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.
Heptagonal Prism
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
Also, since 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
