Problem
Get a cylinder's height or radius back from its surface area
A cylinder's base radius and surface area are known. We need its height.
GeometryExpressions
Your answer
How to solve
Strategy Work Backwards — The surface splits into a part that does not involve the height -- the two bases -- and a part that does. Take the first away and what is left is the side alone, which is one multiplication away from the height.
1STEP 1
Take the two bases off
Their area is fixed by the radius alone.
7 × 7 × 3.14 × 2 = 307.72
The bases are 307.72 cm².
7.G.B.4Convert To AlgebraThe two bases do not depend on the height at all.
Why?
A cylinder's surface is its two bases together with the curved side, and nothing else.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Both bases are circles of the same radius whatever the height, because every point of a base is that one radius from its centre.
🧱A circle's radius is constantEvery point on a circle sits exactly one radius from the center, so all radii of a circle are equal.
2STEP 2
See what is left for the side
Subtract the bases from the whole surface.
1186.92 - 307.72 = 879.2
The curved side is 879.2 cm².
7.G.B.6Identify Subproblems3STEP 3
Divide by the base's circumference
Unrolled, the side is a rectangle as wide as the base's circumference, so its height is the area divided by that width.
7 × 2 × 3.14 = 43.96, 879.2 ÷ 43.96 = 20
The cylinder is 20 cm tall.
6.EE.B.7Work BackwardsAnswer
20 cm
Put it back together: 43.96 × 20 = 879.2 for the side, and 879.2 + 307.72 = 1186.92 cm².
Takeaway
Split a surface into the part the unknown touches and the part it does not.
- Take the two bases off
- See what is left for the side
- Divide by the base's circumference