CK-12 Geometry
Page 41
Introduction
A pyramid is a three-dimensional figure with a single base and a three or more non-parallel sides that meet at a single point above the base. The sides of a pyramid are triangles.
A regular pyramid is a pyramid that has a regular polygon for its base and whose sides are all congruent triangles.
Surface Area of a Pyramid Using Nets
You can deconstruct a pyramid into a net.
To find the surface area of the figure using the net, first find the area of the base:
Now find the area of each isosceles triangle. Use the Pythagorean Theorem to find the height of the triangles. This height of each triangle is called the slant height of the pyramid. The slant height of the pyramid is the altitude of one of the triangles. Notice that the slant height is larger than the altitude of the triangle.
We’ll call the slant height for this problem. Using the Pythagorean Theorem:
Now find the area of each triangle:
As there are triangles:
Finally, add the total area of the triangles to the area of the base.
Example 1
Use the net to find the total area of the regular hexagonal pyramid with an apothem of . The dimensions are given in centimeters.
The area of the hexagonal base is given by the formula for the area of a regular polygon. Since each side of the hexagon measures , the perimeter is or . The apothem, or perpendicular distance to the center of the hexagon is .
Using the Pythagorean Theorem to find the slant height of each lateral triangle.
Now find the area of each triangle:
Together, the area of all six triangles that make up the lateral sides of the pyramid are
Add the area of the lateral sides to the area of the hexagonal base.
Surface Area of a Regular Pyramid
To get a general formula for the area of a regular pyramid, look at the net for this square pyramid.
The slant height of each lateral triangle is labeled (the lowercase letter ), and the side of the regular polygon is labeled . For each lateral triangle, the area is:
There are triangles in a regular polygon—e.g., for a triangular pyramid, for a square pyramid, for a pentagonal pyramid. So the total area, , of the lateral triangles is:
If we rearrange the above equation we get:
Notice that is just the perimeter, , of the regular polygon. So we can substitute into the equation to get the following postulate.
To get the total area of the pyramid, add the area of the base, , to the equation above.
Area of a Regular Pyramid
The surface area of a regular pyramid is
where is the slant height of the pyramid and is the perimeter of the regular polygon that forms the pyramid’s base, and is the area of the base.
Example 2
A tent without a bottom has the shape of a hexagonal pyramid with a slant height of . The sides of the hexagonal perimeter of the figure each measure . Find the surface area of the tent that exists above ground.
For this problem, is zero because the tent has no bottom. So simply calculate the lateral area of the figure.
Example 3
A pentagonal pyramid has a slant height of . The sides of the pentagonal perimeter of the figure each measure . The apothem of the figure is . Find the surface area of the figure.
First find the lateral area of the figure.
Now use the formula for the area of a regular polygon to find the area of the base.
Finally, add these together to find the total surface area.
Estimate the Volume of a Pyramid and Prism
Which has a greater volume, a prism or a pyramid, if the two have the same base and height? To find out, compare prisms and pyramids that have congruent bases and the same height.
Here is a base for a triangular prism and a triangular pyramid. Both figures have the same height. Compare the two figures. Which one appears to have a greater volume?
The prism may appear to be greater in volume. But how can you prove that the volume of the prism is greater than the volume of the pyramid? Put one figure inside of the other. The figure that is smaller will fit inside of the other figure.
This is shown in the diagram on the above. Both figures have congruent bases and the same height. The pyramid clearly fits inside of the prism. So the volume of the pyramid must be smaller.
Example 4
Show that the volume of a square prism is greater than the volume of a square pyramid.
Draw or make a square prism and a square pyramid that have congruent bases and the same height.
Now place the one figure inside of the other. The pyramid fits inside of the prism. So when two figures have the same height and the same base, the prism’s volume is greater.
In general, when you compare two figures that have congruent bases and are equal in height, the prism will have a greater volume than the pyramid.
The reason should be obvious. At the “bottom,” both figures start out the same—with a square base. But the pyramid quickly slants inward, “cutting away” large amounts of material while the prism does not slant.
Find the Volume of a Pyramid and Prism
Given the figure above, in which a square pyramid is placed inside of a square prism, we now ask: how many of these pyramids would fit inside of the prism?
To find out, obtain a square prism and square pyramid that are both hollow, both have no bottom, and both have the same height and congruent bases.
