Have you ever wrapped a gift with wrapping paper? If so, then you are familiar with solids and surface area!
A solid is a three-dimensional (3D) shape. Surface area is the total area for the faces that make up a solid. In other words, for our wrapping paper example, the surface area is the amount of paper it would take to cover the gift! Here you will explore methods and equations for calculating the surface area of solids.
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:
the total surface area
the lateral surface area
Total Surface Area: the sum of the areas of the faces and bases that make up a solid.
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?
Lateral Surface Area: the sum of the faces that make up a solid, excluding the base(s).
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;
\[\begin{align} S& =2B+Ch \\ &=2\pi r^2+2\pi rh. \end{align}\]
To find out more about the surface area of cylinders, see Surface of cylinders.
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
To find out more about the surface area of cones, see Surface of cones.
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:
\[S=4\pi r^2\]
\[r=\text{the radius}\]
To find out more about the surface area of spheres, see Surface of spheres.
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
To find out more about the surface area of pyramids, see Surface of pyramids.
Surface Area of a Rectangular Solid
A rectangular solid is a 3D shape where all of the sides are rectangles.
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.
Find the surface area for the following rectangular solid;
Fig. 7. Example of rectangular solids
Answer:
In order to find the surface area for a rectangular solid, let's first identify each part of the shape.
- \(L = 5\, cm\)
- \(W = 7\, cm\)
- \(H = 10 \, cm\)
Now you can input each value into the formula and simplify:
\[\begin{align} S&=2LW+2LH+2WH\\ &=2(5)(7)+2(5)(10)+2(7)(10) \\ &= 2\cdot 35+2\cdot 50+2\cdot 70 \\&=70+100+140 \\ &=310. \]
Don't forget the units! The surface area is \(310 \, cm^2\).
Surface Area of a Triangular Solid
A triangular solid, also known as a triangular prism, is a type of 3D shape where the bases of the shape are triangles.
A triangular solid looks like this:
Fig. 8. Triangular solid (triangular prism)
There are many different types of prisms, not only the triangular prism.
A prism is a type of solid where both bases are the same shape.
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
To find out more about the surface area of prisms, see Surface of prisms.
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.
Find the total surface area for a solid hemisphere that has a radius of \(5\, cm\).
Answer:
Firstly, you have been told that the solid is a solid hemisphere with a radius of \(5\, cm\). To find the total surface area, you can use the formula for the solid:
\[A=3\pi r^2.\]
Now you can input the information from the question, namely \(r=5\), to get
\[\begin{align} A&=3\pi 5^2 \\ &= 75\pi \\ &\approx 235.6 .\]
Notice the difference between the exact area \( 75\pi \, cm^2\) and the approximation of the area, \( 235.6 \, cm^2\).
Examples of the Surface of a Solid
Here are some examples of finding the surface area of solids.
Find the surface area for the following solid.
Fig. 11. Worked example
Answer:
First, notice that this is a cone. Next, what information do you have in the diagram?
- The radius \(r\) is 5 inches.
- The slant height \(l\) is 10 inches.
Knowing that you have the slant height tells you which formula for the surface area of a cone you should use. In this case, it is
\[S=\pi r^2+\pi r \cdot l.\]
Now you can plug in what you know to the formula:
\[\begin{align} S &=\pi 5^2+\pi (5)(10)\\ &=\pi 5^2+50\pi \\ & = 75\pi .\end{align}\]
When writing your answer don't forget the units! So the surface area of the cone is \(75\) square inches, or \(75\, in^2\).
You may be asked to approximate the surface area. In that case, using an approximation for \(\pi\) and rounding to one decimal place gives you that the surface area is approximately \(235.6 \, in^2\). You could write this as
\[S \approx 235.6 \, in^2.\]
Here is another example.
What formula would you use to find the surface area of the following solid?
Fig. 12. Worked example
Answer:
To find the surface area of this shape you would first need to identify the shape. It is a sphere.
Now you can recall the formula used to find the surface area for a sphere, which is
\[S=4\pi r^2.\]
Surface of Solids - Key takeaways
- A solid is a 3D shape, you can find the surface area of a solid by summing all the faces and bases of the shape.
- You can use different formulas depending on the solid to help you quicker find the surface area;
- Surface area of a prism \[S=2B+Ph = aP+Ph\]
- Surface area of a cylinder \[S=2B+Ch=2\pi r^2+2\pi rh\]
- Surface area of a cone \[S=B+\frac{1}{2}Cl=\pi r^2+\pi rl\]
- Surface area of a sphere \[S=4\pi r^2\]
- Surface area of a pyramid \[ S=B+\frac{1}{2}Pl\]
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