# Heptagonal Prism

A heptagonal prism is a three-dimensional solid consisting of two identical heptagonal bases joined together by seven rectangular faces. It has 9 faces (2 Heptagonal and 7 rectangular), 19 edges, and 14 vertices.

Like all other polyhedrons, we can calculate the surface area and volume of a regular heptagonal prism.

## Formula

### Surface Area

The surface area (or total surface area) of a heptagonal prism is the entire amount of space occupied by all its outer surfaces (or faces). It is measured in square units such as m2, cm2, mm2, and in2.

Total Surface Area (TSA) = ${\dfrac{7}{2}\times a^{2}\times \cot \left( \dfrac{\pi }{7}\right) +7ah}$, here a = base edge, h = height, cot π/7 = 2.0765

As we know,

Total Surface Area (TSA) = 2 × Base Area + Base Perimeter × height

Alsosince Lateral Surface Area (LSA) = Base Perimeter × height

Here, Base Perimeter = 7ah

∴ Lateral Surface Area (LSA) = 7ah

Thus, we can write

Total Surface Area (TSA) = ${\dfrac{7}{2}\times a^{2}\times \cot \left( \dfrac{\pi }{7}\right) +LSA}$

### Volume

The volume of a heptagonal prism is the space it occupies in the three-dimensional plane. It is measured in cubic units such as m3, cm3, mm3, ft3. The formula is given below:

Volume (V) = ${\dfrac{7}{4}\times a^{2}\times \cot \left( \dfrac{\pi }{7}\right) \times h}$, here a = base edge, h = height, cot π/7 = cot 25.71 = 2.0765

## Solved Example

Find the lateral and total surface area, and volume of a heptagonal prism with a base edge of 7 cm and a height of 6 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = 7ah, here a = 7 cm, h = 6 cm, cot π/7 = 2.0765
LSA = 7 × 7 × 6
= 294 cm2
Total Surface Area (TSA) = ${\dfrac{7}{2}\times a^{2}\times \cot \left( \dfrac{\pi }{7}\right) +LSA}$, here a = 7 cm, LSA = 294 cm2, cot π/7 = 2.0765
${\therefore TSA=\dfrac{7}{2}\times 7^{2}\times 2.0765+294}$
= 650.12 cm2

Volume (V) = ${\dfrac{7}{4}\times a^{2}\times \cot \left( \dfrac{\pi }{7}\right) \times h}$ , here a = 7 cm, h = 6 cm, cot π/7 = 2.0765
${\therefore V=\dfrac{7}{4}\times 7^{2}\times 2.0765\times 6}$
= 1068.37 cm3

• More Resources: