How to find the radius of a cylinder
How to find the radius of a cylinder: halve the diameter, or, if you know the volume and height, use r = √(V ÷ (πh)): divide the volume by π times the height, then take the square root. From the circumference, r = C ÷ 2π; from the surface area, solve the area formula for r.
Contents
r = √(V ÷ (π × h)). With V = 500 in³ and h = 8 in: r = √(500 ÷ 25.133) = 4.46 in (diameter 8.92 in).
The calculator above works from the volume, the total or lateral surface area, the circumference or the base area; pick the mode that matches what you know. Each case is explained below with the formula, a worked example and the traps.
Radius of a cylinder formula: every case
The radius is the distance from the center of the circular base to its edge. All the formulas below come from the standard cylinder formulas, solved for r.
| You know | Radius formula | Example | Radius |
|---|---|---|---|
| Diameter d | r = d ÷ 2 | d = 6 in | 3 in |
| Volume V and height h | r = √(V ÷ (πh)) | V = 500 in³, h = 8 in | 4.46 in |
| Circumference C | r = C ÷ (2π) | C = 20 in | 3.18 in |
| Base area B | r = √(B ÷ π) | B = 50 in² | 3.99 in |
| Lateral area L and height h | r = L ÷ (2πh) | L = 150 in², h = 8 in | 2.98 in |
| Total area A and height h | r = (−h + √(h² + 2A ÷ π)) ÷ 2 | A = 300 in², h = 8 in | 3.98 in |
How to find the radius of a cylinder from the volume
This is the most common question. Start from the volume formula, V = πr²h, explained in the guide on how to find the volume of a cylinder. Divide both sides by πh to get r² = V ÷ (πh), then take the square root.
- Check the unitsThe volume must be in the cube of the height's unit: in³ with inches, cm³ with centimeters. Convert gallons or liters first.
- Multiply π by the heightπ × 8 ≈ 25.133.
- Divide the volume by that number500 ÷ 25.133 ≈ 19.894. This is r².
- Take the square root√19.894 ≈ 4.46 in. Double it for the diameter: 8.92 in.
Example in centimeters: a two-liter container
A container must hold 2 L and stand 20 cm tall. Two liters are 2,000 cm³. Then r² = 2,000 ÷ (π × 20) ≈ 31.831, and r ≈ 5.64 cm. The inside diameter is about 11.28 cm.
Example in gallons: a drum 34 in tall
A drum holds 55 US gallons and is 34 in tall inside. Convert gallons to cubic inches: 55 × 231 = 12,705 in³. Then r² = 12,705 ÷ (π × 34) ≈ 118.94, so r ≈ 10.91 in, an inside diameter of about 21.81 in.
Example in meters: one cubic meter, 2 m tall
A tank of 1 m³ with a height of 2 m: r² = 1 ÷ (2π) ≈ 0.15915, so r ≈ 0.399 m, just under 40 cm.
Radius from the diameter or the circumference
If you can measure straight across the end, the radius is simply half of that width: r = d ÷ 2. The page on the volume of a cylinder with the diameter shows how to skip the radius altogether and work with d.
When the cylinder is too large to measure across, wrap a tape around it. The circumference is C = 2πr, so r = C ÷ (2π). A 20 in tape reading gives r = 20 ÷ 6.2832 ≈ 3.18 in. A 100 cm reading around a post gives r ≈ 15.92 cm. Pull the tape square to the axis: a tilted tape reads long and overstates the radius.
Radius from the surface area
From the base area
The base is a circle of area B = πr², so r = √(B ÷ π). For a base area of 50 in²: 50 ÷ π ≈ 15.915, and r ≈ 3.99 in. The same formula gives the radius from any cross-sectional area of a cylinder cut square to its axis.
From the lateral (curved) area
The curved side unrolls into a rectangle 2πr wide and h tall, so L = 2πrh and r = L ÷ (2πh). For L = 150 in² and h = 8 in: 2π × 8 ≈ 50.265, and r = 150 ÷ 50.265 ≈ 2.98 in.
