Geometry Formulas:

Surface Area Surface Area & Volume& Volume

CCS:

• 6.G.4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Objectives:Students will be able to:

•Identify Formulas for Surface Area and Volume

•Find the surface area and volume of prisms

•Find the surface area and volume of cylinders

•Solve real world problems using surface area and volume

A formula is just a set of instructions. It tells you exactly what to do!

All you have to do is look at the picture and identify the parts.

Substitute numbers for the variables and do the math. That’s it!

Let’s start in the beginning…

Before you can do surface area or volume, you have to know the following formulas.

Rectangle A = lw

Triangle A = ½ bh

Circle A = π r²

C = πd

You can tell the base and height of a triangle by finding the right angle:

TRIANGLES

CIRCLESYou must know the difference between RADIUS and DIAMETER.

r

d

Counting Cubes to Find Volume

How many cubes represent the Length? 6cm

How many cubes represent the width? 6cm

How many cubes represent the height? 6cm

So the volume = lwh= 6 x 6 x 6= 216 cm3

Let’s start with a rectangular prism.

Surface area can be done using the formula

SA = 2 lw + 2 wl + 2 lw OR

Either method will give you the same answer.

you can find the area for each surface and add them up.

Volume of a rectangular prism is V = lwh

Example:

7 cm

4 cm

8 cm Surface Area

Top/bottom 2(8)(4) = 64

Left/right 2(4)(7) = 56

Front/back 2(8)(7) = 112

Add them up!

SA = 232 cm²

V = lwh

V = 8(4)(7)

V = 224 cm³

To find the surface area of a triangular prism you need to be able to imagine that you can take the prism apart like so:

Notice there are TWO congruent triangles and THREE rectangles. The rectangles may or may not all be the same.

Find each area, then add.

Example:

8mm

9mm

6 mm 6mm

Find the AREA of each SURFACE

1. Top or bottom triangle:

A = ½ bh

A = ½ (6)(6)

A = 18

2. The two dark sides are the same.

A = lw

A = 6(9)

A = 54

3. The back rectangle is different

A = lw

A = 8(9)

A = 72

ADD THEM ALL UP!

18 + 18 + 54 + 54 + 72

SA = 216 mm²

SURFACE AREA of a CYLINDER.

You can see that the surface is made up of two circles and a rectangle.

The length of the rectangle is the same as the circumference of the circle!

Imagine that you can open up a cylinder like so:

The Soup CanThink of the Cylinder as a soup can.

You have the top and bottom lid (circles) and you have the label (a rectangle – wrapped around the can).

The lids and the label are related.

The circumference of the lid is the same as the length of the label.

EXAMPLE: Round to the nearest TENTH.

Top or bottom circle

A = πr²

A = π(3.1)²

A = π(9.61)

A = 30.1754

Rectangle

C = length

C = π d

C = π(6.2)

C = 19.468

Now the area

A = lw

A = 19.5(12)

A = 234

Now add:

30.2 + 30.2 + 234 =

SA = 294.4 in²

There is also a formula to find surface area of a cylinder.

Some people find this way easier:

SA = 2πrh + 2πr²

SA = 2π(3.1)(12) + 2π(3.1)²

SA = 2π (37.2) + 2π(9.61)

SA = π(74.4) + π(19.2)

SA = 233.616 + 60.288

SA = 299.904 in²

The answers are REALLY close, but not exactly the same. That’s because we rounded in the problem.

Follow These Easy Steps to Find SA of a cylinder:

Find the radius and height of the cylinder.

Then “Plug and Chug”…

Just plug in the numbers then do the math.

Remember the order of operations and you’re ready to go.

The formula tells you what to do!!!!

2πrh + 2πr² means multiply 2(π)(r)(h) + 2(π)(r)(r)

PracticeBe sure you know the difference between a radius and a diameter!

SA = 2πrh + 2πr²

= (2 x 3.14 x 11 x 14) + (2 x 3.14 x 112)

= (967.12) + (2 x 3.14 x 121)

= (967.12) + 2 (379.94)

= (967.12) + (759.88)

= 1727 cm2

More Practice!

SA = 2πrh + 2πr² = (2 x 3.14 x 5.5 x 7) + (2 x 3.14 x 5.52)

= (241.78) + (2 x 3.14 x 30.25)= (241.78) + (2 x 94.985)= (241.78) + 189.97 = 431.75 cm2 11 cm

7 cm

Volume of Prisms or Cylinders

You already know how to find the volume of a rectangular prism: V = lwh

The new formulas you need are:

Triangular Prism V = (½ bh)(H)

h = the height of the triangle and

H = the height of the cylinder

Cylinder V = (πr²)(H)

Try one:

Can you see the triangular bases?

V = (½ bh)(H)

V = (½)(12)(8)(18)

V = 864 cm³

Notice the prism is on its side. 18 cm is the HEIGHT of the prism. Picture if you turned it upward and you can see why it’s called “height”.

V = (πr²)(H)

V = (π)(3.1²)(12)

V = (π)(3.1)(3.1)(12)

V = 362.1048 in³

Volume of a Cylinder

We used this drawing for our surface area example. Now we will find the volume.

optional step!

Try one:

10 m

d = 8 m

V = (πr²)(H)

V = (π)(4²)(10)

V = (π)(16)(10)

V = 502.4 m³Since d = 8,

then r = 4

r² = 4² = 4(4) = 16

Here are the formulas you will need to know:

A = lw SA = 2πrh + 2πr²

A = ½ bh V = (½ bh)(H)

A = π r² V = (πr²)(H)

C = πd

and how to find the surface area of a prism by adding up the areas of all the surfaces

Classwork:

• Try this Surface Area and Volume interactive Practice

• Use this Interactive to find the Volume of rectangular prisms

HOMEWORK:

Reteaching Surface Area and Volume HO