Circle of a radius?

Question

What is the length of the side of a square inscribed in a circle of radius 1?
Also, what is the lenth of the side of a cube inscribed in a sphere of radius 1?

Answer 1

1) if circle has radius 1 then diameter is 2.

therefore the diagonal of the square is 2, therefore the square has side lengths of sqrt(2).

here is the proof:

sqrt(2)^2 + sqrt(2)^2 = 2 ^ 2
2 + 2 = 4
4 = 4

therefore the length of the square is approx. 1.4142....

2)

if a sphere has radius 1 then we can conclude that if we draw lines inside the sphere taking the shape of a cube, then draw additional lines inside the cube, then figuring out the dimensions like so. The sphere has diameter 2 from one end to another through the center of the sphere.
The 3d hypotenuse line going from one corner of the cube to the other corner is the "diameter",,,which is 2.

at the bottom surface of the cube

x^2 (bottom surface hypotenuse) + y^2 (actual cube side length) = 2^2 (3d hypotenuse from one corner to the opposite corner)

2(y ^ 2) (actual cube side lengths) = x^2 (bottom surface hypotenuse)

x = y * root(2)

now, lets go back again and substitute.....

[y*root(2)]^2 (bottom surface hypotenuse) + y^2 (actual cube side length) = 2^2 (3d hypotenuse from one corner to the opposite corner)

3y^2 = 4
y^2 = 4/3
y = sqrt(4/3)

y = 2/sqrt(3)

that is the final answer,,,i hope you fully understood the visualizations, i tried my best to picture it out for you as best as possible...

Answer 2

Cool. Well lets try this...


If r = 1 then from the center to a corner of the square is 1. Which means the hypotenuse is 2r = 2. Since we have a square 2(s^2) = (2)^2

(this is a special case of the whole a2 + b2 = c2 deal (when a = b))

s^2 = 2 Therefore the side (s) of a square in the circle with r = 1 is:

s = SQUAREROOT (2)

================================

Same thing here just in 3 dimensions. The r = a and r = b where c = s of the cube

s = SQUAREROOT (2)

Answer 3

The diagonal of the square inscribed in the circle is 1, so the side of the square is
2 / sqrt(2) = 1.41421.

The diagonal of the cube inscribed in the sphere is found using the extended Pythagorean theorem:

a^2 + b^2 + c^2 = d^2
where d is the diagonal, and a, b, and c are the three sides

Since a = b = c, the length of the cube's side is twice a, or
2 / sqrt(3) = 1.1547

Answer 4

If the side of the square is of length a, and the radius of the circle is r, then the length of the diagonal of the square is:
sqrt(a^2 + a^2) = a sqrt(2)
a sqrt(2) = 2r
a = 2r / sqrt(2) = r sqrt(2)
With r = 1 this gives:
a = sqrt(2).

For the cube, the length of the diagonal is:
sqrt(a^2 + a^2 + a^2) = a sqrt(3).
a sqrt(3) = 2r
With r = 1, this gives:
a = 2 / sqrt(3) = 2 sqrt(3) / 3.

Answer 5

An since the side of the inscribed square is also the diagonal of the face of the sphere inscribed cube.....
(I won't do ALL of your homework for you.)

Answer 6

circle with radius r would have an area of r^2 .pi (pi we can take as 22/7) to give an area of 22/7

the square enclosed would have a diameter across the diagonal the same as the diameter (IE 2r) and the side would be r sqrt2 (using Pythagoras) so a side of 1.41 (2dp)

the length of a side of a cube inside a sphere would be the same as the lengths would be unchanged apart from it being in 3d rather than 2d as before

the area of your new cube would be (sqrt2)^3

Answer 7

The first one is 1.414