Nets, volume, and surface area of prisms
Defining nets, volume, and surface area for different prisms
In this lesson we’ll look at an introduction to three-dimensional geometric figures, specifically nets, volume, and surface area of prisms.
Prism
A prism is a figure that has two congruent bases of any shape, and whose sides are made up of rectangles. Here are some examples of prisms:
The bases of a prism are the pairs of shapes that are congruent, and the height is the length that connects the two bases. In a rectangular prism, any two pairs of congruent rectangles that are connected by a height can be used as the bases.
Net
A net of a polyhedron is a two-dimensional flattened out version of it. We make the net by cutting the polyhedron along one or more of the edges until we can lay out the whole thing flat in a plane.
Once we have the net of a polyhedron, we should be able to reconstruct it by folding the pieces of the net, using the polygons in the net as faces, and using each line segment in the net as the boundary between some pair of faces of the polyhedron. Here is a triangular prism (on the left) and its net (on the right).
Surface area of a rectangular prism
You can also think of each side of a rectangular prism as a rectangle with a specific area, and the surface area of the prism is the sum of the area of all six sides.
The area of each side of the rectangular prism is given in the table:
The formula for surface area of a rectangular prism is
???A=2lw+2wh+2lh???
Surface area of a prism
To find the surface area of a prism, it can be helpful to sketch the net, find the area of each shape in the net, and then add the areas together. To find the surface area of this triangular prism, find the area of the three rectangles and two triangles in its net and add all the areas together.
Volume of a rectangular prism
The volume of a rectangular prism is the length times the width times the height.
???V=lwh???
Volume of a prism
The volume of any prism is the area of the base times the height.
???V=\text{(area of base)(height)}???
The volume of this triangular prism is the area of one of the triangles, multiplied by the length of the side that connects the two triangular bases.
Calculating surface area of prisms
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Example
Which net does not belong to a rectangular prism?
Net C is the net of a triangular pyramid. All of the other nets are examples of prisms, because they have two congruent bases and the rest of the faces are rectangles.
Net A is a triangular prism with two triangular bases.
Net B is another example of a triangular prism.
Net C is a triangular pyramid and not a prism.
Net D is a rectangular prism.
Let’s do an example of surface area.
Example
What is the surface area of the figure?
It can be helpful to draw the net of the figure to calculate the surface area.
Now you can see we have three pairs of shapes. We can find the area of each and add them together. We have:
???A=2(2\cdot 3)+2(3\cdot 6)+2(2\cdot 6)???
???A=2(6)+2(18)+2(12)???
???A=12+36+24???
???A=72\ \text{cm}^2???
You can also think of the surface area of a rectangular box as the sum of the surface area of its six sides.
We'll use the surface area formula.
???A=2lw+2wh+2lh???
Plugging in the dimensions of the box we’ve been given, we get
???A=2(2\cdot 3)+2(3\cdot 6)+2(2\cdot 6)???
???A=2(6)+2(18)+2(12)???
???A=12+36+24???
???A=72\ \text{cm}^2???
Which is what we got by drawing the net.
Let’s do an example with volume.
Example
What is the volume of the figure?
To find the volume of a rectangular prism, multiply the length, width, and height.
???V=lwh???
???V=2(6)(3)???
???V=36\ \text{cm}^2???