- Use nets to represent prisms.
- Find the surface area of a prism.
A prism is a 3-dimensional figure with a pair of parallel and congruent ends, or bases. The sides of a prism are parallelograms. Prisms are identified by their bases.
Look at the names of the prisms above. Each type of prism is named after its base. In the figure on the left, the base is a rectangle, so it is called a rectangular prism. In the middle, the base is a triangle, so the shape is a triangular prism. On the right, the base is a hexagon (6-sided figure), so the prism is called a hexagonal prism.
As you can see, the base is not always on the “bottom” of the prism!
Prisms are named by their ______________________.
If a prism is called a pentagonal prism, then its base is a __________________________.
Surface Area of a Prism Using Nets
The prisms in the picture above are right prisms. In a right prism, the lateral sides are perpendicular to the bases of prism. In the picture below, compare a right prism to an oblique prism, in which sides and bases are not perpendicular:
1. Fill in the blanks:
A prism is a _____-dimensional figure with parallel and congruent ________________ and sides that are __________________________________.
2. In a right prism, what is the relationship between the sides and the bases?
3. What is the difference between a right prism and an oblique prism?
Surface area is the exposed area of a 3-dimensional or solid figure. This means that surface area is a measurement of the outside area of an object. For example, the surface area of a soccer ball is the outside part of the ball that touches the air. The inside of the ball is not included in surface area. Surface area is, as its name describes, the area of the surface of a solid object.
Surface area is the area of the _________________________ of a 3-dimensional object.
Area Addition Postulate
The surface area of a 3-dimensional figure is the sum of the areas of all of its non-overlapping parts.
This means that we find the surface area by adding up all of the face areas of the figure. Since the faces are on the surface of the object, if we add up all of the faces, we will have the entire surface area!
To find the surface area of a shape, you add the areas of its _______________________.
You can use a net and the Area Addition Postulate to find the surface area of a right prism:
From the net of the prism, you can see that that the surface area of the entire prism equals the sum of the shapes that make up the net. Since there are 6 faces of the prism and 6 shapes in the net, there are 6 areas that you add together:
To find the areas of shapes in the net, we must use the formula for the area of a rectangle:
Find the areas of the other rectangles in the net:
Then insert these areas back into the surface area equation above:
If 2 polygons (or plane figures) are congruent, then their areas are congruent.
This means that if 2 shapes are the same, then they have the same area! This may seem simple, and it is!
Use a net to find the surface area of the prism.
The area of the net is equal to the surface area of the figure:
Now insert the areas you found back into the surface area equation on the previous page:
1. True/False: The surface area of a 3-D figure is the same as the area of its net.
2. True/False: You can figure out the area of a net by finding the area of each of the shapes in it (rectangles, triangles, etc.) and multiplying them together.
Surface Area of a Prism Using Perimeter
The hexagonal prism below has 2 regular hexagons for bases and 6 sides. Since all sides of the hexagon are congruent, all of the rectangles that make up the lateral sides of the 3-dimensional figure are also congruent.
You can break down the figure like this:
The word “lateral” means “on the side.”
For example, in football, a lateral pass is when the quarterback throws the ball sideways to a receiver: the ball is passed to the side instead of forwards.
Lateral sides of a solid are the faces on the sides of the shape (not the bases).
Lateral area, which you will learn next, is the area of the sides of the shape.
The surface area of the rectangular sides of the figure is called the lateral area of the figure. The lateral area does not include the area of the bases. To find the lateral area, you can add up all of the areas of the 6 rectangles:
The lateral area is the surface area of the rectangular ____________________ of a right prism.
The lateral area does not include the area of the _____________________.
Find the lateral area of a right prism by multiplying the ________________________ of the base by the ________________________ of the prism.
We can use the lateral area formula to calculate the total surface area of the prism. Remember, lateral area does not include the area of the bases, but we must include the bases to find the total surface area of the prism!
1. True/False: The lateral area includes the area of the sides and the area of the bases.
2. True/False: The perimeter of the base and the area of the base are the same thing.
3. How could you find the total surface area of a 3-dimensional shape if you know the height of the shape and the perimeter of its base? What other information would you need to know? Explain.
Use the formula to find the total surface area of the trapezoidal prism below.
The dimensions of the trapezoidal base are shown in the diagram, but we should list our information so it is easy to see:
Use the formula for total surface area from the previous page:
Now find the perimeter of the base:
Use these values to find the total surface area of the solid: