Math CentralQuandaries & Queries


Question from Beth:

My question is related to a dome I would like to construct, for which I know the circumference of the base: 120ft. I now need to figure out the diameters of two smaller circles, one at 20ft along the arc of the dome form the ground, and the other at 30ft along the arc. Assuming a true hemisphere, or 180 degrees total arc, how can I calculate these two circumferences?

Thanks for the help,

Hi Beth.

The circumference of the base is 120ft, so the radius of the dome is 120/(2π) = 60/π. That's about 19.1 ft.

Thus, from the center of the floor under the dome, it is this distance to any point on the dome, including 20 ft along the arc or 30 ft along the arc. Let's look at 20 ft along.

The arc is 20 ft, which is 20/120 = 1/6 of a full revolution. That means 1/6 of 360 degrees, making 60 degrees. Thus, if you sat on the floor at the center and angled your eye up at 60 degrees, you'd be looking at a point 20 ft along the arc from the floor.

Now consider a triangle composed of the line of sight from the center to some point 20 ft up (along the arc), a vertical line dropped from that point to the floor and a line from there back to the center. It's a right angled triangle, with a 60 degree angle and a hypotenuse of 60/π feet. The circle around the dome 20 ft up would be identical in size to a circle drawn on the ground at that lateral distance from the floor's center. So really we just need that radius, which is the "adjacent" side of the triangle I described.

Using the cosine ratio, we can calculate it: cos 60 = r / ( 60/π ). r = ( 60/π ) cos 60 = 9.55 ft.

The circumference of a circle is just 2πr, so the circumference of the circle which is a ring 20 ft along the arc above the ground on this dome would be 2π(9.55) = 60 ft. Yes, the numbers just work out to that coincidental figure!

Stephen La Rocque

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