How to Find the Surface Area of a Cylinder: Formula & Examples

To find the surface area of a closed cylinder, use the formula A = 2πr(r + t), where r is the base radius and t is the height. For an open cylinder (such as a pipe), use the lateral surface area A = 2πrt. Example: a cylinder with r = 7 cm and t = 10 cm has a total surface area of 2 × (22/7) × 7 × (7+10) = 748 cm².

What is the Surface Area of a Cylinder?

A cylinder is a three-dimensional solid with two circular bases and one curved lateral surface. The surface area of curved solids has been studied since ancient Greek times — Archimedes (287–212 BCE) was the first to prove mathematically the relationship between the lateral surface area of a cylinder and its circular base. A cylinder has two main measurements:

• Radius (r): the radius of the circular base
• Height (t): the height of the cylinder

The surface area of a cylinder is the total area of every face of the cylinder, including the two circular bases and the lateral (curved) surface.

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Surface Area of a Cylinder Formula

There are three formulas depending on what you need:

Lateral surface area (no caps):

Lateral area = 2 × π × r × t (base circumference × height)

Area of one circular base:

Total surface area (with both caps):

Total area = 2 × π × r² + 2 × π × r × t = 2πr(r + t)

💡 Tip for Remembering the Formula: Lateral area = base circumference × height = 2πr × t. Imagine unrolling a can: the side becomes a rectangle whose width equals the circumference of the circular base.

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How to Find the Surface Area of a Cylinder: Worked Examples

Example 1: Total Surface Area

A cylinder has a radius of 7 cm and a height of 10 cm. Find its total surface area. (π = 22/7)

Step 1: r = 7 cm, t = 10 cm

Step 2: Apply the total surface area formula

Answer: Total surface area = 748 cm²

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Example 2: Lateral Surface Area Only (Open Cylinder)

A water pipe in Jakarta is a cylinder with no caps, with a radius of 3.5 cm and a length of 20 cm. What is the area of its outer surface? (π = 22/7)

Step 1: The pipe is open → lateral surface only (no bases)

Answer: Lateral surface area of the pipe = 440 cm²

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Example 3: Word Problem

A milk can made in Surabaya is shaped like a cylinder with a diameter of 14 cm and a height of 20 cm. What is the minimum area of paper needed to wrap the entire surface? (π = 22/7)

Step 1: d = 14 cm → r = 7 cm, t = 20 cm

Answer: Minimum paper area = 1,188 cm²

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Common Mistakes when Calculating the Surface Area of a Cylinder

Mistake 1: Forgetting to multiply the base by 2 A cylinder has two bases (top and bottom), each = πr². A common mistake is to count only one base, giving A = πr² + 2πrt. The correct formula is A = 2πr² + 2πrt. The exception is when the problem specifies an open cylinder (only one cap), in which case A = πr² + 2πrt.

Mistake 2: Using the diameter as the radius If a problem gives a diameter of d = 14 cm, the radius is r = 7 cm (not 14). Using d = 14 directly in the formula gives a surface area four times larger than it should be. Always divide the diameter by 2 to get the radius before calculating.

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Surface Area of a Cylinder Table (π = 22/7)

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Practice Problems: Surface Area of a Cylinder

Problem 1: ⭐ A cylinder has r = 14 cm and t = 5 cm. Find its total surface area. (π = 22/7)

Problem 2: ⭐⭐ An oil drum is a cylinder with a diameter of 70 cm and a height of 1 m. What is the surface area of the drum? (π = 22/7)

Problem 3: ⭐⭐⭐ A cylinder has only a bottom cap (no top cap), with r = 21 cm and t = 30 cm. Find its surface area. (π = 22/7)

Problem 4: ⭐⭐⭐ A cylinder has r = 7 cm. If the lateral surface area is 440 cm², what is its height?

Answer Key:

1. A = 2×(22/7)×14×(14+5) = 88×19 = 1,672 cm²
2. r = 35 cm, t = 100 cm. A = 2×(22/7)×35×(35+100) = 220×135 = 29,700 cm²
3. A = πr² + 2πrt = (22/7)×441 + 2×(22/7)×21×30 = 1,386 + 3,960 = 5,346 cm²
4. 440 = 2×(22/7)×7×t = 44t → t = 10 cm

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Summary

1. The surface area of a cylinder consists of two parts: the two circular bases (2πr²) and the lateral surface (2πrt).
2. Total formula: A = 2πr(r + t); lateral-only formula: A = 2πrt.
3. If you unroll a cylinder's lateral surface, it becomes a rectangle with length = base circumference (2πr) and width = height (t).
4. For an open cylinder (one cap): A = πr² + 2πrt; for a cylinder with no caps: A = 2πrt.
5. This topic falls under Curved Solids in Grade 9 and is foundational for higher-level standardised maths tests.

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See also: How to Calculate the Circumference of a Circle and How to Find the Area of a Triangle.