Total Surface Area of a Cylinder

If you need to wrap a cylindrical mailing tube or estimate paint for a storage tank, you need to know how much material covers the entire outside. The total surface area of a cylinder is the combined area of its two circular bases and its curved side. You calculate it with the formula TSA = 2πr² + 2πrh, where r is the radius of the circular base and h is the height (or length) of the cylinder. Both measurements must use the same unit. The result tells you the exact amount of paper, paint, or sheet metal required to cover the shape completely, including the top and bottom.

Dimensions
Type
Enter radius of the base
Enter height of cylinder

The calculator above accepts the radius and height of any right circular cylinder. It applies the standard formula TSA = 2πr² + 2πrh and returns the combined area of both bases plus the lateral surface. You can enter values in any unit–centimeters, meters, inches, or feet–as long as both measurements share the same unit. The output matches in corresponding square units.

What is the formula for total surface area of a cylinder?

A cylinder has three faces: a top base, a bottom base, and one continuous curved side that connects them.

  • Each circular base has an area of πr². Because there are two bases, their combined area is 2πr².
  • The curved side, if unrolled, becomes a rectangle. One edge equals the height h, and the other edge equals the circumference of the base, which is 2πr. This gives a lateral area of 2πrh.

Add both parts together:

TSA = 2πr² + 2πrh

You can factor this as 2πr(r + h) to enter fewer numbers on a calculator.

How to find the total surface area of a cylinder?

Follow these steps for any solid right cylinder:

  1. Measure the radius r of the circular base and the height h.
  2. Calculate the area of the two bases: multiply π by the square of the radius, then by 2.
  3. Calculate the curved surface area: multiply 2, π, the radius, and the height together.
  4. Add the two results to get the total surface area.

Use the same unit for both measurements. If the radius is 5 cm and the height is 10 cm, the area will be in cm².

Total surface area vs curved surface area

Understanding the difference prevents missing the top and bottom faces.

  • Curved (lateral) surface area covers only the side wall: CSA = 2πrh.
  • Total surface area adds both flat ends: TSA = 2πr² + 2πrh.

For a can with no lids, only the curved area matters. For a sealed drum or closed tank, use the total surface area.

Worked example: cylinder with r = 6 cm and h = 14 cm

Here is a complete numerical walkthrough:

  • Base area: π × 6² = 36π cm². Two bases = 72π cm².
  • Curved area: 2 × π × 6 × 14 = 168π cm².
  • Total surface area: 72π + 168π = 240π cm².

Using π ≈ 3.14159, the final value is approximately 753.98 cm².

Frequently Asked Questions

What is the unit for total surface area of a cylinder?
Surface area is always expressed in square units that match the linear measurements used. If the radius and height are measured in centimeters, the total surface area will be in square centimeters (cm²). For meters, the result is in square meters (m²).
Can you use the diameter instead of the radius in the formula?
Yes. Replace the radius r with d/2. The formula becomes TSA = 2π(d/2)² + πdh, or simplified, TSA = (πd²)/2 + πdh. Using the diameter directly avoids the extra step of dividing by two before squaring, which can reduce rounding errors.
Does total surface area include the inside surfaces of a hollow cylinder?
No. The standard formula TSA = 2πr² + 2πrh gives the exterior surface area of a solid right cylinder. A hollow cylinder has distinct inner and outer radii, so you must calculate the area of each surface–inner wall, outer wall, and both rims–separately and then add them.
What is the difference between total surface area and volume?
Total surface area measures the external coverage of the cylinder using square units, while volume measures the space inside using cubic units. The formulas are TSA = 2πr(r + h) for area and V = πr²h for volume. They answer completely different geometric questions and use different units.
How do you find the height of a cylinder if you know the total surface area and radius?
Rearrange the formula TSA = 2πr² + 2πrh to solve for height. Subtract 2πr² from both sides, giving TSA - 2πr² = 2πrh, then divide by 2πr. The height equals h = (TSA - 2πr²) / 2πr. This is useful when you know how much material you have but not the cylinder’s length.
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