L3 · Surface area of right cylinders

Learning goal

“I can unroll a right cylinder into its net - two circles and one rectangle - and find its total surface area: the rectangle's length is the circumference of the base, and I adjust the census for open tops and given diameters.”

Lateral surface — the curved side surface of a cylinder, not counting the two circular bases. Unrolled flat, it is a rectangle.

Circumference — the distance around a circle: C = 2πr = πd. On the cylinder's net, the circumference becomes the LENGTH of the lateral rectangle - the label must wrap exactly once around.

Net of a cylinder — two circles (the bases) plus one rectangle (the unrolled lateral surface), the rectangle's length equal to the circumference and its width equal to the cylinder's height.

Worked example

A right cylinder has radius 5 cm and height 8 cm. Find its total surface area. Step 1: Census from the net: 2 circular bases + 1 lateral rectangle. Step 2: Bases: each is πr^2 = 3.14 x 25 = 78.5; two of them: 157 cm2. Step 3: The rectangle: its length is the circumference, 2 x 3.14 x 5 = 31.4 cm; its width is the height, 8 cm. Lateral area = 31.4 x 8 = 251.2 cm2. Step 4: SA = 157 + 251.2 = 408.2 cm2. As one line: SA = 2πr^2 + 2πrh.

Check yourself — explain it

The lateral rectangle's length is the circumference because...

Paired problem — example still visible

Radius 3 cm, height 10 cm -> bases 2 x (3.14 x 9) = 56.52; lateral 2 x 3.14 x 3 x 10 = 188.4; SA = 244.92 cm2

Faded completion — last step is yours

"Radius 2 cm, height 6 cm: bases 2 x (3.14 x 4) = 25.12; circumference = 12.56; lateral = 12.56 x 6 = ___; SA = ___ cm2" -> 75.36; 100.48 cm2

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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