Geometry Worksheets

Practice and master geometry concepts with helpful walkthroughs from our elite math educators, then try practice math problems to sharpen your skills.

Square Pyramid

By Ellen Rattin

Square Pyramids are probably the “traditional” pyramid you think of when talking about pyramids! They are formed by a square base and four triangular sides.

Why is this concept useful?

Square pyramids can be used to model a variety of real-world objects! For example, the pyramids in Giza, Egypt are approximately square pyramids. Many architectural features, such as roofs and peaks, can also be modeled with square pyramids. Once we’ve chosen to model an object with a square pyramid, we can then use all the formulas we know for surface area and volume to learn about that object.

Where does this concept fit into the curriculum?

Grade 6, 7, 8, and High School Geometry

How can we use the concept:

When solving problems with square pyramids, we are most often looking for their surface area or volume.

The surface area tells us how much area is needed to cover the outside of the pyramid. It can be found by adding the area of each face - the four triangles and the one square. Or you can use the formula which combines the area of all those sides for you:

SA=b2+2bsSA=b^2+2bs

Where s is the slant height and b is the side length of the square.

The volume tells us how much 3D space this pyramid fills. The formula for volume of a square pyramid is:

V=13b2hV=\frac{1}{3}b^2h

where b is the side length of the square and h is the height of the pyramid.

Sample Math Problems

Question

Find the surface area of the following pyramid:

Solution

The formula for surface area is SA =

b2b^2
+2bs where s is the slant height and b is the side length of the square. In this case, s = 7 cm and b = 4 cm. By placing these numbers in the formula, we get:

SA=42+2(4)(7)=16+56=72cm2 SA = 4^2 + 2(4)(7) = 16 + 56 = 72 cm^2

Question

Find the volume of the following pyramid:

Solution

The formula for volume is V =

13b2h\frac{1}{3}b^2h
, where b is the side length of the square and h is the height of the pyramid. In this case, b = 4 and h = 5. By placing these numbers in the formula, we get:

V=1342(5)=13(16)(5)=13(80)=2623cm3.V=\frac{1}{3}4^2(5) = \frac{1}{3}(16)(5) = \frac{1}{3}(80) = 26\frac{2}{3}cm3.

Question

St. Mary’s in Chebanse, IL recently installed a new church steeple with a copper roof. It is approximately in the shape of a square pyramid with a base length of 6 feet and a slant height of 8.5 feet. They’ve hired a company to patina the copper roof. The company charges $2.30 per square foot. How much will it cost to patina the church’s steeple?

Solution

First we need to find the surface area of the steeple, which we are told is in the shape of a square pyramid. By inputting the values into the formula we find:

SA=62+2(6)(8.5)=36+102=138ft2SA = 6^2+2(6)(8.5) = 36 + 102 = 138ft^2

However, we realize that the metal roofing does not cover the bottom of the square pyramid. So we need to subtract that off. The area of the bottom (a square) is 6 x 6 = 36

ft2ft ^2
So:

138ft2−36ft2=102ft2.138 ft^2-36 ft^2=102 ft^2.

We know that the company charges $2.30 per square foot. So 102 x 2.30 = $234.60 to patina the church steeple.


Question

Square Pyramid #1 has a base length b and height h. Square Pyramid #2 has the same base length b but twice the height (2h). What is the relationship between the volume of Pyramid #1 and Pyramid #2?

Solution

The volume of the first pyramid will be V =

13b2h\frac{1}{3}b^2h
. For the second pyramid, we can see that when we substitute 2h for h, we get V =
13b2h\frac{1}{3}b^2h
. The two can be easily brought out the front, giving us V =
213b2h2\frac{1}{3}b^2h
, showing us that the volume of the second pyramid is twice that of the first one.

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Practice Math Problems

1) Find the surface area of the following pyramid:

2) Find the volume of the following pyramid:

3) The pyramids of Giza, Egypt, are famous for their classic square pyramid shape. The largest one, the Great Pyramid, has a base length of 756 feet and a height of 481 feet.  Assuming that it is completely filled with and made out of limestone, which weighs 150 pounds per cubic foot, what is the weight of the Great Pyramid?

4) Square Pyramid #1 has a base length b and height h. Square Pyramid #2 has the same height h but twice the base length (2b). What is the relationship between the volume of Pyramid #1 and Pyramid #2?

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