From the total surface area
The total area adds the two ends: A = 2πr² + 2πrh. With A and h known, this is a quadratic equation in r: 2πr² + 2πh × r − A = 0. Its positive solution is:
r = (−h + √(h² + 2A ÷ π)) ÷ 2
For A = 300 in² and h = 8 in: 2A ÷ π = 600 ÷ π ≈ 190.99; add h² = 64 to get 254.99; the square root is about 15.968; subtract 8 to get 7.968; halve it: r ≈ 3.98 in. Check: 2π × 3.984² + 2π × 3.984 × 8 ≈ 300 in². The negative root of the quadratic has no physical meaning and is dropped.
How the radius changes the volume
Because the radius is squared, a small change in it has a large effect. Double the radius and the volume is multiplied by four. To double a tank's volume without changing its height, multiply the radius by √2 ≈ 1.414, not by 2.
In calculus terms, the derivative of the volume of a cylinder with respect to the radius, with h held constant, is dV/dr = 2πrh. That is exactly the lateral area: a thin layer added around the side has about that area times its thickness. For r = 3 in and h = 8 in, dV/dr ≈ 150.80 in³ per inch. Growing the radius from 3 to 3.1 in adds about 150.80 × 0.1 ≈ 15.08 in³; the exact increase is 15.33 in³, the small gap coming from the square term.
Common mistakes when finding the radius
Forgetting the square root is the classic error. V ÷ (πh) gives r², not r: for V = 500 in³ and h = 8 in it gives 19.89, while the radius is 4.46 in.
- Mismatched units. A volume in gallons and a height in inches cannot go straight into the formula. Convert to in³ first (× 231 for US gallons) or to cm³ (× 1,000 for liters).
- Giving the diameter as the answer. Formulas from the volume and the area return the radius. Double it if the question asks for the width.
- Dividing by π only. In r = √(V ÷ (πh)), the whole product πh goes below the line.
- Inside or outside radius. For a pipe or a tank, the capacity sets the inside radius; the material of the wall is handled with the hollow cylinder volume formula.
To find the other missing dimension, the guide on how to find the height of a cylinder solves the same formula for h. Once you have the radius, the cylinder volume calculator checks it: enter r and h, and the volume should come back to the number you started from.
FAQ
What is the radius of a cylinder?
The radius is the distance from the center of the circular base to its edge. It is half the diameter, and it is the same at every height of a straight cylinder.
How do I find the radius of a cylinder given its volume?
Divide the volume by π times the height, then take the square root: r = √(V ÷ (πh)). A cylinder of 500 in³ and 8 in tall has r = √(500 ÷ (π × 8)) ≈ 4.46 in.
How do you find the radius given the diameter?
Divide the diameter by 2: r = d ÷ 2. A pipe with a 6 in inside diameter has a 3 in inside radius.
How do you find the radius of a cylinder from the circumference?
Divide the circumference by 2π: r = C ÷ (2π). A tape reading of 100 cm around a post gives r = 100 ÷ (2π) ≈ 15.92 cm.
How do you find the radius from the surface area?
With the total area A and the height h, solve 2πr² + 2πrh = A: r = (−h + √(h² + 2A ÷ π)) ÷ 2. For A = 300 in² and h = 8 in, r ≈ 3.98 in. With the lateral area L only, r = L ÷ (2πh).
What is the derivative of the volume of a cylinder with respect to the radius?
dV/dr = 2πrh, with the height held constant. It equals the curved surface area: a thin extra layer of thickness dr around the side adds about 2πrh × dr of volume.
Related
- Height of a cylinderHow to find the height of a cylinder: h = V ÷ (πr²) from the volume, or from the surface area. Calculator, tank examples in gallons and liters, common traps.
- Volume with diameterVolume of a cylinder with the diameter: V = πd²h ÷ 4. Calculator, worked examples in inches, feet and cm, and how to find the diameter from the volume.
- How to find the volumeHow to find the volume of a cylinder: square the radius, multiply by π and the height. Worked examples in inches, feet, cm and m, plus common mistakes.
Updated October 12, 2026