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Volume of a Triangular Pyramid

A structure with triangular outer faces such that they converge into one point is called a pyramid. The base of the pyramid can be any polygon – triangle, quadrilateral, pentagon etc. The most popular pyramids are the ones found in Egypt at Giza. Pyramids are one of the most ancient structures that were built by humans. Some of the pyramids found in Egypt belong to the prehistoric Egyptian civilization. 

A pyramid is a three-dimensional object that has one base and as many sides face as the sides on the base. All the side faces converging into one single point on the top. It is classified as a polyhedron. The point at which the side faces converge is called its apex. 

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In this article, we are going to talk about a specific type of pyramid, the triangular pyramid. A triangular pyramid is a pyramid with a triangular base. Such a pyramid will have exactly four faces: one base and three side faces. If all the four faces are congruent equilateral triangles, then such a triangular pyramid would be called a tetrahedron. As in all other pyramids, the side faces of a triangular pyramid are all triangular. However, in this case, the base is also triangular. That is why it is called a triangular pyramid. 

Volume of a Triangular Pyramid


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The most important mensuration associated with any three-dimensional object is its volume. However to be able to find the volume of any pyramid we first need to find the area of the base of the pyramid. For a triangular pyramid, the base is a triangle. The area of such a base can be found using various formulas. 

If we know one of the sides of the base triangle and the corresponding altitude, then the area of the base triangle can be calculated using the formula:

$area$ = $\frac{1}{2}$$\times\ side \times\ altitude$

Alternatively, if we know the length of the three sides of the triangle, then the base area can be calculated using the 
Heron’s formula as follows:

$area$ = $\sqrt{s(s\ -\ a)(s\ -\ b)(s\ -\ c)}$

Where, $s$ = $\text{semiperimeter}$ = $\frac{a\ +\ b\ +\ c}{2}$

$a,\ b$ and $c$ are the sides of the triangle

If the base is an equilateral triangle of side ‘a’, then its area can be given by the formula:

$area$ = $\frac{\sqrt{3}}{4}$$a^2$

How to find the Volume of a Triangular Pyramid?

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Now that we have established the various formulas that we can use to find the area of the triangular base, let us look into the volume of a triangular pyramid. In general, the volume of a pyramid is given by the formula:

$Volume$ = $\frac{1}{3}$$\times\ (\text{area of base}) \times\ height$

Now, for our pyramid the base is triangular. So the area of base would be:

$area\ of\ base$ = $\frac{1}{2}$ $ba$

Where, $b$ = length of base and 

$a$ = altitude to that  base

Therefore the volume of the pyramid would be:

$Volume$ = $\frac{1}{3}$ $\times$ $\frac{1}{2}$ $ba$ $\times$ $h$ = $\frac{1}{6}$ $bah$

This is sometimes also written as:

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

where, $B$ = area of base


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Examples 1: Find the volume of the pyramid shown in the following figure:

Volume of a Triangular Pyramid Examples


From the figure we see that the base is a triangle. The base length of this triangle is $12$ $\frac{1}{2}$ ft. The altitude of the base triangle is $10$ ft. Therefore the area of base is:

Area = $B$ = $\frac{1}{2}$ $\times\ b\ \times\ a$

$B = $\frac{1}{2}$ $\times\ 12 $\frac{1}{2}$ $\times\ 10$

$B$ = $\frac{1}{2}$ $\times$ $\frac{25}{2}$ $\times\ 10$

$B = $\frac{250}{4}$ = $62$ $\frac{1}{2}$$ft^2$

Now, let us use this area to find the volume of the pyramid using the volume formula that we studied just a little 
while ago.

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

The height $h$ of the pyramid is 14 ft. Therefore,

$Volume$ = $\frac{1}{3}$ $\times\ 62$ $\frac{1}{2}$ $\times\ 14$

$Volume$ = $\frac{1}{3}$ $\times$ $\frac{125}{2}$ $\times\ 14$

$Volume$ = $\frac{750}{6}$ = $291$ $\frac{2}{3}$$ft^3$

Now let us look at another type of problem pertaining the volume of a triangular pyramid.
Example 2: The base of a triangular pyramid is an isosceles triangle as shown below:
Find the Volume of a Triangular Pyramid

If the volume of the pyramid is 36.66 cm^3, then find the height of the pyramid.


Since we are given all the three side lengths of the base, let us use the Heron's formula to calculate the area of the base:

$\text{Area of base}$ = $B$ = $\sqrt{s(s\ -\ a)(s\ -\ b)(s\ -\ c)}$

First, we need to find the semi-perimeter, 's':

$s$ = $\frac{a\ +\ b\ +\ c}{2}$ = $\frac{5\ +\ 5\ +\ 4}{2}$ = $\frac{14}{2}$ = $7$

Next we use this semi perimeter to find the area using the Heron’s formula:

$B$ = $\sqrt{7(7\ -\ 5)(7\ -\ 5)(7\ -\ 4)}$

$B$ = $\sqrt{7(2)(2)(3)}$ = $\sqrt{84}$ = $9.165\ cm^2$

Now that we have the area let us use that to find the volume of the pyramid:

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

However we are given that volume = 36.66. Thus substituting these values into the above equation we have:

$36.66$ = $\frac{1}{3}$ $\times\ 9.165\ \times\ h$

Solving this equation for 'h' we have:

$36.66\ \times\ 3$ = $9.165\ h$

Dividing both the sides by 9.165 gives us:

$36.66\ \times$ $\frac{3}{9.165}$ = $h$

Evaluating that we have:

$h$ = $12\ cm$ $\rightarrow$ Answer
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