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Volume of a Square Pyramid

Pyramids are one of the oldest structures constructed by humans. People of ancient Egyptian civilization were the forerunners in pyramid construction. Typically a pyramid has the maximum load at the bottom and the weight is reduced as the construction moves upwards. Pyramids of Giza in Egypt are the most popular. In fact, they are one of the wonders of the ancient world! Besides Egypt, pyramids were also found in Central American countries. Ancient pyramidal structures are also found in some south Asian countries as well such as Indonesia. It is believed that the Mesopotamians built the first pyramids, however, the Egyptian pyramids are more colossal and hence more popular today. Some of the pyramids are as old as 3000 BC!

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Definition

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From a mathematician’s point of view, a pyramid is a three-dimensional object that has a base and as many lateral faces as the sides of the base. The lateral faces converge into a single point called the apex of the pyramid. Thus, all the lateral faces of a pyramid are triangular. Unlike a prism, that has two bases, a pyramid has just one base and one apex. The base of a pyramid can be any polygon such as a triangle, a square, a rectangle, a pentagon, etc. Also, the base can be a regular polygon or an irregular polygon. A pyramid with a regular polygon for the base is called a regular pyramid.  A pyramid that has an equilateral triangle as its base is called a tetrahedron. A pyramid that has any triangle as its base is called a triangular pyramid. A pyramid is classified as a type of a polyhedron in geometry. If the base of the pyramid has ‘n’ sides, then that pyramid will have ‘n+1’ vertices. It will have ‘n+1’ faces and ‘2n’ edges. 

A pyramid can be a right pyramid or an oblique pyramid. In a right pyramid the apex of the pyramid is exactly above the centroid of the base of the pyramid; whereas in an oblique pyramid the apex is not directly above the centroid of the base. If two pyramids are such that they have the same base then they are called bipyramids.

All or most of the popular ancient pyramids that we discussed above are square pyramids. That means that the base of the pyramid is a square. Thus they are all regular pyramids. Since a square has four sides, the corresponding pyramid would have 4+1 = 5 vertices, 5 faces, and 4×2=8 edges. 

Formula

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The formulas associated with a pyramid are for its surface area and volume. For a square pyramid since the base is a square, the area of the base can be given by:

area of base = $B$ = $b^2$

Where, b = length of side of base.

Now if each of the lateral faces has slant height of l, then the area of each of the lateral face would be:

area of lateral face = $L$ = $\frac{1}{2}$$bl$

Therefore the total lateral area of all the 4 lateral faces would be:

Total lateral area = $4L$ = $4\ \times$ $\frac{1}{2}$$bl$ = $2\ bl$

(Note: Since the square pyramid has 4 sides on the base, it would have 4 lateral faces.)

This brings us to the total surface area of the square pyramid.

Total Surface area = $B\ +\ 4L$ = $B\ +\ 2\ bl$

Let us say that the perimeter of the square base is 

$P$ = $4\ b$

Then the lateral surface area formula can be rewritten as:

Total lateral area = $4\ b\ \times$ $\frac{1}{2}$$l$ = $\frac{Pl}{2}$

This is another way of finding the surface area of a pyramid. Thus the total area would now be:

Total surface area = $B\ +\ P$$(\frac{l}{2})$

How to find the Volume of a Square Pyramid?

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Another formula associated with any pyramid is that of its volume. We sometimes need to know the amount of space occupied by a given pyramid. That is called the volume. The general formula for volume of any pyramid is given by the formula:

Volume = $\frac{1}{2}$$Bh$

Where, again B = area of base and h=vertical height of the pyramid.

Now, for a square pyramid since the base is a square, the area of base can be given by:

Area of base = $B$ = $b^2$

Therefore the volume of a square pyramid can be given by the formula:

Volume = $\frac{1}{3}$$b^2\ h$

Volume of a Square Pyramid with Slant Height

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Sometimes we are not given the vertical height of the pyramid. In the case of solid pyramids, it may not be possible to exactly measure the vertical height of the pyramid. In such cases, we can use the slant height.

We first use the slant height to find the vertical height and then use the vertical height to find the volume of the pyramid. Consider a square pyramid as shown in the figure below:
 
Square Pyramid with Slant Height

Note that the slant height, the height and the apothem of the square base form a right triangle. These three lines form a right triangle. If we call the apothem as 'a', the slant height as 's' and the height as 'h', then applying the Pythagorean theorem to this right triangle we have:

$l^2$ = $a^2\ +\ h^2$

Solving this for height 'h' we have:

$h^2$ = $l^2\ -\ a^2$

This equation can be used to find the height of the pyramid. Also, the apothem of a square is nothing but the half of 
the base side. Thus, for a square,

$a$ = $\frac{b}{2}$

Examples

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Examples 1: Find the volume of the following square pyramid.

Volume of a Square Pyramid
 
Solution: 

From the figure, we see that the length of the side of the base square is 16. Therefore,

b=16

Thus, area of base,

= $B$ = $b^2$ = $16^2$ = $256$

Next, the vertical height as we can see from the picture is 24. Thus,

$h$ = $24$

The volume of a pyramid is given by:

$Volume$ = $\frac{1}{3}$$Bh$

Therefore for this one it would be:

$Volume$ = $\frac{1}{3}$$(256)24$ = $2048$ $\rightarrow$ units
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