Regular Square Pyramid Formula, Volume, Surface Area

Regular Square Pyramid Formula: A square pyramid, a geometric figure, features a square base and four triangular faces. It is a type of pyramid, a polyhedron characterized by a base and three or more triangular faces meeting at a single point above the base, known as the apex. Specifically, a square pyramid holds a total of five faces, earning its classification as a pentahedron within three-dimensional space .

A regular square pyramid has a square base, 4 triangular faces, 5 vertices, and 8 edges. Its major formulas encompass Volume and Surface Area.

Regular Square Pyramid Formula Volume

Volume Formula for a Regular Square Pyramid: Given a base length 'a' and height 'h', the volume (V) of a regular square pyramid is calculated using the formula:

V= 1/3 ​ ×Base Area× Height

V= 1/3 ​ ×a ² ×h

Regular Square Pyramid Formula Surface Area

Surface Area Formula for a Regular Square Pyramid:

The surface area (SA) of a regular square pyramid is given by the sum of the area of the base and the area of the 4 triangular faces:

SA=Base Area+4×Area of One Triangular Face

SA=a ² +2a× ( 2 a ​ ² ) ^1/2 +h ² ​

Regular Square Pyramid Formula Solved Example

Example1: Find the volume of a regular square pyramid with a base length of 6 cm and a height of 8 cm. Solution:

Given: Base length (a) = 6 cm, Height (h) = 8 cm

Using the volume formula:

V= 1/3 ​ ×a ² ×h

V= 1/3 ​ ×6 ² ×8

V= 1/3 ​ ×36×8

V= 1/3 ​ ×288

V=96 cm ³

Therefore, the volume of the given regular square pyramid is 96 cm ³ .

Example 2: Question: Determine the volume of a regular square pyramid with a base length of 5 cm and a height of 10 cm. Solution:

Given: Base length (a) = 5 cm, Height (h) = 10 cm

Using the volume formula for a regular square pyramid:

V= 1/3 ​ ×a ² ×h

Substituting the given values:

V= 1/3 ​ ×5 ² ×10

V= 1/3 ​ ×25×10

V= 1/3 ​ ×250

V=83.33cm 3 (rounded to two decimal places)

Therefore, the volume of the given regular square pyramid is approximately 83.33 cm ³ 83.33cm ³ .

Example 3: Question: Find the volume of a regular square pyramid with a base length of 5 cm and a height of 9 cm. Solution:

Given: Base length (a) = 5 cm, Height (h) = 9 cm

Using the volume formula:

V= 1/3 ​ ×a ² ×h

V= 1/3 ​ ×5 ² ×9

V= 1/3 ​ ×25×9

V= 1/3 ​ ×225

V=75 cm ³

Hence, the volume of the regular square pyramid is 75 cm ³ .

These examples illustrate how to compute the surface area and volume of a regular square pyramid using the provided formulas and given dimensions.

Understanding the properties and calculations related to a regular square pyramid, specifically its volume and surface area, is crucial in geometry. The formulas for volume and surface area provide a straightforward way to compute these measures based on the given dimensions of the pyramid. The volume of a regular square pyramid, determined by its base length 'a' and height 'h', is given by

V= 1/3 ​ ×a ² ×h, while the surface area is calculated using SA=a 2 +2a×(  2 ​ a ² ) ^1/2 +h ² .

The solved examples illustrate how these formulas are applied to find the volume and surface area of a regular square pyramid when specific measurements are provided. Understanding and applying these formulas enable us to efficiently compute these geometric measures for different regular square pyramids, aiding in various mathematical and real-world applications involving these shapes.

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