Question

A regular square pyramid is \[3\] m height and the perimeter of its base is \[16\] m. Find the volume of the pyramid?

Answer 1

Hint: A Pyramid is a polyhedron that has a base and \[{\text{3}}\]or greater triangular faces that meet at a point above the base (the apex).
In the case of a square pyramid, the base is in square shape and it has four faces. A most famous example of such a pyramid in real life is the Great Pyramid of Giza.
A three-dimensional geometric shape having a square base and four triangular faces/sides that meet at a single point is called a square pyramid. If all the triangular faces have equal edges, then this pyramid is said to be an equilateral square pyramid. If the apex of the pyramid is right above the center of its base, it forms a perpendicular with the base and such a square pyramid is known as the right square pyramid.
\[{\text{Volume of square pyramid}}\; = {\text{ }}\dfrac{1}{3} \times {{\text{a}}^2} \times {\text{height}}\]
Where ‘a’ is the base edge length.
h is the height.
Surface area of square pyramid
\[{\text{A}} = {{\text{a}}^2} + {\text{2a}}\sqrt {\dfrac{{{a^2}}}{4} + {h^2}} \]
Where ‘a’ is the base edge length.
h is the height.
The general formula for the total surface area of a regular pyramid is given by:
\[{\text{T}}.{\text{S}}.{\text{A}}. = \dfrac{1}{2} \times {\text{P}} \times l{\text{ + B}}\]
Where,
P is the perimeter of the base
l the slant height and
B the area of the base

Complete answer:

\[{\text{Volume of square pyramid}}\; = {\text{ }}\dfrac{1}{3} \times {{\text{a}}^2} \times {\text{height}}\]
Given, Perimeter of base square of base 's' is \[{\text{16}}\]m
\[{\text{4s}} = {\text{16}}\]
\[{\text{s}} = {\text{4m}}\]
\[{\text{area of square is }}{{\text{s}}^{\text{2}}}{\text{ = }}{{\text{4}}^{\text{2}}}{\text{ = 16}}\]
Hence, base area is \[16{m^2}\]
Given pyramid height is \[{\text{3}}\]m
Hence, \[{\text{Volume of pyramid is = }}\dfrac{1}{3} \times 16 \times 3 = 16{{\text{m}}^{\text{3}}}\]

Note: Properties of Square Pyramid
(i) It has \[5\]Faces.
(ii) The \[4\] Side Faces are Triangles.
(iii) It has \[5\] Vertices (corner points).
(iv) It has \[8\] Edges.