Cylinder Volume Calculator

Compute the volume, lateral surface area, and total surface boundary of any right circular cylinder with exact π expressions and step-by-step arithmetic.

▮ Cylinder Volume Calculator

Formula: V = πr²h

Base Radius (r) *

Radius of the circular base

Cylinder Height (h) * Perpendicular height between bases

Decimal Precision Display precision for result

Calculated Cylinder Volume

141.3717 cm³

Step-by-Step Mathematical Breakdown:

1. Radius \( r = 3\text{ cm} \), Height \( h = 5\text{ cm} \)

2. Volume Formula: \( V = \pi r^2 h \)

3. Base Area: \( A_{\text{base}} = \pi \times 3^2 = 9\pi \approx 28.2743\text{ cm}^2 \)

4. Multiply by Height: \( V = 9\pi \times 5 = 45\pi \approx \mathbf{141.3717\text{ cm}^3} \)

5. Total Surface Area: \( A = 2\pi r(r + h) = 2\pi(3)(8) = 48\pi \approx \mathbf{150.7964\text{ cm}^2} \)

The Cylinder & Sphere Connection

Understanding the cylinder is foundational to understanding the Sphere Volume Formula. When a sphere of radius \(r\) is enclosed inside a cylinder with height \(h = 2r\):

V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3

Archimedes' famous ratio: \( V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}} = \frac{2}{3}(2\pi r^3) = \frac{4}{3}\pi r^3 \).

Frequently Asked Questions

What is the formula for the volume of a cylinder?

The volume of a right circular cylinder is V = πr²h, where r is the radius of the circular base and h is the height of the cylinder.

How does cylinder volume relate to sphere volume?

A cylinder that circumscribes a sphere of radius r has height h = 2r and volume V = 2πr³. The inscribed sphere has volume V = (4/3)πr³, which is exactly two-thirds (2/3) of the cylinder's volume.

What is the total surface area of a cylinder?

The total surface area includes the curved lateral area (2πrh) and both top and bottom circular bases (2πr²): A = 2πrh + 2πr² = 2πr(r + h).