Table of Contents

Last modified on August 3rd, 2023

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.

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 m^{2}, cm^{2}, mm^{2}, and in^{2}.

**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**

Also**, **since **Lateral Surface Area ( LSA) = **Base Perimeter

Here, Base Perimeter = 7*ah*

∴ **Lateral Surface Area (****LSA) = ****7****ah**

Thus, we can write

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

The volume of a heptagonal prism is the space it occupies in the three-dimensional plane. It is measured in cubic units such as m^{3}, cm^{3}, mm^{3}, ft^{3}. 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

**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 cm^{2}

Total Surface Area (*TSA*) = ${\dfrac{7}{2}\times a^{2}\times \cot \left( \dfrac{\pi }{7}\right) +LSA}$*, *here *a* = 7 cm,* LSA* = 294 cm^{2}, cot π/7 = 2.0765

${\therefore TSA=\dfrac{7}{2}\times 7^{2}\times 2.0765+294}$

= 650.12 cm^{2}

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 cm^{3}

**More Resources:**- Volume of a Prism
- Surface Area of a Prism
- Right Prism
- Oblique Prism
- Rectangular Prism
- Volume of a Rectangular Prism
- Surface Area of a Rectangular Prism
- Triangular Prism
- Volume of a Triangular Prism
- Surface Area of a Triangular Prism
- Hexagonal Prism
- Volume of a Hexagonal Prism
- Surface Area of a Hexagonal Prism
- Pentagonal Prism
- Volume of a Pentagonal Prism
- Surface Area of a Pentagonal Prism
- Trapezoidal Prism
- Volume of a Trapezoidal Prism
- Surface Area of a Trapezoidal Prism
- Square Prism
- Volume of a Square Prism
- Surface Area of a Square Prism
- Octagonal Prism
- Heptagonal Prism
- Decagonal Prism

Last modified on August 3rd, 2023