Barrel Volume Calculator

What is a Barrel?

A barrel is a cylindrical container with bulging sides, typically made of wooden staves bound by metal hoops. In geometry, we consider it as a solid formed by rotating a curved line around an axis, creating a shape similar to two truncated cones joined at their wider ends.

How to Calculate Barrel Volume

Calculating the volume of a barrel involves using the formula for a truncated cone and accounting for the barrel's unique shape. We use the radii at the top, middle, and bottom of the barrel, along with its height, to determine the volume.

Formula

The formula for calculating the volume of a barrel is:

\[ V = \frac{\pi h}{12} (r_1^2 + 4r_2^2 + r_3^2) \]

Where:

• \(V\) is the volume of the barrel
• \(h\) is the height of the barrel
• \(r_1\) is the radius at the top of the barrel
• \(r_2\) is the radius at the middle of the barrel
• \(r_3\) is the radius at the bottom of the barrel
• \(\pi\) (pi) is approximately 3.14159

Calculation Steps

1. Measure the height (h) of the barrel
2. Measure the diameter at the top, middle, and bottom of the barrel
3. Calculate the radii by dividing each diameter by 2
4. Square each radius (r₁², r₂², r₃²)
5. Multiply the middle radius squared by 4
6. Add the three terms: r₁² + 4r₂² + r₃²
7. Multiply by π and h/12
8. The result is the volume of the barrel

Example and Visual Representation

Let's calculate the volume of a barrel with the following measurements:

• Height (h) = 100 cm
• Top radius (r₁) = 30 cm
• Middle radius (r₂) = 40 cm
• Bottom radius (r₃) = 30 cm

Applying the formula:

\[ \begin{align*} V &= \frac{\pi h}{12} (r_1^2 + 4r_2^2 + r_3^2) \\ &= \frac{\pi \cdot 100}{12} (30^2 + 4 \cdot 40^2 + 30^2) \\ &= \frac{100\pi}{12} (900 + 6400 + 900) \\ &= \frac{100\pi}{12} (8200) \\ &\approx 214,675.5 \text{ cm}^3 \\ &\approx 214.68 \text{ liters} \end{align*} \]

Here's a visual representation of this barrel:

In this diagram, you can see the barrel shape with its key measurements labeled. The red line represents the top radius (r₁), the green line the middle radius (r₂), and the blue line the bottom radius (r₃). The purple line shows the height (h) of the barrel. The calculated volume is displayed below the barrel.