A square, with sides of length x cm, is inside a circle. Each vertex of the square is on the circumference of the circle. The area of the circle is 49 cm^2. Work out the value of x. Give your answer correct to 3 significant figures.
The question tells us that the area of the circle is 49cm2, therefore we are able to form the equation πr2=49 (where r = radius of the circle). We can now work out the radius of the circle by rearranging our equation:r2=49/π r= √(49/π) = 3.9493...As each vertex of the square touches the circumference of the circle, we can see that the diameter of the circle is equal to the diagonal length of the square. We can simply calculate the diameter by doubling the radius, this gives us a value of 7.89865....Next, we can use pythagoras's theorem to calculate the value of x, we can do this as the diagonal line (which equals the diameter) cuts the square into two identical right angled triangles. As the diameter is the hypotenuse of these triangles, we can set up the equation:a2+b2=(7.89865)2, and as a and b are both equal to x:2x2=(7.89865)2x2=31.1944x=±5.585.... However, as we know a length cannot be negative, we can state x = 5.59 (question asks for answer correct to 3 sig figs)
