How to find the volume of a triangular pyramid calculator

Calculating the Volume of a Right-Triangular Pyramid:

A right-triangular pyramid is a three dimensional shape with a right-angle triangle at its base extruding up to a single point. The volume of a right-triangular pyramid can be calculated according to the following formula:

V = 1/3(Ah)
where A = 1/2(bh) (standard right-triangle)
so : V = 1/6(bhH)

Explanation: The general formula used to find the volume of a pyramid is one-third times area of the base times height of the pyramid. Because it is known that this pyramid has a right-triangle at its base the formula for a right-triangle is used to calculate the area. The formula one-half base times height is inserted into the volume formula in the place of area. After multiplying out the formula now reads one-sixth time base of triangle (b) times height of triangle (h) times height of pyramid (H).

Below is an example of how to find the volume of this type of triangular pyramid. In this example the triangle at the base of the pyramid must have a ninety degree (right) angle. Lets say the triangle has a base of 3 meters, a height of 4 meters and that the pyramid is 5 meters tall:

• V = 1/6(bhH)
• = 1/6 (3 x 4 x 5)
• = 1/6 (60)
• = 10 meters

Let's take an example triangular pyramid and try this formula out. Let's say the height of the pyramid is 8, and the triangular base has a base of 6 and a height of 4.

First we need #A#, the area of the triangular base. Remember that the formula for the area of a triangle is #A=1/2bh#.
(Note: don't get this base confused with the base of the whole pyramid- we'll get to that later.)

So we just plug in the base and height of the triangular base:

#A=1/2*6*4#
#A=12#

Okay now we plug this area #A# and the height of the pyramid (8) for #h# in the main formula for the volume of a triangular pyramid, #V = 1/3Ah#.
#V = 1/3*12*8#.
#V=32#

There we go- now if you're given area of the triangular base it's even easier, just plug it and the pyramid height directly into the formula.

A triangular pyramid is solid with a triangular base and triangles having a shared vertex on all three lateral faces. It’s a tetrahedron with equilateral triangles on each of its four faces. It has a triangle base and four triangular faces, three of which meet at one vertex. A right triangular pyramid’s base is a right-angled triangle, with isosceles triangles on the other faces. All of the faces of a regular triangular pyramid are equilateral triangles and it contains six symmetry planes.

Volume of a Triangular Pyramid

The amount of space occupied by a triangular pyramid in a 3D plane is called its volume. To put it another way, volume specifies the confined area or region of the pyramid. Knowing the base area and height of a triangular pyramid is enough to calculate its volume. Its formula equals one-third the product of base area and height. It is measured in units of cubic meters (m3). 

V = 1/3 × B × h

Where,

V is the volume,

B is the base area,

h is the height of pyramid.

If we are given a regular triangular pyramid consisting of equilateral triangles, its volume is given by the formula,

V = a3/6√2

Where,

V is the volume,

a is the side length.

How to Find the Volume of a Triangular Pyramid?

Let’s take an example to understand how we can calculate the volume of a triangular pyramid.

Example: Calculate the volume of a triangular pyramid of base area 90 sq. m and height 6 m.

Step 1: Note the base area and height of a triangular pyramid. In this example, the base area of the pyramid is 90 sq. m and height is 6 m.

Step 2: We know that the volume of a triangular pyramid is equal to 1/3 × B × h. Substitute the given value of base area and height in the formula.

Step 3: So, the volume of triangular pyramid is calculated as, V = (1/3) × 90 × 6 = 180 cu. m

Sample Problems

Problem 1: Calculate the volume of a triangular pyramid with a base area of 50 sq. m and a height of 4 m.

Solution:

We have,

B = 50

h = 4

Using the formula we get,

V = 1/3 × B × h

= (1/3) × 50 × 4

= 66.67 cu. m

Problem 2: Calculate the volume of a triangular pyramid with a base area of 120 sq. m and a height of 10 m.

Solution:

We have,

B = 50

h = 4

Using the formula we get,

V = 1/3 × B × h

= (1/3) × 120 × 10

= 400 cu. m

Problem 3: Calculate the base area of a triangular pyramid if its volume is 300 cu. m and height is 15 m.

Solution:

V = 300

h = 15

Using the formula we get,

V = 1/3 × B × h

=> B = 3V/h

=> B = 3 (300)/15

=> B = 60 sq. m

Problem 4: Calculate the base area of a triangular pyramid if its volume is 600 cu. m and height are 5 m.

Solution:

V = 600

h = 5

Using the formula we get,

V = 1/3 × B × h

=> B = 3V/h

=> B = 3 (600)/5

=> B = 360 sq. m

Problem 5: Calculate the height of a triangular pyramid if its volume is 200 cu. m and the base area is 60 sq. m.

Solution:

We have,

V = 200

B = 60

Using the formula we get,

V = 1/3 × B × h

=> h = 3V/B

=> h = 3 (200)/60

=> h = 10 m

Problem 6: Calculate the height of a triangular pyramid if its volume is 150 cu. m and the base area is 50 sq. m.

Solution:

We have,

V = 150

B = 50

Using the formula we get,

V = 1/3 × B × h

=> h = 3V/B

=> h = 3 (150)/50

=> h = 9 m

Problem 7: Calculate the volume of a regular triangular pyramid if the side length is 10 m.

Solution:

We have,

a = 10

Using the formula we get,

V = a3/6√2

= (10)3/6√2

= 117.85 cu. m

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