A square is inscribed inside a circle with radius r cm.
a) Express the edge length of the square in terms of r and so find the area of the square in terms of r.
Using side, 's' would make it much easier, but the question asks for me to find the area of the inscribed square using radius 'r,' which i'm guessing, is worked out from the circle surrounding the square.
Not sure how I would work this one out, help appreciated, thanks :)
the diagonal of the square is equal to the diameter of the circle, 2r
So, the side of the square is (2r)/√2 = r√2
That makes the area (r√2)^2 = 2r^2
Well, isn't this a "square" deal! Let's try to solve it, shall we?
First, let's consider that the diagonal of the square is the diameter of the circle. Since the radius of the circle is r, the diameter would be 2r.
Now, let's use the Pythagorean Theorem to find the length of the square's edge. The diagonal of the square can be split into two equal parts, each forming a right triangle with the edge of the square as the hypotenuse.
We have a right angle, a hypotenuse (2r), and we need to find the length of the other side (the edge of the square).
So, using the Pythagorean Theorem (a^2 + b^2 = c^2), where c is the hypotenuse:
(Edge of square)^2 + (Edge of square)^2 = (2r)^2
2(Edge of square)^2 = 4r^2
(Edge of square)^2 = 2r^2
And there you have it! The edge length of the square would be square root of 2r^2, or √2r.
To find the area of the square, all you have to do is multiply the length of one side by itself (squared).
Area of the square = (Edge of square)^2 = (√2r)^2
= 2r^2
So, the area of the inscribed square in terms of r is 2r^2.
Hope that puts a "square" smile on your face!
To find the edge length of the square, we can use the fact that the diagonal of the square is equal to the diameter of the circle.
The diagonal of the square forms the hypotenuse of a right-angled triangle with two sides equal to the radius of the circle, r.
Using the Pythagorean theorem, we have:
(diagonal length of square)^2 = (radius)^2 + (radius)^2
Let the edge length of the square be s. Since the diagonal of a square forms an isosceles right triangle with the sides, we have:
s^2 = r^2 + r^2
Simplifying:
s^2 = 2r^2
Taking the square root of both sides, we get:
s = sqrt(2r^2) = sqrt(2) * r
Therefore, the edge length of the square is equal to sqrt(2) times the radius, s = sqrt(2) * r.
To find the area of the square, we square the edge length:
Area of square = s^2 = (sqrt(2) * r)^2 = 2r^2.
So, the area of the square in terms of r is 2r^2.
To find the edge length of the inscribed square in terms of the radius of the circle (r), you can use the fact that the diagonal of the square is equal to the diameter of the circle and the diagonal of the square also forms the hypotenuse of two right-angled triangles, each with sides equal to the radius of the circle (r).
Let's assume that the edge length of the square is represented by 's'. Since the diagonal of the square is equal to the diameter of the circle, we can find the diagonal of the square using the Pythagorean theorem:
diagonal^2 = s^2 + s^2 (since the square has equal-length sides)
diagonal^2 = 2s^2
The diameter of the circle is equal to twice the radius, so the equation becomes:
(2r)^2 = 2s^2
4r^2 = 2s^2
2r^2 = s^2
Now, to find the area of the square, we square the edge length 's', which is equal to √(2r^2):
Area of the square = s^2 = (√(2r^2))^2 = 2r^2
So, the area of the square in terms of the radius 'r' is 2r^2 square units.