In geometry, different types of shapes are studied. Most commonly, we come across with two-dimensional and there-dimensional shapes. The plane or two-dimensional figures may include circle, square rectangle, triangle, parallelogram, rhombus etc. The three-dimensional figures are the object which have 3 dimensions - length, breadth and height. The examples of 3D shapes are sphere, hemisphere, cone, pyramid, parallelopiped, cube, cuboid and many more.

Let us focus on pyramids. A pyramid is said to be a 3D shape in which the base has to be a polygon. It has triangular lateral faces that meet at one single point lying outside the base. A pyramid is demonstrated in the following diagram.

There are different types of pyramids. Actually they can be classified on the basis of shape of base. We commonly see a triangular pyramid (triangle base), square pyramid (square base - as shown above), pentagonal pyramid (pentagon base). Though, the base may be any polygon.

The pyramids can also be classified in two major forms - right pyramid and oblique pyramid. A right pyramid is one in which the line joining midpoint of base and apex (point where lateral edges meet) is at right angles to the base. While an oblique pyramid is tilted towards one side.

Let us go ahead in this article and learn about oblique pyramids in detail.

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One can easily recognize an oblique pyramid by just having a look at it. A pyramid that seems tilted in any direction, is an oblique triangle. Following figure shows an oblique pyramid.

The apex of an oblique pyramid is not right over the center of the base. We can say that in an oblique pyramid, line joining the apex and the center of base is not perpendicular to the base, unlike a right pyramid; i.e. the altitude (or height) and line joining apex to midpoint of base are different in an oblique pyramid. Have a look at the following diagram illustrating height of such pyramid.

Eventually unlike a right pyramid, all the lateral faces of oblique pyramid are not essentially isosceles triangle.
For an oblique pyramid, the formula for calculating surface area for right pyramid does not work. We need to break the pyramid into different parts and then find their surface area separately and then add them. On the other hand, the formula for finding volume of oblique pyramid remains same as the formula of volume in an oblique pyramid.

Therefore, the volume of an oblique pyramid is equal to one third of the product of area of base and height. This formula can be written as below :

$V$ = $\frac{1}{3}$ $b\ \times\ h$

where, V denotes volume of right or oblique pyramid, b represents area of base and h stands for perpendicular height of pyramid.

An oblique pyramid having a circular base represents an oblique circular pyramid. Its lateral surfaces are not triangles since the base is not polygon. Hence, it is characterized as a totally different shape in three-dimensional geometry. This kind of pyramid is actually classified as a cone. We can say that an oblique circular pyramid is in fact an oblique cone. It is shown in the following diagram.

**Example 2 :**

An oblique square pyramid of altitude $9 cm$, measures each base side of $4 cm$. Calculate the volume of this pyramid.

Solution :

Here, base is a square of $4 cm$ side, therefore

Area of base, $b\ =\ side^{2}\ =\ 4^{2}\ =\ 16\ sq\ cm$

Height, $h = 9 cm$.

Volume of an oblique pyramid $V$ = $\frac{1}{3}$ $b\ \times\ h$

The apex of an oblique pyramid is not right over the center of the base. We can say that in an oblique pyramid, line joining the apex and the center of base is not perpendicular to the base, unlike a right pyramid; i.e. the altitude (or height) and line joining apex to midpoint of base are different in an oblique pyramid. Have a look at the following diagram illustrating height of such pyramid.

Eventually unlike a right pyramid, all the lateral faces of oblique pyramid are not essentially isosceles triangle.

Therefore, the volume of an oblique pyramid is equal to one third of the product of area of base and height. This formula can be written as below :

$V$ = $\frac{1}{3}$ $b\ \times\ h$

where, V denotes volume of right or oblique pyramid, b represents area of base and h stands for perpendicular height of pyramid.

An oblique pyramid having a circular base represents an oblique circular pyramid. Its lateral surfaces are not triangles since the base is not polygon. Hence, it is characterized as a totally different shape in three-dimensional geometry. This kind of pyramid is actually classified as a cone. We can say that an oblique circular pyramid is in fact an oblique cone. It is shown in the following diagram.

The examples based on oblique pyramid are given below.

**Example 1 :**

Find the volume of an oblique polygonal pyramid whose area of base is 51 square meters and height is 8 meters.

Solution :

The formula is :

Volume of an oblique pyramid $V$ = $\frac{1}{3}$ $b\ \times\ h$

$V$ = $\frac{1}{3}$ $\times\ 51\ \times\ 8$

= $136 cm^{3}$

Find the volume of an oblique polygonal pyramid whose area of base is 51 square meters and height is 8 meters.

Solution :

Given that

Area of base, $b\ =\ 51\ sq\ m$

Height, $h\ =\ 8\ m$

The formula is :

Volume of an oblique pyramid $V$ = $\frac{1}{3}$ $b\ \times\ h$

$V$ = $\frac{1}{3}$ $\times\ 51\ \times\ 8$

= $136 cm^{3}$

An oblique square pyramid of altitude $9 cm$, measures each base side of $4 cm$. Calculate the volume of this pyramid.

Solution :

Here, base is a square of $4 cm$ side, therefore

Area of base, $b\ =\ side^{2}\ =\ 4^{2}\ =\ 16\ sq\ cm$

Height, $h = 9 cm$.

Volume of an oblique pyramid $V$ = $\frac{1}{3}$ $b\ \times\ h$

$V$ = $\frac{1}{3}$ $\times\ 16\ \times\ 9$

= $48cm^{3}$

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