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High School Precalculus: Help and Review32 chapters | 297 lessons

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Lesson Transcript

Instructor:
*Miriam Snare*

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

This lesson will teach you about the slant height of cones and pyramids. You will learn the formula to calculate the slant height. After the lesson, you will be able to test your knowledge with a quiz.

Figures such as cones and pyramids have two measurements that indicate how tall the figure is. One of these measurements is called the **slant height** and the other is called the **altitude**.

Let's consider a traffic cone to help us learn the difference between these two measurements. If you measure the distance along the outside of the cone from the top to the base, that measurement is the **slant height**. To measure it, you have to place the yardstick at a slant.

If you drop a yardstick straight down through the hole in the top of the cone until it hits the bottom, that measurement is the **altitude**. The yardstick in this case will make a right angle at the center of the base. Take a look at the picture to see these measurements labeled.

Note that the dashed segment of the altitude indicates that it is inside of the cone.

Our traffic cone is a little different from the geometric shape called a cone. In geometry, the base of a cone is only a circle that does not extend beyond the opening of the cone. The point of a cone in geometry is called the **vertex point**. The slant height and the altitude always meet at that vertex point in a cone. On the traffic cone, the two segments did not meet because the tip is flat and does not come to one point.

In a pyramid, you also have a slant height and an altitude. Imagine you are standing at the top of a pyramid at the vertex point, and you want to get back to ground level. You have two options. You can slide down the center of the triangular side of the pyramid - the slant height - to the outside of the pyramid. Otherwise, you can drop straight down through a hole in the top and land in the middle of the inside of the pyramid - the altitude.

To calculate the slant height of either a cone or a pyramid, you need to imagine that you can look inside of the figure. Let's cut open a cone as an example. First, we cut down through the cone from vertex point *A* to segment *BC* to get two halves. The cut surface of either half is now in the shape of an **isosceles triangle**, which is a triangle with two sides that are the same length. Those two sides were the slant height of the cone. We now have triangle *ABC*, where sides *AB* and *AC* have the same length.

Now, we draw in the altitude of the cone from A straight down to *BC*, so that a right angle is created. Point *M* is the point where the altitude meets *BC*. Triangle *AMC* is a right triangle where *AM* is the altitude and *AC* is the slant height of the original cone. *AC* is also the **hypotenuse** of the right triangle, since it is the side opposite the right angle.

Every cone and pyramid contains a right triangle if we cut up the figure like we did in this example.

We can use the **Pythagorean theorem**, ** a^2 + b^2 = c^2**, to calculate the slant height. For both cones and pyramids,

Let's go through an example where we calculate the length of the slant height of a pyramid. In this example, we are given that the altitude of the pyramid measures 8 inches and each side of the base is 12 inches long. The altitude is the dashed red segment *DM* and the slant height is the purple segment *DY*.

To calculate the length of the slant height, we need to find the right triangle inside the pyramid. That triangle is made up of points *D*, *M*, and *Y*. Let's draw and label the lengths of its sides.

The red segment *DM* measured 8 inches and that same segment is one side of the triangle. The purple segment *DY* was the slant height of the pyramid, and it forms the hypotenuse of the triangle. *DY* is the length we are trying to calculate, so we will give it the variable *c*.

Finally, we need to know the length of *YM* to have enough information to solve this problem. *YM* was not labeled on the pyramid. We were given that the sides of the base of the pyramid measure 12 inches, so we had labeled *RA* with that measurement. *YM* is half the length of a side because *M* is in the middle of the square base. So, *YM* is 6 inches.

Now, we have all of the information we need to find the slant height of this pyramid. We use the formula *a*^2 + *b*^2 = *c*^2. As we discussed previously, *a* stands for the altitude, so we substitute 8 for the *a*. *b* is half the length of the side of the base, so we use 6 for *b*. Substituting those values, our equation becomes 8^2 + 6^2 = *c*^2.

Then, we have to simplify: 64 + 36 = *c*^2.

Then, we add: 100 = *c*^2

Finally, we take the square root of both sides of the equation to get *c* = 10.

So, the slant height of the pyramid is 10 inches.

- The
**vertex point**is the point of a cone or pyramid. - The
**slant height**of a cone or pyramid is the length of a segment from the vertex point to the base along the outside of the shape. - The
**altitude**of a cone or pyramid is the length of a segment from the vertex point to center of the base inside of the shape, forming a right angle at the base. - The slant height and the altitude are both sides of a right triangle that can be visualized inside every cone and pyramid.
- The
**slant height**can be calculated using the formula*a*^2 +*b*^2 =*c*^2. In the formula,*a*is the altitude,*b*is the distance from the center of the base to the point where the slant height segment starts, and*c*stands for the slant height.

Slant height measures how tall a cone or a pyramid is from the outside. You can use the Pythagorean theorem (a^2 + b^2 = c^2) to determine the slant height, where: *a* = altitude

*b* = distance from the center of the base to the edge of the base

*c* = slant height

After reviewing this lesson on slant height, you should be able to

- Contrast slant height and altitude
- Recall the formula for slant height
- Calculate slant height

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High School Precalculus: Help and Review32 chapters | 297 lessons

- Solving Equations & Inequalities Involving Rational Functions 7:07
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