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.Â TheÂ slant heightÂ 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 lateralÂ surface areaÂ of a regularÂ pyramidÂ is the sum of the areas of the lateral faces of the pyramid.

\(L.S.A. = \frac{1}{2} Ã— p Ã— l\)

p | the perimeter of the base |

l | slant height |

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

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

\(T.S.A.= \frac{1}{2} Ã— p Ã— l + B\)

p | the perimeter of the base |

l | slant height |

B | the area of the base |

**3] Volume of the Pyramid**

VolumeÂ of aÂ 3Â -dimensional solid is the amount of space occupies by the solid.

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

**Types of Pyramid**

**1] Square Pyramid**

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

- Base Area of a Square Pyramid = b
^{2} - Surface Area of a Square Pyramid = 2 Ã— b Ã— s +b
^{2} - Volume of a Square Pyramid=\(\frac{1}{3}\) Ã— b
^{2 }Ã— 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. TheÂ Triangular Pyramid formulasÂ are,

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

a | apothem length |

b | base length |

s | slant height |

h | height |

**3] Pentagonal Pyramid**

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

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

**Solved Example forÂ ****Pyramid Formula**

Q.1: Find the volume of a regular square pyramid with base sidesÂ 10Â cm and altitudeÂ 18Â cm.

Solution: Volume of a pyramid is,

\(V=\frac{1}{3}Base area Ã— h\)

base area= 10 Ã— 10 =Â 100â€‰â€‰cm^{2}.

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

Therefore, the volume of the regular square pyramid isÂ 600â€‰cm^{3}

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

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

Perimeter of the base = 3 Ã— (8) = 24â€‰cm

L.S.A. =\( \frac{1}{2} p Ã— lÂ = \frac{1}{2} (24) Ã— (5) = 60â€‰â€‰cm^2\)

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