No, this isn’t a homework question. I’m about twenty years past homework.
For my own information, as reference to a story I’m working on, I’m trying to figure out the dimensions in feet–that is, area, diameter, and circumference–of a field that is one **mile ** square in area. The field is perfectly circular.
Here’s where I am:
One mile = 5280 feet, so one square mile = 27,878,400 square feet.
The equation for the area of a circle is
A=Pi * the square of the radius
The equation for the circumference of a circle is
C=Pi * the radius * 2
Pi is approximately 3.1416
so…
27,878,400 square feet = 3.1416 * r-squared
Dividing both sides by Pi gets me
8,873,950= R squared
Taking the square root of I get
Radius = 2979 feet
So the diameter of this circle would be 5958 feet.
Looks good to me. Why do they seem wrong to you? You’d expect a circular field to be slightly longer end to end than a square-shaped one, wouldn’t you? Because it’s missing the corners…
Simple recheck: consider if the field were square - the perimiter should go up slightly, because the ratio of perimiter to area is a bit higher for a square than a circle, but not by much.
Perimiter of a square mile: 5280 * 4 feet, or 21,120 feet.
So yeah, your result of 18718 feet for the perimiter (circumference) of the circle looks right on that basis too.
It doesn’t look wrong so much as it FEELS wrong. Because the diameter of this one-mile square circle is obviously more than a mile; it’s close to 6000 feet. I guess I’m just looking for reassurance in my old age.
A = [symbol]p[/symbol]r[sup]2[/sup], so r = sqrt(A/[symbol]p[/symbol]). C = 2[symbol]p[/symbol]r, so C = 2[symbol]p[/symbol]sqrt(A/[symbol]p[/symbol]), or C = 2sqrt([symbol]p[/symbol]A).
If A = 5280[sup]2[/sup], C = 10560*sqrt([symbol]p[/symbol]), which is 18717 and change. That’s close enough to what you got.
Well, consider this… the square is an object with a variable diameter, compared to the constant-diameter circle. At it’s minimum, the diameter of the square is 5280 feet, but at it’s maximum, corner to corner, the square’s “diameter” is 5280 times the square root of two, 7467 feet (plus a bit more than half an inch.)
The diameter of the circle should be somewhere in between these two values. 5958 feet looks quite right to me.
A nitpick: the circumference to the nearest foot is 18717 ft, not 18818 ft. You’ve probably used the rounded value of the radius, instead of the full-precision value. It’s best to use un-rounded values throughout the calculation, and then do all of the rounding at the end.
I’m an Algebra teacher. All your math looks good. A gold star!
To look at the “wrongness” and make it feel right: Image a square that is 1 mile on each side. From its center, draw a line to the middle of one of its sides. That line is 2640 ft. Square this, and you’ll get 6,969,600 ft[sup]2[/sup], which is the area of one quadrant of the square. To get the area of the whole square based on the quadrant’s area, you’d multiply by 4, getting 27,878,400 ft[sup]2[/sup]. This corresponds to A[sub]S[/suB]=4*r[sup]2[/sup].
Now imagine a circle inscribed within the original square. It’s center is the same place as the square’s. Its radius is colinear with the quadrant side we drew from the square’s center to the middle of a side. But the area of the circle (based on the radius) is A[sub]C[/suB]=π*r[sup]2[/sup]=21,895,644.16ft[sup]2[/sup]. This area is about 78.5% (π/4) of the area of the circumscribed square, which makes sense since the circle’s area is less because the corners were cut.
Therefore, to make a circle of the same area as a square, its diameter must be √(1/78.5%) the size of the square’s side. √(1/78.5%)=112.8%. 5280*112.8% = 5957.84 ft.
So Pleonast corrects an error I didn’t make, and in correcting him I make a different error.
What is that–Gaudere squared? Thank God you got it right, or it would have ripped open a hole in space time and[Jenn Broznek](www.galactanet.com/ comic/Strip568.png) would have fallen naked into my lap.
Wait…
(Warning: image may not be work-safe, if you work for, like, Focus on the Family or something.)
That’s because his link had a hole in space in it (I don’t know about time) :D. Edit out the space, and you get this (the lady in question is a cartoon, and she’s in a bathtub with nothing naughtier than knees visible).
And I think that what happened was iterative Gaudere. And we all know that iterative phenomena often lead to chaos.
