Pentagonal Pyramid

Definition

A pentagonal pyramid is a pyramid with a pentagonal base bounded by five lateral faces meeting at a common point, known as the apex. The lateral faces are in the shape of triangles.

How many faces, edges and vertices does a pentagonal pyramid have?

A pentagonal pyramid consists of 6 faces, 10 edges, and 6 vertices.

A regular pentagonal pyramid has a base in the shape of a regular pentagon, and its lateral sides are equilateral triangles.

Like all other polyhedrons, we can calculate the surface area and the volume of a pentagonal pyramid.

Formulas

Volume

The formula is:

Volume (V) = ${\dfrac{5}{6}abh}$, here a = apothem, b = base h = height

Let us solve an example to understand the concept better.

Find the volume of a pentagonal pyramid with an apothem of 4.81 cm, a  base length of 7 cm, and a height of 11 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{5}{6}abh}$, here a = 4.81 cm, b = 7 cm, h = 11 cm
∴ V = ${\dfrac{5}{6}\times 4.81\times 7\times 11}$
= 308.64 cm³

Surface Area

The formula is:

Surface Area (SA) = ${\dfrac{5}{2}b\left( a+s\right)}$, here a = apothem, b = base, s = slant height,

Also ${\dfrac{5}{2}bs}$ = lateral surface area (LSA)  

∴ ${SA=\dfrac{5}{2}ab+LSA}$

Let us solve some examples to understand the concept better.

Find the lateral and total surface area of a pentagonal pyramid with an apothem of 6.9 in, a  base length of 10 in, and a slant height of 14 in.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\dfrac{5}{2}bs}$, here b = 10 in, s = 14 in
∴ LSA = ${\dfrac{5}{2}\times 10\times 14}$
= 350 in²
Total Surface Area (TSA) = ${\dfrac{5}{2}ab+LSA}$, here a = 6.9 in, b = 10 in, LSA =350 in²
∴TSA =  ${\dfrac{5}{2}\times 6.9\times 10+350}$
=522.5 in²

Find the lateral area of a pentagonal pyramid with a base of 5 cm and a slant height of 6.5 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\dfrac{5}{2}bs}$, here b = 5 cm in, s = 6.5 cm
∴ LSA = ${\dfrac{5}{2}\times 5\times 6.5}$
= 81.25 cm²