If the pyramid is cut with a plane perpendicular to the base but not through the top vertex, what is the cross section? trapezoid
Sketch the shape of the plane surface at the cut of this solid figure.
pentagon
Prisms
Learning Objectives
Use nets to represent prisms.
Find the surface area of a prism.
Find the volume of a prism.
Introduction
A prism is a three-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.
Surface Area of a Prism Using Nets
The prisms above are right prisms. In a right prism, the lateral sides are perpendicular to the bases of prism. Compare a right prism to an oblique prism, in which sides and bases are not perpendicular.
Two postulates that apply to area are the Area Congruence Postulate and the Area Addition Postulate.
Area Congruence Postulate:
If two polygons (or plane figures) are congruent, then their areas are congruent.
Area Addition Postulate:
The surface area of a three-dimensional figure is the sum of the areas of all of its non-overlapping parts.
You can use a net and the Area Addition Postulate to find the surface area of a right prism.
From the net, you can see that that the surface area of the entire prism equals the sum of the figures that make up the net:
Using the formula for the area of a rectangle, you can see that the area of rectangle is:
Similarly, the areas of the other rectangles are inserted back into the equation above.
Example 9
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. To find the area of the triangle, we use the formula:
where is the height of the triangle and is its base.
Note that triangles and are congruent so we can multiply the area of triangle by .
Thus, the surface area is .
Surface Area of a Prism Using Perimeter
This hexagonal prism has two regular hexagons for bases and six sides. Since all sides of the hexagon are congruent, all of the rectangles that make up the lateral sides of the three-dimensional figure are also congruent. You can break down the figure like this.
The surface area of the rectangular sides of the figure is called the lateral area of the figure. To find the lateral area, you could add up all of the areas of the rectangles.
Notice that is the perimeter of the base. So another way to find the lateral area of the figure is to multiply the perimeter of the base by , the height of the figure.
Substituting , the perimeter, for , we get the formula for any lateral area of a right prism:
Now we can use the formula to calculate the total surface area of the prism. Using for the perimeter and for the area of a base:
To find the surface area of the figure above, first find the area of the bases. The regular hexagon is made of six congruent small triangles. The altitude of each triangle is the apothem of the polygon. Note: be careful here—we are talking about the altitude of the triangles, not the height of the prism. We find the length of the altitude of the triangle using the Pythagorean Theorem,
So the area of each small triangle is:
The area of the entire hexagon is therefore:
You can also use the formula for the area of a regular polygon to find the area of each base:
Now just substitute values to find the surface area of the entire figure above.
You can use the formula to find the surface area of any right prism.
Example 10
Use the formula to find the total surface area of the trapezoidal prism.
The dimensions of the trapezoidal base are shown. Set up the formula. We’ll call the height of the entire prism to avoid confusion with , the height of each trapezoidal base.
Now find the area of each trapezoidal base. You can do this by using the formula for the area of a trapezoid. (Note that the height of the trapezoid, is small .)
Now find the perimeter of the base.
Now find the total surface area of the solid.
Volume of a Right Rectangular Prism
Volume is a measure of how much space a three-dimensional figure occupies. In everyday language, the volume tells you how much a three-dimensional figure can hold. The basic unit of volume is the cubic unit—cubic centimeter, cubic inch, cubic meter, cubic foot, and so on. Each basic cubic unit has a measure of for its length, width, and height.
Two postulates that apply to volume are the Volume Congruence Postulate and the Volume Addition Postulate.
Volume Congruence Postulate
If two polyhedrons (or solids) are congruent, then their volumes are congruent.
Volume Addition Postulate
The volume of a solid is the sum of the volumes of all of its non-overlapping parts.
A right rectangular prism is a prism with rectangular bases and the angle between each base and its rectangular lateral sides is also a right angle. You can recognize a right rectangular prism by its “box” shape.
You can use the Volume Addition Postulate to find the volume of a right rectangular prism by counting boxes. The box below measures in height, in width, and in depth. Each layer has or .
