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Volume and Surface Area of Three-dimensional Figures

Three-dimensional Figures

The most common three-dimensional figures are prisms, cylinders, pyramids, cones and spheres. They all have two important qualities: volume and surface area. Here is a list of the figures and their properties.

Prism

Prism and formulas for calculation

Example 1

A prism has a triangle with the area $B = 12 {cm}^{2}$ as its base, and height $h = 5 cm$ . Calculate the volume of the prism.

You’ll find the volume by multiplying the area of the base $B$ by the height $h$ :

\begin{array}{l} Volume & = B \cdot h = 12 {cm}^{2} \cdot 5 cm \\ = 60 {cm}^{3} \end{array}

Volume = B \cdot h = 12 {cm}^{2} \cdot 5 cm = 60 {cm}^{3}

Cylinder

Formula of the surface area of cylinder

Example 2

A cylinder has a radius $r = 2.0 cm$ and height $h = 8.0 cm$ . Calculate the volume and the surface area of the cylinder.

Volume

The formula for the volume of a cylinder is $V = B \cdot h$ . You have the height $h$ , but you need to find the area of the base $B$ . Since the area of the base in a cylinder is a circle, you’ll have to find the area of a circle:

\begin{array}{l} B & = π r^{2} = π \cdot 2.0 cm \cdot 2.0 cm \\ \approx 12.6 {cm}^{2} \end{array}

B = π r^{2} = π \cdot 2.0 cm \cdot 2.0 cm \approx 12.6 {cm}^{2}

Now, you can find the volume of the cylinder by multiplying the base area

B

by the height

h

\begin{array}{l} Volume & = B \cdot h \\ \approx 12.6 {cm}^{2} \cdot 8.0 cm \\ \approx 101 {cm}^{3} \end{array}

\begin{array}{l} Volume = B \cdot h \approx 12.6 {cm}^{2} \cdot 8.0 cm \approx 101 {cm}^{3} \end{array}

Surface Area

To find the surface area of a cylinder, you add together the areas of all the sides. The blue rectangle you see in the drawing comes from cutting out, then flattening, the wall of the cylinder. First, you calculate the area of the top and bottom of the cylinder (they have the same area), and then the area of the rectangle which is the height times the length of the rectangle. The length is the circumference of the circle (see the figure):

\begin{array}{l} Area of a circle & = π r^{2} \\ = π \cdot 2.0 cm \cdot 2.0 cm \\ \approx 12.6 {cm}^{2}, \\ Area of a rectangle & = l \cdot h \\ = 2 π r \cdot h \\ = 2 π \cdot 2.0 cm \cdot 8.0 cm \\ \approx 101 {cm}^{2} \end{array}

\begin{array}{l} Area of a circle & = π r^{2} \\ = π \cdot 2.0 cm \cdot 2.0 cm \\ \approx 12.6 {cm}^{2}, \\ Area of a rectangle & = l \cdot h \\ = 2 π r \cdot h \\ = 2 π \cdot 2.0 cm \cdot 8.0 cm \\ \approx 101 {cm}^{2} \end{array}

The surface area of the cylinder then becomes

\begin{array}{l} Surface area & = 2 \cdot Area of a circle \\ + Area of a rectangle \\ \approx 2 \cdot 12.6 {cm}^{2} + 101 {cm}^{2} \\ = 126.2 {cm}^{2} \end{array}

\begin{array}{l} Surface area & = 2 \cdot Area of a circle + Area of a rectangle \\ \approx 2 \cdot 12.6 {cm}^{2} + 101 {cm}^{2} \\ = 126.2 {cm}^{2} \end{array}

Pyramid

Pyramid and formulas for calculation

Example 3

A pyramid has a triangle with area $B = 9 {cm}^{2}$ as its base. The height $h$ of the pyramid is $4 cm$ . Calculate the volume of the pyramid.

You’ll find the volume of the pyramid by multiplying the area of the base $B$ by the height $h$ , and then dividing by 3:

\begin{array}{l} Volume & = \frac{B \cdot h}{3} = \frac{9 {cm}^{2} \cdot 4 cm}{3} \\ = 12 {cm}^{3} \end{array}

Volume = \frac{B \cdot h}{3} = \frac{9 {cm}^{2} \cdot 4 cm}{3} = 12 {cm}^{3}

Cone

Formula of the surface area of cone

Example 4

A cone has a circle with radius $r = 3.0 cm$ as its base. The height $h$ of the cone is $7.0 cm$ and the side $s$ is $9.0 cm$ . Calculate the volume and the surface area of the cone.

Volume

The volume of a cone is determined using the formula $V = B \cdot h$ , so you first need to find the area of the base $B$ , which is a circle. That area is:

\begin{array}{l} B & = π r^{2} = π \cdot 3.0 cm \cdot 3.0 cm \\ \approx 28.3 {cm}^{2} \end{array}

B = π r^{2} = π \cdot 3.0 cm \cdot 3.0 cm \approx 28.3 {cm}^{2}

Then you’ll find the volume of the cone by multiplying the area of the base

B

by the height

h

, and then dividing by 3:

\begin{array}{l} Volume & = \frac{B \cdot h}{3} \\ = \frac{28.3 {cm}^{2} \cdot 7.0 cm}{3} \\ \approx 66.0 {cm}^{3} \end{array}

Surface Area

You’ll find the surface area by using the formula for the surface area of a cone. Here you can put your values directly in to the formula and calculate.