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# Pyramids

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 Sub Topics When 3 or more triangular faces share a common vertex and make a Geometry, which is called as a pyramid. The base of a pyramid may be any triangle shape, square shape or any other Polygon shape. If base of pyramid have triangle shape, then the given pyramid has 4 faces. Pyramid is a simplest polyhedron, which is also called as a tetrahedron. The shape of pyramid is taking from famous ancient Egyptian pyramids. Each pyramid has only one base and the other faces are all congruent Triangles and all Congruent Triangles share a common vertex, which is the top Point. Now we discuss formulas of pyramids for calculating volume and surface area of pyramids– Volume of pyramids = ($\frac{1}{3}$) * B * H, Where ‘B’ is the base of pyramid and H is the height of pyramids. Surface area of pyramid = B + ($\frac{1}{2}$) * P * l, Here ‘B’ is base length, ‘P’ is perimeter and l is a side length of pyramid. Now we discuss types of pyramids: There are following types of pyramids: Triangular pyramid - A pyramid, which has 4 faces and which base made a triangle shape is called as a triangular pyramid. All 3 side faces of triangular pyramid made a triangular shape, which is also another property of triangular pyramid. Square pyramid - A pyramid, which has 5 faces and which base made a Square shape is called as a square pyramid. All 4 side faces of square pyramid made a triangle shape, that’s why we call this pyramid as a square pyramid. Pentagonal pyramid - A pyramid, which has 6 faces and which base made a pentagonal shape is called as a pentagonal pyramid. All 5 side faces of square pyramid made a triangle shape, that’s why we call this pyramid as a pentagonal pyramid. These are 3 types of pyramids, which are define in geometry and each pyramid has their different property like right and oblique pyramid or regular and irregular pyramid. All these pyramids are depending on their shapes.

## LSA of Pyramids

The lateral surface area of a pyramid is defined as the sum of all the areas of the given faces. The formula for finding the lateral surface area of a pyramid is: The lateral surface area of a pyramid = $\frac{1}{2}$ * p * l;

If we find lateral surface area of pyramid with a triangular base then the formula is:

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

We see some of the steps for finding the lateral surface area of a pyramid:

Step 1: Firstly we find the perimeter of a triangular pyramid.

We know that the perimeter of a triangular perimeter is 3s,

Where, s is the length of all sides of a triangular Prism.

Step 2: And then find the length of a triangular pyramid.

Step 3: Both perimeter and length of a pyramid is multiply by $\frac{1}{2}$.

Step 4: when we solve these values we get the lateral surface area of a pyramid.

Now assume 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:

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

Step 1: first we find the perimeter of a triangular prism by adding all the side values of a pyramid.

The given base is a triangular base,

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

Perimeter = 15 + 15 + 15;

Perimeter = 45 inch.

Step 2: we 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 = $\frac{1}{2}$ * p * l;

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

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

The lateral surface area of a pyramid = 31.5 inch2.

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

## TSA of Pyramids

The total surface area of a pyramid is defined as the sum of all the areas of the given faces and including base value in it. The formula for finding the total surface area of a pyramid is:

The total surface area of pyramid = area of the base + 4 * area of triangular face;

The number of a triangular faces depend upon the number of the sides of the base.

If we find the total surface area of pyramid with a Square base then the formula is:

The total surface area of pyramid = area of the base + 4 * area of triangular face.

We see some of the steps for finding the total surface area of a pyramid:

Step 1: First we select the triangle and find the triangle hypotenuse value with the help of pythagoras theorem.

Step 2: Then find the area of a triangle, when we find the area of a triangle we have to find the height of a triangle.

Step 3: After then we find the area of a square base.

Step 4: Then we put all the values in the given formula and we get the total surface area of a pyramid.

Now consider a pyramid which has perpendicular height of 19 inch and base edge of 24 inch then we see the total surface area of a pyramid.

We know that:

The total surface area of pyramid = area of the base + 4 * area of triangular face.

Step 1: First we find the triangle hypotenuse value with the help of pythagoras theorem.

H2 = A2 + B2,

H2 = (19)2 + (12)2,

H2 = 361 + 144

H2 = 505,

H = $\sqrt{505}$,

H = 22.47

In step 2 we find the Area of Triangle:

We know that the area of triangle = $\frac{1}{2}$ BH,

Area = $\frac{1}{2}$ * 24 * 22.47.

## Volume of Pyramid

The volume of a pyramid is the sum of the bases and the height of a pyramid and both base and height are multiplied by $\frac{1}{3}$. The formula for finding the volume of a pyramid is:

The volume of pyramid = area of the base * height * $\frac{1}{3}$;

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

If we find the volume pyramid with a rectangular base then the formula is:

The volume of pyramid = length of base * width of base * height * $\frac{1}{3}$;

Now we see some of the steps for finding the volume of a pyramid:

Step 1: Firstly we find the area of the base of a given pyramid.

Step 2: And then find the height of a pyramid.

Step 3: Both area and height of a pyramid is multiply by $\frac{1}{3}$.

Step 4: when we solve these values we get the volume of a pyramid.

When we have Square base then the area of a square = l2.

Where l is length of all sides of a square,

Now assume 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.

We know that the volume of a pyramid = $\frac{1}{3}$ * area of the base * height.

Step 1: first we find the area of the base.

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: we 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 = $\frac{1}{3}$ * area of the base * height.

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

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

Volume = 180 inch3.

The volume of a pyramid is 180 inch3.

## Pentagonal Pyramids

Pyramid is a Solid object where the base of a pyramid is a Polygon and all sides of a pyramid is triangle which joins at the top Point. Now we discuss about Pentagonal Pyramids, it can be defined as " A pentagonal base, which has triangular faces, and all faces meet at a point is known as Pentagonal Pyramids". Now we will see some properties of Pentagonal Pyramids which are given below:

1. Six faces are present in a Pentagonal Pyramid.

2. Five side faces of a Pentagonal Pyramid are Triangles.

3. The base of a Pentagonal Pyramid is pentagon.

4. Six vertices or we can say corner points are present in a Pentagonal Pyramid.

5. Number of edges in a Pentagonal Pyramid is 10.

These all are the properties of Pentagonal Pyramid, if a pyramid follows all these properties then we can say that the pyramid is Pentagonal Pyramid.

Now we see some formulas of Pentagonal Pyramid which are given below:

If we want to find the volume of a Pentagonal Pyramids then the formula is:

Volume = $\frac{1}{3}$ * b * h.

Where ‘b’ is the base area;

And ‘h’ is the height of a Pentagonal Pyramid.

By using this formula we find the Volume of Pyramid.

If all the faces of Pentagonal Pyramid are same then the surface area of a Pentagonal Pyramid is given by:

Surface area = [base area] + $\frac{1}{2}$ * p * l.

Where ‘p’ is the perimeter, and ‘l’ is the side length.