Mathematical Animation: Derivation of the Formula for the Surface Area of a Cylinder

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BUY THIS ANIMATION of surface area of a cylinder!

A cylinder is unwrapped to illustrate how the formula cylinder surface area = 2 π r² + 2 π rh can be understood.

The animation starts with a translucent cylinder of height r and capped by two ends of radius r. The circumference of the cylinder is 2 π r. The wall of the cylinder opens up to form a rectangle whose length is the circumference of the cylinder and whose height is h. This gives the area of the cylinder wall as 2 π rh. The two ends of the cylinder hinge backwards to demonstrate their two areas. Each circle has an area of π r² (see area of a circle animation) giving a total area of the two ends as 2 π r².

In the particular case of a cylinder whose height is the same as its diameter (i.e. the height is equal to 2 r ) that cylinder will perfectly inscribe a sphere of radius r. The formula becomes:
surface area of cylinder = π r² + π r² + (2 π r x 2 r) =
2(π r²) + (4 π r²) =
6 π r²

The area of the cylinder is 6 π r² and that of its circumscribed sphere is 4 π r² see animation of the surface area of a sphere. In other words, the sphere has 4/6 or two thirds the area of its enclosing cylinder. Now, this is interesting because it is the same ratio as the volume a sphere to the volume of its circumscribing cylinder. Archimedes discovered these relationships between a cylinder and its enclosed (circumscribed) sphere.

Try our circle and area calculator to derive various values from different starting points.

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