The difference of two regions

A square of side 20 centimetres. Here it is. And a circle of radius 10, with its centre at the corner B. Region X is the part of the square outside the circle. Region Y is the part of the circle outside the square. X is blue and Y is pink. The piece in the middle, in both shapes at once, is left plain. Find the difference between X and Y. Take pi as three point one four. Before any number, give the middle a name. Call it S. S is in the square, and S is in the circle. The square is X and S together, so the area of X is the area of the square with the area of S taken out. And the area of Y is the area of the circle with the same area of S taken out. Two lines, ending on the same area of S. Take one from the other and the two go together. So the area of X minus the area of Y is the area of the square minus the area of the circle. The square first. The area of the square is 20 times 20. Four hundred square centimetres. Then the circle. The area of the circle is three point one four, times 10, times 10 gives three hundred and fourteen square centimetres. So the area of X minus the area of Y is four hundred minus three hundred and fourteen and it gives 86 square centimetres. And neither X nor Y was ever measured on its own.

Your turn. A square of side 18 centimetres. A quarter circle of radius 18 is drawn from the corner A. And the diagonal AC is drawn across it. X is the quarter circle outside triangle ABC, and Y is the triangle outside the quarter circle. Find the difference between X and Y. Take pi as three point one four. Try it first. Pause if you like. Look at where the diagonal crosses the arc. There is no length there we can work out. So do not go looking for one. Neither X nor Y can be measured on its own. Same move. Name the plain middle S. The area of X is the area of the quarter circle with the area of S taken out. And the area of Y is the area of the triangle with the same area of S taken out. Take one from the other, and the two go. So the area of X minus the area of Y is the area of the quarter circle minus the area of the triangle. Now the 2 whole shapes. The area of the quarter circle is one quarter, times three point one four, times 18, times 18. Cross out the 4 and the first 18. Half of 18 is 9. Cross out the other 18 as well. Half of that is 9 too, and the 4 is used up. That leaves three point one four, times 9, times 9 equals two hundred and fifty four point three four square centimetres. The diagonal cuts the square in half, so the area of the triangle is half, times 18, times 18. Cross out the 2 and the first 18. Half of 18 is 9. That leaves 9 times 18. One hundred and sixty two square centimetres. Now the line we wrote before any of that. The area of X minus the area of Y is two hundred and fifty four point three four, minus one hundred and sixty two, gives ninety two point three four square centimetres. Two regions, and the same piece inside both of them. That piece. It belongs to X's shape and to Y's shape. So it cannot change the difference, and it never has to be worked out. Write the two regions before you write a number, and the difference is the difference of the two whole shapes.