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Rectangular Pyramid

By Alison Rodriguez

There are numerous 3-dimensional shapes that we can explore. With 3-dimensions, we can find more than just the area. With 2-dimensional shapes, we can find perimeter and area. With 3-D shapes we are able to find what is called the surface area and volume.

Why is this concept useful?

A rectangular pyramid is what you often think about when the word pyramid is mentioned. The base is a rectangle and all of the remaining 4 sides are triangles that meet together at a point called the apex. Since the rectangular pyramid is one of the most common pyramids, we often find the surface area or the volume of it.

Where does this concept fit into the curriculum?

6th grade and Geometry

How can we use rectangular pyramids?

There are two main calculations for a rectangular pyramid, surface area and volume.

To find the surface area of a pyramid, we can look at it in two ways.

  1. Surface area - using the net.

To find the surface area using the net, we are finding individual areas of the faces of the pyramid. In other words, you will find the area of the rectangle and separately find the area of the triangles. You will need to know the dimensions of the base as well as the height of each of the triangles.



2. Surface area - using the formula

If the base is a square, we can use the formula

SA=B+12×P\timeslSA = B + \frac{1}{2} \times P\timesl
,  where
BB
is the area of the base,
PP
is the perimeter of the base and
ll
is the slant height (the height of the triangles, not the pyramid. Note that this formula only applies for regular pyramids. This means the base has to be a regular shape, like a square.

The volume of a rectangular pyramid is exactly one-third the amount of it’s corresponding rectangular prism. To find the volume of a pyramid, we can use the following formula:

V=13BhV = \frac{1}{3}Bh

Where B is the area of the base and h is the height of the pyramid from the apex to the center of the base.

Sample Math Problems

Question

Find the surface area:

Solution

This is a rectangular pyramid, we need to find individual areas and add them up.

Area of the base:  6 x 12 = 72  square inches

Area of the triangle whose base is 6:    ½ x 6 x 10 = 30 square inches

Area of the triangle whose base is 12:  ½ x 12 x 8.5 = 51 square inches

Surface area = 72 + 2(30) + 2(51) = 234 square inches

Note that we are doubling the 30 and 51 because there are two triangles of equal areas.

Question

Find the surface area:

Solution

This is a square pyramid  so we can find the surface area using the formula

SA=B+12SA = B + \frac{1}{2}
×Pl\times Pl

Area of the base = 10 x 10 = 100 square feet

P = 4(10) = 40 feet

Slant height, l = 12.1

SA = 100 + ½ (40)(12.1) = 342 square feet

Question

Find the volume, rounded to 2 decimal places.

Solution

Area of the base, B = 10 x 7 = 70

Height (from the apex perpendicular to the base) = 8

V = ⅓ (70)(8) = 186.67  cubic centimeters

Question

The volume of the pyramid is 2,000 cubic inches. The base is a square whose side length is 20 inches. Find the slant height of the pyramid.

Solution

Given the formula is

V=13BhV = \frac{1}{3}Bh
, we can fill in the information that we know.

Area of the base, as a square, is 20 x 20 = 400 square inches

2000=13×(400)×l2000 = \frac{1}{3} \times (400) \times l

3×2000=13×400×l×33\times2000 = \frac{1}{3}\times400\times l \times3

60000=400×l60000 = 400 \times l

15=l15 = l

The slant height is 15 inches.

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Practice Math Problems

1. Find the surface area.

2. Find the surface area.

3. Find the volume.

4.  The volume of a pyramid is 144 square feet. The base of the pyramid is a square. The slant height of this pyramid is 12 feet long.  What is the side length of the square base?

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