Surface Area of a Solid

Formulas for Surface of Solids

The faces of a shape are the flat surfaces that make up the solid, the bases are the top and bottom surfaces of a solid.

Fig. 1. Identifying faces and bases of a solid.

When finding the surface area of a solid, there are two different types of surface area that can be found:

1. the total surface area

2. the lateral surface area

To find the total surface area of any solid, you sum the area of all the faces and bases of the solid.

What about lateral surface area?

To find the lateral surface area, you sum the area of the faces of the solid excluding the base(s).

When finding the surface area of any solid, you will have to break down the shape: this can be done differently depending on the solid that you have been given. To help you find the surface area of a solid, there are formulas that can be used, these depend on the type of solid that you have!

Let's take a look at some types of solids and the formulas that you could use to find the surface area.

Surface Area of a Cylinder

A cylinder is a type of solid that has no straight edges, it is similar to a prism where both bases are the same shape and the surface area can be calculated in a similar way.

Fig. 2. Example of a cylinder

In general, the variables used will be:

• \(B\) - area of the base;

• \(C\) - circumference of the base;

• \(r\) - radius of the base;

• \(h\) - height of the cylinder; and

• \(S\) - surface area of the cylinder.

There is a formula that can be used to find the surface area of a cylinder;

\[\begin{align} S& =2B+Ch \\ &=2\pi r^2+2\pi rh. \end{align}\]

Surface Area of a Cone

A cone is a type of solid that has a base and a vertex. A cone has a height and a slant height, the height is the distance from the center of the base to the top of the cone, the vertex. While the slant height is the distance from the edge of the base to the vertex.

Fig. 3. Example of a cone

There is a formula that can be used to help you find the surface area of a cone:

\[S=B+\frac{1}{2}Cl=\pi r^2+\pi r\cdot l\]

where

• \(B\) - area of the base

• \(C\) - circumference of the base

• \(r\) - radius of the base

• \(l\) - slant height

Surface Area of a Sphere

A sphere is a type of solid that is a 3D circle, for example a ball. A sphere has a center point and the radius is the distance from the center point to the outer point on the sphere.

Fig. 4. Example of a sphere

There is a formula that can be used to help you find the surface area of a sphere:

\[S=4\pi r^2\]

\[r=\text{the radius}\]

Surface area of a pyramid

A pyramid is a type of solid that has a base and triangular faces all coming to a vertex. There are different types of pyramids, that are all named based on the type of base that they have:

• Square pyramid

• Rectangular pyramid

• Triangular pyramid

• Hexagonal pyramid

Here are a few diagrams showing what these pyramids look like;

Fig. 5. Examples of pyramids

There is a formula that can be used to find the surface area of a pyramid:

\[S=B+\frac{1}{2}Pl\]

where

• \(B\) - area of the base
• \(P\) - perimeter of the base
• \(l\) - slant height

Surface Area of a Rectangular Solid

Here is an example of what a rectangular solid may look like.

Fig. 6. A rectangular solid

To understand how to find the surface area of a rectangular solid it can be helpful to break the shape down, into its different sections. In the diagram above you can see that there are two faces with sides \(L\) and \(W\). There are two faces with the side lengths \(L\) and \(H\) and there are two faces with the side lengths \(W\) and \(H\).

Since the surface area is the sum of the area of each of the shapes' faces, to find the surface area of a rectangular solid you can find the area of each of these faces and add them together.

This can be put into a formula to help you find the total surface area of the rectangular solid:

\[S=2LW+2LH+2WH.\]

Let's look through an example of how this formula may be used.

Surface Area of a Triangular Solid

A triangular solid looks like this:

Fig. 8. Triangular solid (triangular prism)

There are many different types of prisms, not only the triangular prism.

When a prism is cut in half you are left with two identical shapes, there are different types of prisms:

• Hexagonal prism

• Triangular prism

• Rectangular prism

• Square prism

Here are a few diagrams showing what these prisms look like:

Fig. 9. Examples of prisms

The apothem of a base is the distance from the midpoint of the shape to the outer side.

No matter the type of prism you have, you can find the surface area of a prism by using the formula:

\[S=2B+Ph = aP+Ph\]

where

• \(B\) - area of the base

• \(a\) - apothem of the base

• \(P\) - perimeter of the base

• \(h\) - height

Calculating Surface Area of Solid Hemisphere

A solid hemisphere looks like a sphere that has been cut in half. It looks like this;

Fig. 10. A solid hemisphere

To find the total surface area of a solid hemisphere, you have to find the area of the circle base as well as the area of the curved face. To help you do this in just one calculation, there is a formula that can be used:

\[A=3\pi r^2\]

where \(r\) is the radius.

This formula is very similar to the formula that you use to find the surface area of a sphere, \(4\pi r^2\). When you are finding the surface area of a solid hemisphere you are finding the surface area of half a sphere, therefore you half the formula to give you \(2\pi r^2\). You also need to add the area for the circle base \(\pi r^2\), adding these together gives you the formula for a solid hemisphere!

Let's look at an example using this formula.

Examples of the Surface of a Solid

Here are some examples of finding the surface area of solids.

Here is another example.