# 5th Grade - Surface Area Of Boxes

 Grade Level: 5th Skill: Shapes and Geometry Topic: Surface Area of Boxes Goal: Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects. Skill Description: Be able to visualize how a flat (2-dimensional) net folds up into a 3-dimensional cube or box and also find the surface area of the cube or rectangular prism from its 2D net.

## Building Blocks/Prerequisites

### Sample Problems

 (1) What figure has this net? (a cube) (2) What is the surface area of this figure? (106 cm2) (3) If the surface area of a cube is 24 cm2, what is the length of an edge? (2 cm2) (4) If the sides of a cube are in centimeters, what units will the surface area of that cube be in? (centimeters squared, or cm2) (5) How many different nets are there for a cube? (11 nets)

### Learning Tips

 (1) Vocabulary Net– a two-dimensional pattern for a three-dimensional object, which indicates lines for folding. Cube- a figure that looks like a dice; its height, width, and depth are all the same length, so all six of its sides are the same size. Rectangular box- a rectangular prism. Surface area- the amount of material needed to cover a 3-dimensional figure. (2) 2-D and 3-D Something that is 2-dimensional is flat like a piece of paper. It only has 2-dimensions because it only have a height (that you measure up and down) and a length (that you measure left and right); it doesn’t have any thickness. Something that is 3-dimensional has a height, length, and thickness. These dimensions are also called height, width, and depth. Are most of the objects around you 2D or 3D? Well, you are 3D, your computer is 3D, the chair you are sitting on is 3D, a pencil is 3D, and so on…. (3) 2-D Patterns for 3-D Shapes A net is a flat pattern that will fold up to create a 3-dimensional shape. For example, this is a net of a cube: If you cut it out and fold it along the lines, like you see below, then you will get a cube. (4) Don’t Be Fooled by Flaps Sometimes you will see nets with flaps on the sides. This is for nets that you cut out and tape together. The flaps are there so you can have a little extra piece of paper to fold underneath and keep the sides from gaping. That extra flap is NOT counted in the surface area of the figure. Can you tell which are the flaps? They have the slanted edges instead of all straight edges. (5) Why Nets are Useful Nets are helpful because we like to figure out things like how much wrapping paper it will take to cover a box. Surface area is the amount of material needed to cover a 3-dimensional figure. Nets helps us figure out the surface area of cubes and rectangular prisms. To figure out the total surface area, find the area of each square or rectangle. Then add all six areas up to find the total. Since a cube has six identical faces, you can figure out the area of one face and multiply it by six. For example: Each side or edge of the cube is 4 cm long. Here is a net of the cube, with each side labeled: How many faces does a cube have? Six, of course. So how many squares are in the cube’s net? You got it- six! If you count the squares inside of a face, there are 16. So the area of a face is 16 cm2. With six identical faces, the total surface area is: 6 * 16 cm2= 96 cm2. (6) The Surface Area of a Rectangular Box The more formal name for a rectangular box is a rectangular prism. To find the surface area of a rectangular prism, you need to find the area of each face and then add those areas together. Here is an example. The rectangular prism’s height is 1 cm, length is 2 cm, and depth is 3 cm. Here is the net of this box: Notice how each side is labeled in centimeters, and each area is labeled in centimeters squared. Let’s add the area of each face together: 6 cm2 + 3 cm2 + 6 cm2 + 3 cm2 + 2 cm2 + 2 cm2 = 22 cm2 Did you notice that there are pairs of faces with the same area? That’s because the front and back are the same, the top and bottom are the same, and the two sides match as well.

### Extra Help Problems

 (1) If the sides of a cube are in inches, what units will the surface area of that cube be in? (inches squared, or in2) (2) True or False: A rectangular prism has six faces that are identical. (false) (3) Which 3-D figure has six faces that are identical? (a cube) (4) What is the name of this figure? (rectangular prism) (5) Is this figure 2 or 3 dimensional? (3-dimensional) (6) What figure has this net? (Technically, it’s a square prism because the bases (in yellow) are squares, but it’s okay to say rectangular prism too) (7) What is the surface area of a rectangular prism with edge lengths of 2 cm, 2 cm, and 3 cm? (32 cm2) (8) If the area of one face of a cube is 4 cm2, what is the surface area of that cube? (24 cm2) (9) If the area of one face of a cube is 9 cm2, what is the surface area of that cube? (54 cm2) (10) If the area of one face of a cube is 25 cm2, what is the surface area of that cube? (150 cm2) (11) If the area of one face of a cube is 1 cm2, what is the surface area of that cube? (6 cm2) (12) If the area of one face of a cube is 100 cm2, what is the surface area of that cube? (600 cm2) (13) If the area of one face of a cube is 100 cm2, what is the length of an edge? (10 cm) (14) If the area of one face of a cube is 9 cm2, what is the length of an edge? (3 cm) (15) If the area of one face of a cube is 4 cm2, what is the length of an edge? (2 cm) (16) If the surface area of a cube is 54 cm2, what is the area of one face? (9 cm2) (17) If the surface area of a cube is 150 cm2, what is the area of one face? (25 cm2) (18) If the surface area of a cube is 150 cm2, what is the length of an edge? (5 cm2) (19) True or False: If you know the area of one face of a cube, you can multiply it by six to find the surface area of the cube. (true) (20) True or False: If you know the area of one face of a cube, you can find the length of an edge by figuring out what number times itself gives that area. (true) (21) True or False: If you know the surface area of a cube, you can find the length of an edge by dividing the surface area by six and then figuring out what number times itself gives that number. (true) (22) True or False: A rectangular prism only has one net. (false) (23) True or False: A cube only has one net. (false) (24) If the surface area of a cube is in centimeters squared (cm2), what units will the length of an edge be in? (centimeters) (25) If the surface area of a rectangular prism is in inches squared or square inches (in2), what units will the length of the edges be in? (inches)