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Calculate the surface area of cylinders

Surface area of a right circular cylinder is understood as the total area of its two congruent circular bases and its curved lateral surface, which can be represented by an unwrapped rectangle; consequently, S=2πr2+2πrhS=2\pi r^2+2\pi rh, with diameter converted to radius when necessary. Calculations use appropriate square units and exact π\pi or a specified approximation, and distinguish surface area from volume; oblique, composite, and calculus-based treatments are beyond this scope.

Detailed Explanation: Calculate the surface area of cylinders

A cylinder has three surface parts:

  • Two circular bases: 2πr22\pi r^2
  • One curved surface: 2πrh2\pi rh

So the total surface area is

S=2πr2+2πrhS=2\pi r^2+2\pi rh

Here, rr is the radius and hh is the height. Surface area is measured in square units.

Example: Find the surface area of a cylinder with diameter 10 cm10\text{ cm} and height 12 cm12\text{ cm}. Leave the answer in terms of π\pi.

Step 1: Convert the diameter to a radius.

The radius is half the diameter:

r=102=5 cmr=\frac{10}{2}=5\text{ cm}

The height is

h=12 cmh=12\text{ cm}

Step 2: Substitute into the formula.

S=2πr2+2πrhS=2\pi r^2+2\pi rh S=2π(5)2+2π(5)(12)S=2\pi(5)^2+2\pi(5)(12)

Step 3: Calculate each part.

Area of the two circular bases:

2π(5)2=2π(25)=50π2\pi(5)^2=2\pi(25)=50\pi

Area of the curved surface:

2π(5)(12)=120π2\pi(5)(12)=120\pi

Step 4: Add the areas.

S=50π+120π=170π cm2S=50\pi+120\pi=170\pi\text{ cm}^2

Therefore, the surface area is

170π cm2\boxed{170\pi\text{ cm}^2}

If π≈3.14\pi\approx3.14, then

170(3.14)=533.8170(3.14)=533.8

so the surface area is approximately 533.8 cm2\boxed{533.8\text{ cm}^2}. Surface area measures the outside covering of the cylinder; it is not the volume inside the cylinder.

Learn by doing: Calculate the surface area of cylinders

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