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Pyramids

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A pyramid is made by using 3 or more triangular faces. The vertex of pyramid is common for all the triangular faces. Pyramid base can be in any shape like square, triangle, polygon and so on. Its base is depends on how many faces of triangle are included in pyramid, if 4 faces are included then the base of pyramid will be triangle. Every pyramid has one base and many number of faces with common vertex. 

Pyramid

In the above figure the most important part of pyramids are clearly mentioned. The base, apex and face. As we already discuss about face and base of the pyramid. Here we are going to discuss about apex of a pyramid, "The apex of a pyramid is single point where all the triangular faces are meet". 

Types of Pyramids

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There are many types of pyramids as discussed below:
Triangular Pyramid: The triangular pyramid has four faces, one vertex and a triangular base.

Square Pyramid: The square pyramid has five faces, one vertex and a square base.  

Pentagonal Pyramid: The pentagonal pyramid has six faces, one vertex and a pentagonal base. .
The above define pyramids are in geometry where each pyramid has their different property. Below find the figure of different kinds of pyramid.

Types of Pyramids

LSA of Pyramids

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The sum of all the are of pyramid faces are known as lateral surface area. 
The formula of LSA is,
lateral surface area of a pyramid = $\frac{1}{2}$ $\times$ p $\times$ l;Below fins some steps to find LSA of a pyramid,

Step 1: First find the perimeter of a triangular pyramid using formula,"perimeter of a triangular perimeter is 3s".
            Where,
            s = The length of all sides of a triangular Prism.

Step 2: Find the length of a triangular pyramid.

Step 3: Multiply perimeter and length from above steps by $\frac{1}{2}$.

Step 4: Calculate the value of lateral surface area of a pyramid.

Example:


A pyramid has each triangular base measure is 15 inch and slant height of a pyramid is 18 inch then we see the lateral surface area of a pyramid. We know that:

Solution:

The lateral surface area of a pyramid = $\frac{1}{2}$ $\times$ p $\times$ l;

Step 1: Find the perimeter of a triangular prism.

So the perimeter of a triangular prism = a + b + c;

Where, 
a, b, c are the size each side of triangular base. 

 Perimeter = 15 + 15 + 15;

Perimeter = 45 inch.

Step 2: Find the height of a triangular pyramid,

Here the height of pyramid is 18 inch.

So put the value in the given formula:

The lateral surface area of a pyramid = $\frac{1}{2}$ $\times$ p $\times$ l;

The lateral surface area of a pyramid = $\frac{1}{2}$ $\times$ 45 $\times$ 18;

The lateral surface area of a pyramid = $\frac{1}{2}$ $\times$ 63;

The lateral surface area of a pyramid = 31.5 inch2.

So the lateral surface area of a pyramid is 31.5 inch2.

Volume of Pyramid

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The amount of space which is occupied by a pyramid is known as volume of pyramid. In other words "Both height and base multiplied by $\frac{1}{3}$ is given the volume value of pyramid". 

Formula of volume of pyramid is,
The volume of pyramid = area of the base $\times$ height $\times$ $\frac{1}{3}$The volume pyramid when the base  is rectangular:
The volume of pyramid = length of base $\times$ width of base $\times$ height $\times$ $\frac{1}{3}$;
Below find some steps to find the volume of a pyramid,

Step 1: First find the area of the base pyramid.

Step 2: Find the height of a pyramid.

Step 3: Multiply area and height by $\frac{1}{3}$.

Step 4: Solve the equation and get the volume of a pyramid.
If the base of pyramid is square then the area of a square = l$^{2}$.

Where,
l = The length of all sides of a square,

Example:

A pyramid has a square base of side 6 inch and height of a pyramid is 15 inch then what is the volume of a pyramid.

Solution: 

The volume of a pyramid = $\frac{1}{3}$ $\times$ area of the base $\times$ height.

Step 1: The given base is a square base,

So the area of a square = l2,

Area of a square = (6)2,

Area of a square = 36 inch2,

Step 2: Find the height of a pyramid,

Here the height of pyramid is 15 inch.

So put the value in the given formula:

The volume of a pyramid = $\frac{1}{3}$ $\times$ area of the base $\times$ height.

Volume = $\frac{1}{3}$ $\times$ 36 $\times$ 15,

Volume = $\frac{1}{3}$ $\times$ 540,

Volume = 180 inch3.

The volume of a pyramid is 180 inch3.

Pentagonal Pyramids

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A solid object which has a polygon base and triangular sides is known as Pentagonal pyramids. There are some properties of pentagonal pyramid given below,
1) A pentagonal pyramid has six faces.

2) Five faces of pentagonal are in triangular shape. 

3) It has a pentagon base.

4) It has six vertices or corner points.

5) It has 10 edges.

Volume formulas of Pentagonal Pyramid is, 
Volume = $\frac{1}{3}$ $\times$ b $\times$ h.
Where, 
b = The base area;
h = The height of a Pentagonal Pyramid.


Surface area of a Pentagonal Pyramid is,

Surface area = [base area] + $\frac{1}{2}$ $\times$ p $\times$ l.
Where,
p = The perimeter.
l = The side length.