How To Find The Volume Of A Square Pyramid

Pyramids, one of the oldest structures created by man, also support one of the most fascinating architectures. Have you ever wondered how many types of pyramids are there? How their volumes are calculated? Well, if you have, this article is definitely going to grab your attention.

What are pyramids?

A 3-D solid that has a flat polygonal base and straight vertices and edges is called a pyramid. In Greek terminology, the figure is known as a polyhedron. Each edge and vertex of the figure forms a triangle. 

There are multiple types of pyramids used in geometry:

1. Irregular pyramid
2. Pentagonal pyramid
3. Triangular pyramid
4. Square pyramid
5. Oblique pyramid
6. Regular pyramid

All you need to know about the square pyramids and their volume

The square pyramid represents a 3-dimensional solid supported by a square base and slanted triangular sides that join at a common spot above the base. In this article, the alphabet ‘L’ is going to define the length of the square base side; the alphabet ‘H’ is going to define the height of the pyramid i.e. the perpendicular distance from the common spot to the base, and the alphabet ‘V’ is going to define the volume of the three-dimensional figure. 

The simple formula for calculating the volume of a square-based pyramid is stated as:

V = ⅓ L²H

This formula is effective in finding the volume of a square-based pyramid irrespective of the fact that it is the size of a paperweight or as large as the great Egyptian pyramids. In certain scenarios, the slant height of the pyramid is also used to calculate the volume of the pyramid. 

Case 1: Using the area of the base and height of the pyramid to find the volume of the square pyramid

Step 1: Measure the length of the side of the base of the pyramid. 

As stated above, the square pyramid has a proper square base and hence all the sides of the base are of equal length. All you have to do is find the length of one base side. 

For example: 

Let us consider a pyramid with the length of the base side as L=6cm. This value will help you to determine the area of the base. 

Step 2: It is time to calculate the area of the pyramid base.

This is the initial step of finding the volume of a square-based pyramid. The formula for calculating the area of the square is to multiply the length of the side by its width. Since the value of a square’s length and width is the same so the area is calculated by squaring the value of the length. Always remember that the area of a 2-D solid is expressed in square units.

For example: 

Taking the above-mentioned value into consideration, 

Area of the base of the square pyramid = L² = 6cm x 6cm = 36cm²

Step 3: Now multiply the calculated base area by the height of the pyramid.

Now it is time to multiply the perpendicular distance of the common spot from the base (height) by the base area of the square-based pyramid. Remember that the volume is 

For example:

The distance from the base to the common point is 8cm. In this example, the calculation should go as:

36cm² x 8cm = 288cm³

Step 4: Divide the acquired answer by 3 to derive the desired volume. 

As per the formula, the volume of a square-based pyramid is derived only by dividing the derived multiplication result by 3. This step will lead you to the final answer to the volume of the pyramid. 

For example: 

Now divide the acquired result 288cm³ by 3 to get the volume. 

V= 288/3 = 96cm³

Case 2: Using the slant height of the pyramid to find the volume of the square pyramid

Step 1: Calculate the slant height of the square pyramid.

Sometimes the questions do not state the value of the perpendicular distance. In such a case, you will have to either measure the slant height of the pyramid or be told the same.

The slant height of the square-based pyramid is the distance from the center point of one of the base sides to the apex of the pyramid.

it is calculated by using the values of the distance of the center point of the base side joined at the midpoint of any side of the base to the apex of the pyramid. You will have to apply the Pythagorean theorem to derive the perpendicular height of the pyramid. 

For example: 

The slant height of a square-based pyramid is given to be 13 cm and the base length measures at 10cm.

Using the Pythagoras theorem, where: 

 p² + b² = q²

Here the alphabets p and b represent the perpendicular sides of the right-angle triangle while q represents the hypotenuse. The value of b will be ½ of the entire length of the side of the square-based pyramid. 

Let the slant height of the square-based pyramid be represented by the alphabet G. 

As per the formula,  the slant height will be calculated as:

p² + (b/2)² = G²

This formula is derived from the foundation of the Pythagoras theorem. 

Step 2: Add the desired values to calculate the perpendicular height of the square-based pyramid. 

Considering the formula derived above,

p² = G² - (b/2) ² 

Square root both the side and you will get the desired formula for calculating your perpendicular height.

p = √G² - (b/2) ²

Adding values from the above example:

p= √13² - (10/2) ² = √169-25 = √144= 12cm

After simplifying the square root, we derived the value of perpendicular height as 12 cm.

Step 3: Use the derived perpendicular height and the actual value of the base length to calculate the volume of the square-based pyramid. Keep in mind that you will have to follow the same formula as used in the first scenario to calculate the volume of the pyramid. Make sure you label the derived answer in cubic units.

The formula for calculating the volume of a square-based pyramid is stated as:

V = ⅓ L²H

V = ⅓  x (10)² x 12

V = ⅓ x 1200

V = 400 cm³

Case 3: Using the edge height to find the volume of the square pyramid

Step 1: calculate the edge height of the square-based pyramid.

The length of the edge of a pyramid calculated from the apex to one of the corners of the base is defined as the edge height of the square-based pyramid. In this scenario as well, the Pythagoras theorem is used to calculate the perpendicular height of the pyramid.

Let us assume the edge height comma denoted by the alphabet e, to be measured at 13cm and the perpendicular height to be 5cm. 

Step 2: Get the image of the right triangle straightened up.

Imagine a right-angle triangle to apply the Pythagoras theorem. In the given scenario, you are aware of the values of the base of the pyramid as well as the perpendicular height and the edge height.

Now assume that you are dividing the pyramid diagonally from one corner to the opposite end and the face emerged as a triangle. The height of the triangle is going to be equal to the perpendicular height of the pyramid and will divide the emerged triangle into two equivalent parts.

Whereas the hypotenuse of the emerged triangle will be equivalent to the measurement of the edge height of the square-based pyramid. 

Let the perpendicular height be denoted by the alphabet p, the edge height of the pyramid be denoted by alphabet e and the diagonal of the base be denoted by the alphabet d. Based on these variables the equation for the Pythagoras Theorem will look like this:

p² + d² = e²

Step 3: Carefully assign the values to the derived equation and calculate the value for the unknown base diagonal of the pyramid. Revise the equation to suit your required scenario. 

d² = e² - p²

d = √e² - p² = √13² - 5² = √169-25 = √144 = 12cm

Remember that this is only half of the actual value of the base diagonal of the pyramid, hence, you need to multiply the derived outcome by 2 to get the length of the complete diagonal of the pyramid’s base.

12 x 2 = 24cm

Step 4: Use the diagonal base of the pyramid to calculate the side of the base of the square-based pyramid.

Since we all know that this is a square pyramid the diagonal of any square is equivalent to the value of the side of the square multiplied by the square root of 2. We can easily use the values derived above to find the length of the side of the base of the pyramid.

For example: 

The diagonal has been calculated at 24 cm. therefore the side will be:

S = 24 / √2 = 16.97cm

Step 5: Now that you have the value of all the variables, calculate the volume of the square-based pyramid using the basic volume formula

V = ⅓ L²H

V = ⅓  x (16.97)² x 5 = ⅓ x 287.98 x 5 = 479.96cm

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