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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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Lesson Transcript

Instructor:
*Eric Istre*

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

This lesson is about detailing the characteristics of a rectangular pyramid and how to find its surface area and volume. Examples of each are included along with the step by step calculations and sample pyramid diagrams to help with visualization.

Imagine that you lived several thousand years ago. The pharaoh has asked you to find out how much treasure he can possibly store inside one of his pyramids. He also wants to seal the surface of the pyramid so that it will last forever, and he wants to know how much surface he will need to seal. He will not accept failure! What will you do? How are you going to be able to do what he wants?

The first and most important thing to know is exactly what is the shape of a pyramid. Pyramids are named by the type of base that they have. A **rectangular pyramid** has a rectangle for a base with triangles for the other faces. Sometimes the rectangle at the base is actually a square. The pyramids in Egypt are rectangular pyramids. A pyramid culminates at a peak, sometimes called the **apex**. A pyramid can be a **right pyramid** in which the apex is directly over the center of the base or an **oblique pyramid** in which the apex is not directly over the center of the base.

Think about the first request the pharaoh made. The pharaoh wants to know how much space is inside the pyramid. This is known as the **volume**. Volume is the amount of space inside a three-dimensional object, like a pyramid, and it must always be given in cubed units. The Great Pyramid in Egypt is 455 feet tall and each side of the base is 756 feet long, making it a square pyramid with the length and width both being the aforementioned 756 feet.

The volume of a pyramid, right or oblique, has the following formula:

Volume = (length * width * height) / 3

To solve for the volume, plug the value of each of the dimensions into the formula and calculate.

Volume = (756 ft * 756 ft * 455 ft) / 3

Volume = 86,682,960 ft3

The pharaoh will have to have a lot of treasure to fill all of that space up!

The pharaoh's second request was what would it take to seal the pyramid? To answer this question, the surface area of the pyramid needs to be determined. **Surface area**, or the total area of all the outside faces of the object, can be a bit trickier to calculate than volume. The formula for surface area is:

Surface area = (1/2 * base perimeter * slant height) + base area

The slant height is needed to solve this equation, which was not required for the volume calculation. If the slant height is unknown, use the **Pythagorean theorem** to figure it out. The Pythagorean theorem is:

a2 + b2 = c2

For example, without the slant height of the pharaoh's pyramid, construct a right triangle with the pyramid's height and half the length of the base as the legs of the right triangle. The longest side, or **hypotenuse**, of the triangle would be your slant height. Using the Pythagorean theorem, follow these steps:

c2 = (455 ft)2 + (756 / 2 ft)2

c2 = 349,909 ft2

c = âˆš349,909ft2 = 591.5 ft

That's how to find slant height in case it's not given.

Now, even though the base of the pyramid would probably not need to be sealed, it will be included in the calculations for surface area in this example. Now that the slant height is known, plug all of the values into the formula:

Surface area = (1/2 * (756 + 756 + 756 + 756) ft * 591.5 ft) + (756 ft * 756 ft)

Surface area = (0.5 * 3024 ft * 591.5 ft) + (756 ft * 756 ft)

Surface area = (894,348 ft2) + (571,536 ft2)

Surface area = 1,465,884 ft2

The pharaoh better hire a lot of camels to carry in all of the sealant needed for that!

If the base is not a regular figure like a square or if the pyramid is oblique, then the slant height will be different on some of the faces. This will mean that the area of each face will need to be calculated individually, and then the sum of all of the faces will be the total surface area. For example, if the length and width of the base are different, the surface area calculation will look different. This time the base is not a square; the lengths of the sides of the base this time are 600 feet and 756 feet.

Don't forget that the area of a triangle is (base * height) / 2.

Total surface area = base area + area of right face + area of left face + area of front face + area of back face

Total surface area = (756 * 600) + (756 * 591.5) / 2 + (756 * 591.5) / 2 + (600 * 545) / 2 + (600 * 545) / 2

Total surface area = 453,600 + 223,587 + 223,587 + 163,500 + 163,500

The sum of these numbers is the total surface area for a pyramid that does not have a square base.

Total surface area= 1,227,774 ft2

A rectangular pyramid is a common three dimensional shape. Its volume is calculated the same way whether it is **oblique** (meaning tilted) or right:

Volume = (length * width * height) / 3

Its surface area is easier to calculate if it is a right pyramid and has a square base:

Surface area = (1/2 * base perimeter * slant height) + base area

If the pyramid is oblique or doesn't have a square base, then the faces have to be calculated individually. Always remember to put the proper units on your calculation - squared for surface area and cubed for volume. This wasn't all that hard, was it? Of course, we didn't actually have the pharaoh breathing down our necks either.

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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

- Properties of Shapes: Rectangles, Squares and Rhombuses 5:46
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