Now turn the figures upside down. Fill the pyramid with liquid. How many full pyramids of liquid will fill the prism up to the top?
In fact, it takes exactly three full pyramids to fill the prism. Since the volume of the prism is:
where stands for the area of the base and is the height of the prism, we can write:
or:
And, since the volume of a square prism is , we can write:
This can be written as the Volume Postulate for pyramids.
Volume of a Pyramid
Given a right pyramid with a base that has area and height :
Example 5
Find the volume of a pyramid with a right triangle base with sides that measure , , and . The height of the pyramid is .
First find the area of the base. The longest of the three sides that measure , , and must be the hypotenuse, so the two shorter sides are the legs of the right triangle.
Now use the postulate for the volume of a pyramid.
Example 6
Find the altitude of a pyramid with a regular pentagonal base. The figure has an apothem of , sides, and a volume of .
First find the area of the base.
Now use the value for the area of the base and the postulate to solve for .
Review Questions
Consider the following figure in answering questions 1 – 4.
What type of pyramid is this?
Triangle is what part of the figure?
Segment AE is _______________ of the figure.
Point is the ________________________
How many faces are there on a pyramid whose base has ?
A right pyramid has a regular hexagon for a base. Each edge measures . Find
The lateral surface area of the pyramid
The total surface area of the pyramid
The volume of the pyramid
The Transamerica Building in San Francisco is a pyramid. The length of each edge of the square base is and the slant height of the pyramid is . What is the lateral area of the pyramid? How tall is the building?
Given the following pyramid:
With , and what is the value of ?
Review Answers
Rectangular pyramid
Lateral face
Edge
Apex
Cones
Learning Objectives
Find the surface area of a cone using a net or a formula.
Find the volume of a cone.
Introduction
A cone is a three-dime
nsional figure with a single curved base that tapers to a single point called an apex. The base of a cone can be a circle or an oval of some type. In this chapter, we will limit the discussion to circular cones. The apex of a right cone lies above the center of the cone’s circle. In an oblique cone, the apex is not in the center.
The height of a cone, , is the perpendicular distance from the center of the cone’s base to its apex.
Surface Area of a Cone Using Nets
Most three-dimensional figures are easy to deconstruct into a net. The cone is different in this regard. Can you predict what the net for a cone looks like? In fact, the net for a cone looks like a small circle and a sector, or part of a larger circle.
The diagrams below show how the half-circle sector folds to become a cone.
Note that the circle that the sector is cut from is much larger than the base of the cone.
Example 1
Which sector will give you a taller cone—a half circle or a sector that covers three-quarters of a circle? Assume that both sectors are cut from congruent circles.
Make a model of each sector.
The half circle makes a cone that has a height that is about equal to the radius of the semi-circle.
The three-quarters sector gives a cone that has a wider base (greater diameter) but its height as not as great as the half-circle cone.
Example 2
Predict which will be greater in height—a cone made from a half-circle sector or a cone made from a one-third-circle sector. Assume that both sectors are cut from congruent circles.
The relationship in the example above holds true—the greater (in degrees) the sector, the smaller in height of the cone. In other words, the fraction is less than , so a one-third sector will create a cone with greater height than a half sector.
Example 3
Predict which will be greater in diameter—a cone made from a half-circle sector or a cone made from a one-third-circle sector. Assume that the sectors are cut from congruent circles
Here you have the opposite relationship—the larger (in degrees) the sector, the greater the diameter of the cone. In other words, is greater than , so a one-half sector will create a cone with greater diameter than a one-third sector.
Surface Area of a Regular Cone
The surface area of a regular pyramid is given by:
where is the slant height of the figure, is the perimeter of the base, and is the area of the base.
Imagine a series of pyramids in which , the number of sides of each figure’s base, increases.
As you can see, as increases, the figure more and more resembles a circle. So in a sense, a circle approaches a polygon with an infinite number of sides that are infinitely small.
In the same way, a cone is like a pyramid that has an infinite number of sides that are infinitely small in length.
Given this idea, it should come as no surprise that the formula for finding the total surface area of a cone is similar to the pyramid formula. The only difference between the two is that the pyramid uses , the perimeter of the base, while a cone uses , the circumference of the base.
