.
Step 1: Think about the net. The net has:
Step 2: Find the area of the square base.
\[A_{\textrm{base}} = s \times s = 4 \times 4 = 16 \textrm{ m}^2.\]
Step 3: Find the area of one triangular face.
The base of each triangle is 4 m, and the triangle’s height is 3 m, so
\[A_{\textrm{triangle}} = \frac{1}{2} \times 4 \times 3 = 6 \textrm{ m}^2.\]
There are four such triangles, so
\[A_{\textrm{four triangles}} = 4 \times 6 = 24 \textrm{ m}^2.\]
Step 4: Add the base and side areas for total surface area.
\[A_{\textrm{total}} = A_{\textrm{base}} + A_{\textrm{four triangles}} = 16 + 24 = 40 \textrm{ m}^2.\]
The surface area of this square pyramid is \(40 \textrm{ m}^2\).
If you only needed material for the four sides of the roof (and not the base), the area would just be \(24 \textrm{ m}^2\).
Many solids have more than one possible net. For example, a rectangular prism can be unfolded in several ways, but all valid nets share some important rules, just like the one in [Figure 1]:
Some drawings are not nets of a shape because they miss a face, have extra faces, or connect edges that do not match in the 3D shape. For example, if you draw a plus-shaped net for a box but forget the top rectangle, it will not close up into a solid.
When you are checking whether a drawing is a net:
These ideas also apply to other shapes like triangular prisms and pyramids, as shown in [Figure 2] and [Figure 3].
Nets and surface area show up all around you in real life. Here are some important examples:
In all of these situations, people do not always memorize special surface area formulas. Instead, they think in terms of faces: find each face’s area, then add them up.
