3-Dimensional Figures

Maybe you are reading this in front of your computer. Or maybe you have a glass of water next to you.

If you look at any of these objects that surround you, it is clear that they are objects in 3d. But, what is the math definition for a three-dimensional figure?

In this article, we will learn more about 3-dimensional figures and their applications.

What is a 3-dimensional figure?

As you may have observed with the box, these three-dimensional shapes have volume. In math, we define volume as the quantity of space inside a closed surface.

Grabbing the box again and if you open it now, the volume would be the quantity of space inside the box. We will learn later how to compute this volume.

These geometric shapes generally, except for some exceptions we will use, have faces which are the surfaces with a certain surface area that delimitate the figure. These faces join in vertices, which are points of union.

Finally, the lines that delimitate these surfaces and the contour of the geometric figure are named edges. We would compare them with the sides of the 2-dimensional shapes.

3-dimensional figures examples

Taking the look away from this article and looking around you, you will probably identify a lot of three-dimensional figures with different structures. From the bed to the chair, to the table or even to the books you use to study. All of them are 3d shapes as they have the 3 dimensions we mentioned before; length, width, and depth, and also because they have volume.

We distinguish between regular and irregular 3d shapes. We will focus on the regular three-dimensional figures, as they are more common in math.

Cone

A cone is a three-dimensional figure that we would obtain if we make a right triangle (that has one angle equal to 90º) turn around with one of its sides fixed, so we get a shape in 3d. This figure normally has a circular base and a vertex where the lateral surface of the cone tapers to.

The base does not have to necessarily be a circle, it can also be another two-dimensional circular figure such as an oval. You can observe this shape in the real world when you look at the traffic cones.

Pyramid

This figure is similar to the cone, but in this case, the base does not have a circular form. The base is a two-dimensional figure with three or more sides such as a triangle, square, rectangle, etc.

As the geometric form of the base can vary, it also changes the number of edges. All of its surfaces, no matter how much it has, taper to a vertex.

The famous pyramids of Egypt are one example of these geometric shapes, in this case, they have a squared base.

Cube

This geometric figure consists of six faces of the equal-area meeting three of them in a single vertex, with a total of eight vertices and a total of twelve edges.

An example of a cube is a dice. If you observe it, all of the faces of a regular dice have the same surface and each vertex of it works as a union for three different faces.

Rectangular prism

It is similar to the cube, as it also has eight vertices, twelve edges, and six faces, but in this case, all of the faces are not equal. Each face is equal to its opposite, therefore we have pairs of equal faces.

An example of a rectangular prism could be a drawer or even a box, although sometimes they have the shape of a cube.

There exist other kinds of prisms, regarding the shape of its base and the opposite face. For example, if these faces have the shape of a triangle, it is a triangular prism that will have five faces in total instead of the six faces that the rectangular prism has. But this base (and the opposite face) can have another 2-dimensional figure that gives different types of prisms: pentagonal prisms, hexagonal prisms, etc.

Cylinder

The shape of this figure can remind you of a rectangular prism, but in this case, it has two surfaces, which are called the top and bottom (or base) of the figure, that consist of two-dimensional circular figures.

This figure does not have any vertex. The surface that connects those two faces is essentially a rectangle but curved.

You can find these kinds of geometric shapes in cans or some glasses.

Sphere

A football, a basketball, or maybe, if we do not want to just limit ourselves to the sports world: a bubble. All of these objects share one common thing: they are spheres.

These geometric shapes are obtained if we make a circle, which is a two-dimensional figure, turned around its diameter. The volume this revolution describes is defined as a sphere.

As it happens with the circle in two dimensions, all of the points of the surface are equally distanced from the point in the center of the figure. This distance is called the radius. If we trace a distance between two points of the surface of the sphere that goes through the center of it, this distance is called the diameter of the sphere, which corresponds to two times the radius.