Surface Area of a Right Cone
The surface area of a right cone is given by:
Since the circumference of a circle is :
You can also express as to get:
Any of these forms of the equation can be used to find the surface area of a right cone.
Example 4
Find the total surface area of a right cone with a radius of and a slant height of .
Use the formula:
Example 5
Find the total surface area of a right cone with a radius of and an altitude (not slant height) of .
Use the Pythagorean Theorem to find the slant height:
Now use the area formula.
Volume of a Cone
Which has a greater volume, a pyramid, cone, or cylinder if the figures have bases with the same "diameter" (i.e., distance across the base) and the same altitude? To find out, compare pyramids, cylinders, and cones that have bases with equal diameters and the same altitude.
Here are three figures that have the same dimensions—cylinder, a right regular hexagonal pyramid, and a right circular cone. Which figure appears to have a greater volume?
It seems obvious that the volume of the cylinder is greater than the other two figures. That’s because the pyramid and cone taper off to a single point, while the cylinder’s sides stay the same width.
Determining whether the pyramid or the cone has a greater volume is not so obvious. If you look at the bases of each figure you see that the apothem of the hexagon is congruent to the radius of the circle. You can see the relative size of the two bases by superimposing one onto the other.
From the diagram you can see that the hexagon is slightly larger in area than the circle. So it follows that the volume of the right hexagonal regular pyramid would be greater than the volume of a right circular cone. And indeed it is, but only because the area of the base of the hexagon is slightly greater than the area of the base of the circular cone.
The formula for finding the volume of each figure is virtually identical. Both formulas follow the same basic form:
Since the base of a circular cone is, by definition, a circle, you can substitute the area of a circle, for the base of the figure. This is expressed as a volume postulate for cones.
Volume of a Right Circular Cone
Given a right circular cone with height and a base that has radius :
Example 6
Find the volume of a right cone with a radius of and a height of .
Use the formula:
Example 7
Find the volume of a right cone with a radius of and a slant height of .
Use the Pythagorean theorem to find the height:
Now use the volume formula.
Review Questions
Find the surface area of
Find the surface area of
Find the surface area of a cone with a height of and a base area of
In problems 4 and 5 find the missing dimension. Round to the nearest tenth of a unit.
Cone:
Cone:
A cone shaped paper cup is high with a diameter of . If a plant needs of water, about how many paper cups of water are needed to water it?
In problems 7 and 8 refer to this diagram. It is a diagram of a yogurt container. The yogurt container is a truncated cone.
What is the surface area of the container?
What is the volume of the container?
Find the height of a cone that has a radius of and a volume of
A cylinder has a volume of and a base radius of . What is the volume of a cone with the same height but a base radius of ?
Review Answers
(approximately)
Spheres
Learning Objectives
Find the surface area of a sphere.
Find the volume of a sphere.
Introduction
A sphere is a three-dimensional figure that has the shape of a ball.
Spheres can be characterized in three ways.
A sphere is the set of all points that lie a fixed distance from a single center point .
A sphere is the surface that results when a circle is rotated about any of its diameters.
A sphere results when you construct a polyhedron with an infinite number of faces that are infinitely small. To see why this is true, recall regular polyhedra.
As the number of faces on the figure increases, each face gets smaller in area and the figure comes more to resemble a sphere. When you imagine figure with an infinite number of faces, it would look like (and be!) a sphere.
Parts of a Sphere
As described above, a sphere is the surface that is the set of all points a fixed distance from a center point . Terminology for spheres is similar for that of circles.
The distance from to the surface of the sphere is , the radius.
The diameter, , of a sphere is the lengt
h of the segment connecting any two points on the sphere’s surface and passing through . Note that you can find a diameter (actually an infinite number of diameters) on any plane within the sphere. Two diameters are shown in each sphere below.
A chord for a sphere is similar to the chord of a circle except that it exists in three dimensions. Keep in mind that a diameter is a kind of chord—a special chord that intersects the center of the circle or sphere.
A secant is a line, ray, or line segment that intersects a circle or sphere in two places and extends outside of the circle or sphere.
A tangent intersects the circle or sphere at only one point.
In a circle, a tangent is perpendicular to the radius that meets the point where the tangent intersects with the circle. The same thing is true for the sphere. All tangents are perpendicular to the radii that intersect with them.