Cypress College Math Department – CCMR Notes Surface Area and Volume, Page 1 of 38

Surface Area and Volume

Objective 1: Surface Area of Polyhedra

Polyhedra is the plural form for a polyhedron. A polyhedron is a solid that is made up of multiple polygons

that are called faces. The faces will enclose a space where the edges of the polyhedron are the line

segments. Where the polygons meet at a single point is called a vertex.

Here are a few diagrams of common polyhedra.

Triangular Prism Rectangular Prism / Cube Pyramid

To find the surface area of a polyhedron, we need to find the area of each face and then add the areas of all

the faces.

Cypress College Math Department – CCMR Notes Objective 1: Surface Area of Polyhedra, Page 2 of 38

Ex: Find the surface area of the rectangular prism.

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Ex: Find the surface area of the rectangular prism.

Cypress College Math Department – CCMR Notes Objective 1: Surface Area of Polyhedra, Page 4 of 38

Ex: Find the surface area of the pyramid.

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Ex: Find the surface area of the triangular prism.

Cypress College Math Department – CCMR Notes Extra Practice, Page 6 of 38

Extra Practice

Objective 1: Surface Area of Polyhedra

Pause the video and try these problems.

Find the surface area.

1)

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2)

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3)

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4)

Restart the video when you are ready to check your answers.

Cypress College Math Department – CCMR Notes Objective 2: Surface Area of Cylinders, Cones, and Spheres, Page 10 of 38

Objective 2: Surface Area of Cylinders, Cones, and Spheres

Right Cylinder

The cylinder is made up of 2 circles and a rectangle

where the length of the side of the rectangle is the

same length as the circumference of the circle.

The surface area can be calculated by adding the

areas of the 2 circles and the area of the rectangle.

Surface Area for a Right Cylinder

( )

( ) ( )circle rectangle

2

2

2 2

where and

Surface Area Area Area

Surface Area r r h

h height r radius

= +

= +

= =

Cypress College Math Department – CCMR Notes Right Cylinder, Page 11 of 38

Ex: Calculate the surface area of the cylinder.

Cypress College Math Department – CCMR Notes Right Circular Cone, Page 12 of 38

Right Circular Cone

Sometimes the problem may not give us the slant

height or the height. In which case we can use the

Pythagorean Theorem to find them:

(𝒔𝒍𝒂𝒏𝒕 𝒉𝒆𝒊𝒈𝒉𝒕)𝟐 = (𝒉𝒆𝒊𝒈𝒉𝒕)𝟐 + (𝒓𝒂𝒅𝒊𝒖𝒔)𝟐

Now, this cone is made up of a circle and a part of a circle called a sector (lateral side). The surface area of

the cone is found by adding the area of the circular base to the area of the sector whose partial circumference

is the same length as the circumference of the circle and whose radius is the slant height.

So the Surface Area of the Cone is given by this formula:

Cypress College Math Department – CCMR Notes Right Circular Cone, Page 13 of 38

Ex: Find the surface area of the right circular cone.

To find the surface area of this cone, we will need to

find the radius and the slant height. The radius is given,

so we will need to use the Pythagorean Theorem to find

the slant height.

Surface Area of a Right Circular Cone 2

Cone

where and

Surface Area r rl

r radius l slant height

= +

= =

Cypress College Math Department – CCMR Notes Right Circular Cone, Page 14 of 38

Ex: Find the surface area.

Cypress College Math Department – CCMR Notes Sphere, Page 15 of 38

Sphere

Ex: Find the surface area.

Surface Area of a

Sphere 2

Sphere 4

where

Surface Area r

r radius

=

=

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Cylinder Cone Sphere

2 2 2Surface Area r rh = +

2 Surface Area r rl = +

2 4Surface Area r=

Summary of Surface Area

Cypress College Math Department – CCMR Notes Extra Practice, Page 17 of 38

Extra Practice

Objective 2: Surface Area of Cylinders, Cones, and Spheres

Pause the video and try these problems.

Calculate the surface area.

1)

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2)

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3)

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4)

Restart when you are ready to check your answers.

Cypress College Math Department – CCMR Notes Objective 3: Volume of Solids, Page 21 of 38

Objective 3: Volume of Solids

In the last two objectives you have learned to calculate the surface area of polyhedra such as the pyramid,

rectangular prism, and triangular prism, and other solids including the cone, sphere, and cylinder. In this

objective you will learn to find the volume of those solids. In finding the volume of solids, we will classify the

solids into three categories: the solids with both top and bottom faces that are parallel, the solids with only the

top or the bottom face, and the sphere. This will help us remember the general formula to finding their volume

and not feel overwhelmed with having to memorize so many formulas.

