Math CentralQuandaries & Queries


Question from angela, a student:

how can i make a cone with a 6cm of height and has 2cm radius in the opening??

Hi Angela. A cone is made by cutting a pie piece out of a circle, then pulling the remainder together into the third dimension. If you remove a slim pie piece from the full circle, you will have a broad, short cone. Removing a large pie piece (even more that half the pie) makes a narrow, tall cone with the remainder.

The slant height of the cone will become the radius of the circle pattern. You can calculate this by using Pythagoras, because the height of the cone and the radius of the opening are right angles and the slant is the hypotenuse. For your cone,

S² = 6² + 2²
S = √40 which is about 6.32 cm.

The circumference of a big circle like this is π times the diameter (twice the radius). For your cone,

C = π(2)(√40) which is about 39.74 cm.

You said the radius of the opening should be 2cm. That means the circumference of the opening is π(2)(2) which is about 12.57 cm.

Okay, we calculated these two circumferences because we need to cut away a pie slice from the circle until only 12.57cm of its circumference remains. Since it has 39.74cm initially, we are cutting away more than half the circle. In fact, the ratio tells us what percentage we want to have left:

12.57 / 39.74 is about 31.6 %.

Still, this is hard to work with for a pattern for the cone. Easiest is to use a protractor and measure the angle. A full circle is 360 degrees, so we want 31.6% of 360 degrees:

31.6% x 360° is about 114 degrees.

And that finishes the problem: You want to make a circle with radius 6.32 cm and remove all but 114 degrees of it as a pie slice. Curl that up and you have a cone with height of 6 cm and an opening whose radius is 2cm.

Here's the "general" solution for any similar cone. Let's say you know the height H and the opening you want has a radius of R.

Let P = the radius of the circle. Then P = √(R² + H²).
Let D = the number of degrees of the circle to keep (as a pie slice). Then D = 360(R/P).

Stephen La Rocque.

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