Calculating the Side Length of an Equilateral Triangular Prism

When dealing with geometrical shapes and their properties, understanding the relationships between different dimensions becomes crucial. This article will guide you through a step-by-step process to find the side length of an equilateral triangular cross-section of a prism, given its volume and height.

Prism Volume and Its Components

Consider a prism with an equilateral triangular cross-section. The volume ( V ) of any prism can be calculated using the formula:

V A cdot h

where A is the area of the cross-section and h is the height or length of the prism.

Given Data

The volume of the prism, V, is 270 cm3. The height or length of the prism, h, is ( 10sqrt{3} ) cm.

Step-by-Step Solution

Step 1: Rearranging the Volume Formula

First, we rearrange the given volume formula to solve for the area A:

A frac{V}{h}

Substituting the known values:

A frac{270}{10sqrt{3}} frac{27}{sqrt{3}} 9sqrt{3} , text{cm}^2

Step 2: Area of an Equilateral Triangle

The area A of an equilateral triangle can be expressed in terms of its side length s using the formula:

A frac{sqrt{3}}{4} s^2

Setting this equal to the area we found:

frac{sqrt{3}}{4} s^2 9sqrt{3}

Step 3: Isolating s^2

To eliminate sqrt{3} from both sides, we divide both sides by sqrt{3}:

frac{1}{4} s^2 9

Next, multiply both sides by 4:

s^2 36

Step 4: Solving for s

Finally, we take the square root of both sides to find the side length:

s 6 , text{cm}

Therefore, the length of the side of the equilateral triangular cross-section is boxed{6} , text{cm}.