Solids with both top and bottom faces that are parallel

Solids with only a top or bottom face

Sphere

𝑽𝒐𝒍𝒖𝒎𝒆 = (𝑨𝒃𝒂𝒔𝒆)(𝒉𝒆𝒊𝒈𝒉𝒕)

V𝒐𝒍𝒖𝒎𝒆 =𝟏

𝟑(𝑨𝒃𝒂𝒔𝒆)(𝒉𝒆𝒊𝒈𝒉𝒕)

𝑽𝒐𝒍𝒖𝒎𝒆 =𝟒

𝟑𝝅𝒓𝟑,

where 𝑟 is the radius.

Cypress College Math Department – CCMR Notes Objective 3: Volume of Solids, Page 22 of 38

Ex: Calculate the volume of the solid.

**Note: Volume is always measured in cubic units. **

Ex: Calculate the volume of the solid.

`

Cypress College Math Department – CCMR Notes Objective 3: Volume of Solids, Page 23 of 38

Ex: Calculate the volume of the solid.

For this particular solid, an isosceles triangle is

the shape of the base. To calculate the volume

of this prism, we first need to find the area of the

triangle.

Cypress College Math Department – CCMR Notes Objective 3: Volume of Solids, Page 24 of 38

Ex: Find the volume of the solid.

Ex: Calculate the volume of the solid.

𝟐𝟖𝒌𝒎

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Ex: Find the volume of the solid.

Ex: Calculate the volume.

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Extra Practice

Objective 3: Volume of Polyhedra

Pause the video and try these problems.

Ex) Calculate the volume of the solid.

1)

2)

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3)

4)

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5)

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6)

7)

Cypress College Math Department – CCMR Notes Objective 4: Constructing an open-top box, Page 30 of 38

8)

Restart when you are ready to check your answers.

Objective 4: Constructing an open-top box

Suppose we are given a sheet of paper and need to construct an open-top box. We need to cut squares out of

each corner so we could fold up the sides. The size of the square depends on the height of each side of the

box.

For example, given a 24 𝑖𝑛 × 10 𝑖𝑛 sheet of paper, let’s cut out a 3 3 inch square from each corner.

Once you cut out those corner squares, the paper should look like this:

Cypress College Math Department – CCMR Notes Objective 4: Constructing an open-top box, Page 31 of 38

We can calculate each edge by subtracting a total of 6in from each original side length.

Cypress College Math Department – CCMR Notes Objective 4: Constructing an open-top box, Page 32 of 38

After calculating, you should have the following dimensions.

Let’s start folding the sides up to make our open-top box along the dotted lines.

Cypress College Math Department – CCMR Notes Objective 4: Constructing an open-top box, Page 33 of 38

From here, we can calculate the surface area and volume.

Cypress College Math Department – CCMR Notes Objective 4: Constructing an open-top box, Page 34 of 38

Ex: Construct an open-top box out a piece of 46 54cm cm cardboard sheet by cutting out 6 6cm cm squares from each corner. Calculate the surface area and volume of the open-top box.

Calculate each side length. Notice we are subtracting ( )2 6 12cm cm= from each side.

Cypress College Math Department – CCMR Notes Objective 4: Constructing an open-top box, Page 35 of 38

Fold the sides up on the dotted lines. We now have an open-top box with the following dimensions:

Ex: Construct an open-top box out a piece of 60 30in in cardboard sheet by cutting out ( )x x in squares from

each corner. Calculate the surface area and volume of the open-top box.

Cypress College Math Department – CCMR Notes Objective 4: Constructing an open-top box, Page 36 of 38

From each corner, cut out squares with side lengths of x inches .

Calculate the side lengths by subtracting 2x from the total length of each side.

Cypress College Math Department – CCMR Notes Extra Practice, Page 37 of 38

Now, along the perforated line, fold up the sides. You should have an open-top box that looks similar to this:

Calculate the surface area and volume. Leave you answer in factored form.

Extra Practice

Objective 4: Constructing an open-top box

Pause the video and try these problems.

1) Construct an open-top box out a piece of 28 40 cm cm cardboard sheet by cutting out 5 5 cm cm squares

from each corner. Calculate the surface area and volume of the open-top box.

Cypress College Math Department – CCMR Notes Objective 4: Constructing an open-top box, Page 38 of 38

2) Construct an open-top box out a piece of 38 66 in in cardboard sheet by cutting out x in x in squares

from each corner. Calculate the surface area and volume of the open-top box.

Restart when you are ready to check your answers.