Together, you get three groups of so the total volume is:
The volume is .
This same pattern holds for any right rectangular prism. Volume is giving by the formula:
Example 11
Find the volume of this box.
Use the formula for volume of a right rectangular prism.
So the volume of this rectangular prism is .
Volume of a Right Prism
Looking at the volume of right prisms with the same height and different bases you can see a pattern. The computed area of each base is given below. The height of all three solids is the same, .
Putting the data for each solid into a table, we get:
Solid Height Area of base Volume
Box
Trapezoid
Triangle
The relationship in each case is clear. This relationship can be proved to establish the following formula for any right prism:
Volume of a Right Prism
The volume of a right prism is .
where is the area of the base of the three-dimensional figure, and is the prism’s height (also called altitude).
Example 12
Find the volume of the prism with a triangular equilateral base and the dimensions shown in centimeters.
To find the volume, first find the area of the base. It is given by:
The height (altitude) of the triangle is . Each side of the triangle measures . So the triangle has the following area.
Now use the formula for the volume of the prism, , where is the area of the base (i.e., the area of the triangle) and is the height of the prism. Recall that the "height" of the prism is the distance between the bases, so in this case the height of the prism is . You can imagine that the prism is lying on its side.
Thus, the volume of the prism is
Example 13
Find the volume of the prism with a regular hexagon for a base and sides.
You don’t know the apothem of the figure’s base. However, you do know that a regular hexagon is divided into six congruent equilateral triangles.
You can use the Pythagorean Theorem to find the apothem. The right triangle measures by by , the apothem.
Thus, the volume of the prism is given by:
Review Questions
For each of the following find the surface area using
a. the method of nets and
b. the perimeter.
The base of a prism is a right triangle whose legs are and and show height is . What is the total area of the prism?
A right hexagonal prism is tall and has bases that are regular
hexagons measuring on a side. What is the total surface area?
What is the volume of the prism in problem #4?
For problems 6 and 7:
A barn is shaped like a pentagonal prism with dimensions shown in feet:
How many square feet (excluding the roof) are there on the surface of the barn to be painted?
If a gallon of paint covers , how many gallons of paint are needed to paint the barn?
A cardboard box is a perfect cube with an edge measuring . How many cubic feet can it hold?
A swimming pool is wide, long and is uniformly deep. How many cubic feet of water can it hold?
A cereal box has length , width and height . How much cereal can it hold?
Review Answers
of paint
(be careful here. The units in the problem are given in inches but the question asks for feet.)
Cylinders
Learning Objectives
Find the surface area of cylinders.
Find the volume of cylinders.
Find the volume of composite three-dimensional figures.
Introduction
A cylinder is a three-dimensional figure with a pair of parallel and congruent circular ends, or bases. A cylinder has a single curved side that forms a rectangle when laid out flat.
As with prisms, cylinders can be right or oblique. The side of a right cylinder is perpendicular to its circular bases. The side of an oblique cylinder is not perpendicular to its bases.
Surface Area of a Cylinder Using Nets
You can deconstruct a cylinder into a net.
The area of each base is given by the area of a circle:
The area of the rectangular lateral area is given by the product of a width and height. The height is given as . You can see that the width of the area is equal to the circumference of the circular base.
To find the width, imagine taking a can-like cylinder apart with a scissors. When you cut the lateral area, you see that it is equal to the circumference of the can’s top. The circumference of a circle is given by
the lateral area, , is
Now we can find the area of the entire cylinder using .
You can see that the formula we used to find the total surface area can be used for any right cylinder.
Area of a Right Cylinder
The surface area of a right cylinder, with radius and height is given by , where is the area of each base of the cylinder and is the lateral area of the cylinder.
Example 1
Use a net to find the surface area of the cylinder.
First draw and label a net for the figure.
Calculate the area of each base.
Calculate .
Find the area of the entire cylinder.
Thus, the total surface area is approximately
Surface Area of a Cylinder Using a Formula
You have seen how to use nets to find the total surface area of a cylinder. The postulate can be broken down to create a general formula for all right cylinders.
