Math CentralQuandaries & Queries


Question from destini, a student:

the volume of a sphere is equal to its surface area. what is its radius


You must be careful with questions that compare apples to oranges -- volume is measured in cubic units while areas come in square units. They cannot be equal. For a sphere

Volume = (4/3) × π × r3 cubic units
Area = 4 × π × r2 square units.

For example, a sphere whose radius is one foot long has a surface area of 4 × π square feet = 576 × π square inches; please note that 4 does not equal 576, even though both numbers represent the area of that sphere. Similarly, its volume is 4 × π/3 cubic feet = 2304 π cubic inches.

So, your question really has no meaning: By choosing units appropriately, every sphere will have the same numerical value for its volume and surface area! My guess is that your questioner wants you to ignore the units and solve the equation (4/3) × π × r3 = 4 × π × r2 for r. Almost everything cancels, so the solution is easily obtained (I got r = 3), but it it is just a number that has no meaning. I fail to see how 36 π square units equals 36 π cubic units.



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