A pyramid is a polyhedron that has a polygonal base and triangles for sides. The three main parts of any pyramid are the apex, face, and base. The base of a pyramid may be of any shape such as triangle, square, pentagon. Faces are in the shape of an isosceles triangle. Apex is a point at the top of the pyramid where all the triangle of the pyramid meets. Let us now discuss the different pyramid formula in detail.

**What is a Pyramid?**

A 3-dimensional shape formed by joining all the corners of a polygon to a central point or apex is named as a pyramid. The slant height is the diagonal height from the center of one of the base edges to the apex denoted by l. Height is the perpendicular distance from the apex to the base. Height is denoted by h.

**Pyramid Formula**

**1] Surface area of the Pyramid**

The lateral surface area of a regular pyramid is the sum of the areas of the lateral faces of the pyramid.

\(L.S.A. = \frac{1}{2} × p × l\)

p | the perimeter of the base |

l | slant height |

**2] Total Surface Area of a Regular Pyramid**

It is the sum of the areas of lateral faces of the pyramid and base of the pyramid.

\(T.S.A.= \frac{1}{2} × p × l + B\)

p | the perimeter of the base |

l | slant height |

B | the area of the base |

**3] Volume of the Pyramid**

Volume of a 3 -dimensional solid is the amount of space occupies by the solid.

The volume of a Pyramid=\(\frac{1}{2} × Base Area × Height\)

**Types of Pyramid**

**1] Square Pyramid**

A square pyramid has a square base, triangular faces as 4 and an apex. The Square Pyramid formulas are,

- Base Area of a Square Pyramid = b
^{2} - Surface Area of a Square Pyramid = 2 × b × s +b
^{2} - Volume of a Square Pyramid=\(\frac{1}{3}\) × b
^{2 }× h

b | base length |

s | slant height |

h | height |

**2] Triangular Pyramid**

A triangular pyramid is a pyramid, which has triangular faces, and a triangular base. The Triangular Pyramid formulas are,

- Base Area of a Triangular Pyramid=\(\frac{1}{2} × a × b\)
- Surface Area of a Triangular Pyramid=\(\frac{1}{2} × a × b+\frac{3}{2} × b ×s\)
- The volume of a Triangular Pyramid=\(\frac{1}{6} × a × b × h\)

a | apothem length |

b | base length |

s | slant height |

h | height |

**3] Pentagonal Pyramid**

The Pentagonal pyramid has a pentagonal base, triangular faces as 5 and an apex. The Pentagonal Pyramid Formulas are,

- Base Area of a Pentagonal Pyramid=\(\frac{5}{2} × a × b\)
- Surface Area of a Pentagonal Pyramid=\(\frac{5}{2} (a × b + b × s)\)
- The volume of a Pentagonal Pyramid=\(\frac{5}{6} × a × b × h\)

**Solved Example for ****Pyramid Formula**

Q.1: Find the volume of a regular square pyramid with base sides 10 cm and altitude 18 cm.

Solution: Volume of a pyramid is,

\(V=\frac{1}{3}Base area × h\)

base area= 10 × 10 = 100 cm^{2}.

\(V= \frac{1}{3} × (100) × (18) = 600\)

Therefore, the volume of the regular square pyramid is 600 cm^{3}

Q.2: Find the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures 88 cm and the slant height is 55 cm.

Solution: The perimeter of the base is the sum of the sides.

Perimeter of the base = 3 × (8) = 24 cm

L.S.A. =\( \frac{1}{2} p × l = \frac{1}{2} (24) × (5) = 60 cm^2\)

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