In mathematics, the problem of solid of revolution is sometimes presented. For a plane figure, we have to calculate the volume or surface area of the figure after one rotation.

How to solve such a solid of revolution problem? For complex solid of revolution, we need to learn high school mathematics integration to be able to calculate them. On the other hand, simple figures such as triangles and squares in solid of revolution can be solved with simple math knowledge.

However, when solving problems with solid of revolution, we have to understand what the shape of the solid of revolution is supposed to look like. Also, there are application problems that combine various shapes. For this reason, the problems in a solid of revolution are very challenging.

In this article, we will explain the basics of solid of revolution in mathematics and how to solve the problems.

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## A Figure Made with One Rotation on an Axis Is Solid of Revolution

Space figures include prisms, cylinders, pyramids, cones, and spheres. However, there is a problem that must be considered as a space figure, even though it is a plane figure. That is solid of revolution.

A solid of revolution refers to a figure that is completed by a single rotation of an axis, as shown below.

This is the solid of revolution. We have to create a space figure from a plane figure. You have to imagine in your mind what kind of figure will be completed.

Then we calculate the volume or surface area of the solid of revolution. For example, what would the volume and surface area of the following solid of revolution?

The volume of the cylinder can be calculated by multiplying the base area by the height. Therefore, the volume of this solid of revolution is $128π$ cm^{3}.

- $4×4×π×8=128π$

The surface area, on the other hand, can be calculated by adding the bottom areas and the side area. The sum of the base area is as follows.

- $4×4×π×2=32π$

Also, the side area of the cylinder is as follows.

- $8×π×8=64π$

Therefore, the surface area of the solid of revolution is $32π+64π=96π$ and the answer is $96π$ cm^{2}. After rotating the plane figure with a line like this, we calculate the volume and surface area.

### The Base of a Solid of Revolution Will Always Be a Circle

Solid of revolution have one thing in common: they are always circular. That is, it will never be a prismatic or pyramidal space figure. A solid of revolution always completes a curved shape.

As an example, here are the triangular and semicircular solid of revolutions.

Earlier, we explained that a rectangular solid of revolution becomes a cylinder. On the other hand, a triangular solid of revolution becomes a cone. A half-circle solid of revolution becomes a sphere.

When you revolve a triangle or a rectangle, its base will always be a circle. On the other hand, there is no base for a half-circle solid of revolution. However, the same is true for the solid of revolution, which is a curved shape.

### Use Formulas for Cylinders, Cones, and Spheres

In this way, we can imagine a three-dimensional object with space figures. Then, use the formulas to solve the problems.

We already know how to calculate cylinders. Multiplying the base area by the height, we can get the cylinder volume. For a cone, we can also calculate it by multiplying the volume of the cylinder by 1/3.

- Cylinder volume = Base area × Height
- Cone volume = Base area × Height × 1/3

The surface area, on the other hand, is calculated by determining the base area and side area of each, and then adding them together.

In the case of spheres, use the formulas to find the volume and surface area. Although the calculation of spheres is infrequent, if you do not remember the formula, you will not be able to solve the problem. If you want to solve the problem, try to remember the following formulas.

**-The formula for the sphere volume**

If we know the radius, we can calculate the volume of the sphere by substituting the numbers into the formula.

**-The formula for the sphere surface area**

Similarly, if we know the radius, we can substitute it into the surface area formula. We can calculate the surface area of a solid of revolution.

## The Application Problem is Combining Figures.

One of the application problems for solid of revolution is the combination of figures. As there are several different figures, the shape of the solid of revolution becomes more complex.

For example, let’s consider the following solid of revolution.

In this figure, triangles and squares are mixed together. In this type of figure, the solid of revolution mixes a cone and a cylinder. In such cases, separate the figures and calculate the volume and surface area.

**-Calculation of volume**

Let’s calculate the volume of a cone and a cylinder, respectively. The volume of the cone is as follows.

- $4×4×π×3×\displaystyle\frac{1}{3}=16π$

And the volume of the cylinder is as follows.

- $4×4×π×4=64π$

By adding the volumes of the cone and the cylinder, we find that the answer is $16π+64π=80π$ cm^{3}.

**-Calculation of surface area**

In calculating surface area, we need to think about the net. In other words, we need to think about the space figure, and then convert it to a plane. The result is as follows.

After calculating the area of each, make sure to add them together. The result is as follows.

- Area of a sector: $5×4×π=20π$
- Area of a rectangle: $8×π×4=32π$
- Area of a circle: $4×4×π=16π$

For reference, we use the following formula for the sector area.

- Side area of a cone = Generatrix × Radius of the base × $π$

The answer for the surface area of the solid is $68π$ cm^{2} by adding these areas.

- $20π+32π+16π=68π$

### Solid of Revolution Shape with Plane Away from Axis

Incidentally, there are some cases where the plane is away from the axis when making a solid of revolution. In that case, a solid of revolution with a hollow space is created.

We have already explained that a rectangular solid of revolution becomes a cylinder. On the other hand, if the rectangle is away from the line of rotation, the solid of revolution will be a donut shape as shown below.

When calculating the volume or surface area of this figure, we have to consider the two cylinders. In any case, consider that a hollow can be created if the plane is away from the axis of rotation.

## Exercise: Volume and Surface Area of a Solid of Revolution

**Q1.** For the following figure, calculate the volume and surface area of the figure formed by making a full revolution around a straight line. Let pi be $π$.

**A1.** answer.

The solid of revolution of this figure is as follows.

Earlier, we explained how to calculate a figure that combines a cone and a cylinder. The idea is the same as this method. The difference is that there is a cone inside a cylinder. So, let’s calculate the cylinder and cone separately.

**-Calculation of volume**

If we subtract a cone from a cylinder, we can get the volume. The volume of the cylinder is as follows.

- $3×3×π×4=36π$

On the other hand, the volume of the cone is as follows.

- $3×3×π×4×\displaystyle\frac{1}{3}=12π$

Therefore, the volume of the solid is $24π$ cm^{3}.

- $36π-12π=24π$

**-Calculation of surface area**

The calculation method is the same as that of the triangle and rectangle solid of revolution explained earlier. If we consider the net, we can see three shapes: a sector, a rectangle, and a circle. So, let’s calculate the area of each.

- Area of a sector: $5×3×π=15π$
- Area of a rectangle: $6×π×4=24π$
- Area of a circle: $3×3×π=9π$

If we add the areas of each, we get $48π$ cm^{2}.

- $15π+24π+9π=48π$

## Learning the Basics of Solid of Revolution in Space Figures.

In Space Figures, we learn about the concept of solid of revolution. A solid of revolution is a space figure that is created when a plane exists and the axis is rotated by one revolution.

In the case of complex solid of revolution, we need to use integrals in high school mathematics to solve the problem. On the other hand, simple solid of revolutions such as triangles and squares can be solved without the use of integrals. So, use the formulas for cones, cylinders, and spheres to do the calculations.

In application problems, multiple figures will be combined. Or, the figures may leave from the axis. However, the basic idea is the same.

Among the space figures, the problem of finding the volume and surface area of a solid of revolution is more difficult. Try to imagine what kind of solid of revolution you can make and calculate the volume and surface area.