A circle is touching a square. The area of the square is 64 cm^2. Work out the area of the circle.

The first step of looking at a question like this is to highlight all the important information in the question. We are told that the square has an area of 64 cm² so what can we infer about the square based on this? 
  
1) We know that area of a square = base x height. Because we have a square, all the sides are the same length, this means that the base and the height will be the same too. Therefore we are multiplying the same number by itself which gives a square number. To figure out the length of each side of the square we therefore need to square root 64 which equals 8.
2) Because the circle is touching all sides of the square we can use the square to figure out the length from the top to the bottom of the circle. This is the diameter of the circle. We know that each side of the square is 8cm therefore the diameter is 8cm.
3) Because the radius is half of the diameter, we know that the radius is 4. We know that area of a circle = π x radius². Therefore area = π x 4². If we put this in our calculator it gives us 50.3 cm² to three significant figures.