Notice that the base, , of any cylinder is:
The lateral area, , for any cylinder is:
Putting the two equations together we get:
Factoring out a from the equation gives:
The Surface Area of a Right Cylinder
A right cylinder with radius and height can be expressed as:
or:
You can use the formulas to find the area of any right cylinder.
Example 2
Use the formula to find the surface area of the cylinder.
Write the formula and substitute in the values and solve.
Example 3
Find the surface area of the cylinder.
Write the formula and substitute in the values and solve.
Example 4
Find the height of a cylinder that has radius and surface area of .
Write the formula with the given information and solve for .
Volume of a Right Cylinder
You have seen how to find the volume of any right prism.
where is the area of the prism’s base and is the height of the prism.
As you might guess, right prisms and right cylinders are very similar with respect to volume. In a sense, a cylinder is just a “prism with round bases.” One way to develop a formula for the volume of a cylinder is to compare it to a prism. Suppose you divided the prism above into slices that were thick.
The volume of each individual slice would be given by the product of the area of the base and the height. Since the height for each slice is , the volume of a single slice would be:
Now it follows that the volume of the entire prism is equal to the area of the base multiplied by the number of slices. If there are slices, then:
Of course, you already know this formula from prisms. But now you can use the same idea to obtain a formula for the volume of a cylinder.
Since the height of each unit slice of the cylinder is , each slice has a volume of , or . Since the base has an area of , each slice has a volume of and:
This leads to a postulate for the volume of any right cylinder.
Volume of a Right Cylinder
The volume of a right cylinder with radius and height can be expressed as:
Example 5
Use the postulate to find the volume of the cylinder.
Write the formula from the postulate. Then substitute in the values and solve.
Example 6
What is the radius of a cylinder with height and a volume of ?
Write the formula. Solve for .
Composite Solids
Suppose this pipe is made of metal. How can you find the volume of metal that the pipe is made of?
The basic process takes three steps.
Step 1: Find the volume of the entire cylinder as if it had no hole.
Step 2: Find the volume of the hole.
Step 3: Subtract the volume of the hole from the volume of the entire cylinder.
Here are the steps carried out. First, use the formula to find the volume of the entire cylinder. Note that since , the diameter of the pipe, is , the radius is half of the diameter, or .
Now find the volume of the inner empty “hole” in the pipe. Since the pipe is thick, the diameter of the hole is less than the diameter of the outer part of the pipe.
The radius of the hole is half of or .
Now subtract the hole from the entire cylinder.
Example 7
Find the solid volume of this cinder block. Its edges are thick all around. The two square holes are identical in size.
Find the volume of the entire solid block figure. Subtract the volume of the two holes.
To find the volume of the three-dimensional figure:
Now find the length of the sides of the two holes. The width of the entire block is . This is equal to:
So the sides of the square holes are by .
Now the volume of each square hole is:
Finally, subtract the volume of the two holes from the volume of the entire brick.
Review Questions
Complete the following sentences. They refer to the figure above.
The figure above is a _________________________
The shape of the lateral face of the figure is _____________________________
The shape of a base is a(n) _____________________________
Segment LV is the ___________________________
Draw the net for this cylinder and use the net to find the surface area of the cylinder.
Use the formula to find the volume of this cylinder.
Matthew’s favorite mug is a cylinder that has a base area of and a height of . How much coffee can he put in his mug?
Given the following two cylinders which of the following statements is true:
Suppose you work for a company that makes cylindrical water tanks. A customer wants a tank that measures in height and in diameter. How much metal should you order to make this tank?
If the radius of a cylinder is d
oubled what effect does the doubling have on the volume of this cylinder? Explain your answer.
Review Answers
Cylinder
Rectangle
Circle
Height
The volume will be quadrupled
Pyramids
Learning Objectives
Identify pyramids.
Find the surface area of a pyramid using a net or a formula.
Find the volume of a pyramid.
CK-12 Geometry Page 40