Octagonal Prism

An octagonal prism is a three-dimensional solid consisting of two identical octagonal bases joined together by eight rectangular faces. It has 10 faces (2 octagonal and 8 rectangular), 24 edges, and 16 vertices.

The shape of birdhouses and chocolate boxes are some examples of octagonal prism in real life.

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

Formulas

Surface Area

The surface area, also known as the total surface area, of an octagonal prism is the entire space occupied by its outermost layer (or faces). It is measured in square units such as m2, cm2, mm2, and in2. The formula is given below:

As we know,

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

Alsosince Lateral Surface Area (LSA) = BasePerimeter × height

Here, Base Perimeter = 8ah

∴ Lateral Surface Area (LSA) = 8ah

Thus, we can write

Total Surface Area (TSA) = 4(1+√2)a2 + LSA

Let us solve an example to understand the concept better.

Find the lateral and total surface area of an octagonal prism with a base edge of 2 cm and height of 5 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = 8ah, here a = 2 cm, h = 5 cm
∴ LSA = 8 × 2 × 5
= 80 cm2
Total Surface Area (TSA) = ${4\left( 1+\sqrt{2}\right) a^{2}+LSA}$, here a = 2 cm, LSA = 80 cm2
${\therefore TSA=4\left( 1+\sqrt{2}\right) 2^{2}+80}$
= 118.63 cm2

Volume

The volume of an octagonal 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:

Let us solve some examples to understand the concept better.

Find the volume of an octagonal prism whose base edge is 6 cm and height is 7 cm.

Solution:

As we know,
Volume (V) = ${2\left( 1+\sqrt{2}\right) a^{2}h}$, here a = 6 cm, h = 7 cm
${\therefore V=2\left( 1+\sqrt{2}\right)\times 6^{2}\times 7}$
= 1216.76 cm3

Finding the volume of an octagonal prism when APOTHEM, BASE EDGE and HEIGHT are known

Find the volume of an octagonal prism whose base edge is 4 cm, apothem is 4.828 cm, and height is 5 cm.

Solution:

Here we will use an alternative formula.
Volume (V) = 4 × base edge × apothem × height, here base edge = 4 cm, apothem = 4.828 cm, height = 5 cm
V = 4 × 4 × 4.828 × 5
= 386.24 cm3

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