Formulas of 3-dimensional figures

When working with 3d shapes, there are some things we might want to know about them. In particular, there are two characteristics we are interested in.

To obtain the total surface area of the figure we have to sum the areas of each face of the shape. We should not confuse the surface area of the figure with its volume. The area consists only of the surface of the faces, independently of what is inside of them.

On the other hand, we have the volume of the figure.

If we grab again the box we have talked about in this article, you can see that the surface of the cardboard used for all of the faces corresponds to the surface area of the box, but the space there which is inside the box corresponds to its volume.

Let’s see how some of the math equations for the 3d shapes we have seen before.

Area and volume of a cone

The surface area of a three-dimensional figure is the sum of the areas of its faces.

For a cone, the surface area of its base is $A_b=\mathrm{\pi }\times {\mathrm{r}}^2$, where r is the radius of the circle. The area of the lateral face is $A_l=\mathrm{\pi }\times \mathrm{r}\times \mathrm{g}$, being g the distance between any point of the edge of the base to the vertex. Therefore, the surface area of a cone can be generally expressed as,

$A_{cone}=\mathrm{\pi }\times \mathrm{r}\times (\mathrm{r}+\mathrm{g})$.

The volume for a cone is given by the following formula,

$V_{cone}=\frac{(\mathrm{\pi }\times \mathrm{h}\times {\mathrm{r}}^2)}{3}$,

where h is the distance from the center of the base to the vertex.

Area and volume of a pyramid

In this case, the formulas of the area and volume will depend on the number of edges the base has.

For example, if the pyramid has a squared base, the surface area of the pyramid will be the sum of the area of the square $A_s=l^2$ with the sum of the areas of each triangle that connects the vertices $A_t=\frac{1}{2}b\times h$. In general, we can express the surface area of a pyramid as,

$A_{pyramid=\hspace{0.17em}}{A}_{base}+A_{triangles}$

Be careful, as the base does not have to be regular, and the surface area of the triangles that connect with the vertex does not have to be either.

$V_{pyramid}=\frac{h\times l^2}{3}$

being

• h the distance from the center of the base to the vertex

• l the length of the edges of the base.

Area and volume of a rectangular prism and a cube

In this case, as the rectangular prism and the cube are formed by six faces, to obtain the total surface area of the figure we just have to sum the areas of each face.

For the cube, all six faces will have the same area, but for the rectangular prism, as each face is equal to its opposite, there are three different values. A general math expression for the surface area of a rectangular prism is,

$A_{r.p}=2·A_1+2·A_2+2·A_3$

where A₁ , A₂ , and A₃ are the three different values of those areas. The area of a rectangle is $A_r=b·h$.

The volume for those shapes is the multiplication of the three edges; the length, the width, and the depth of the prism, such as,

$V_{r.p}=a\times b\times h$

In the case of the cube, as all of the sides are equally long, we have,

$V_{cube}=l^3$

Area and volume of a cylinder

The cylinder consists of two circles that are the top and bottom of the figure and a curved rectangle. Therefore, if the area for a circle is $A_c=\mathrm{\pi }\times {\mathrm{r}}^2$, the sum of all the areas is,

$A_{cyl}=2\times \mathrm{\pi }\times {\mathrm{r}}^2+2\times \mathrm{\pi }\times \mathrm{h}$

where h is the height from one point of the bottom to the point in the top at the same position.

The volume for the cylinder is described by the following equation,

Area and volume of a sphere

The sphere we know is a different type of geometrical figure, as it is not formed by the union of different faces. That is why we need a math expression to compute its surface area,

$A_{sphere}=4\times \mathrm{\pi }\times {\mathrm{r}}^2$

And the volume for the sphere is determined by the following formula,

$V_{sphere}=\frac{4}{3}\times \mathrm{\pi }\times {\mathrm{r}}^3$.

Examples of problems on 3-dimensional figures

Now, let us look into some examples of problems you may encounter on 3-dimensional figures.