My strategy to work out the area of a tilted square is to draw a square border around the tilted one and it should be one with straight vertical lines. The square border is also supposed to be a straight 4-sided closed shape and as small as it can be while still acting as a border for the tilted one.Then work out the 4 triangles areas' ( (base times height) divided by two ) and minus it from the border square's total area.
My point is not about this but what my topic really it is about the difference when actually switching from one tilted square to 2 and then to 3 and eventually to 4. As you can see from the screenshot attached, when drawing the square border, you can note that there are 4 triangles in the corner, and they are all RIGHT ANGLE TRIANGLES for a fact. With this in mind, lets start of by tilting the perfect square which has an area of 1 cm squared. Now tilted, we draw a border square around it and we now have the tilted square and our 4 right angle triangles. Now I will work out the one of the right-angled triangles - ( (base times height) divided by two ) = ( (one times one) divided by two = 0.5)
This is my base point which I will pull my solution out of. If you have the triangle's area the next square which is tilted and following the next square number (1,4);so from our previous tilted square example, the right-angled triangle had an area of 0.5 cm squared so we add the triangle's area from the tilted square for 1 cm by 1 cm ( 0.5 cm) on which means 1 of the right-angled triangle's areas' will be 1 cm squared (0.5 + 0.5). Just to make sure you understand, let's go onto the next tilted square which follows the next square number (1,4,9) since our last right-angled triangle's area was 1 we then add the triangle's area from the tilted square for 1 cm by 1 cm (0.5 cm ) on which means 1 of the right-angled triangle's areas' will be 1.5 cm squared (1 + 0.5).
0.5 is really and truly the star number here as when moving onto 2 tilted squares, we add 0.5. Meaning now the area from the tilted square for 1 cm by 1 cm is 1 cm squared (0.5 + 0.5.) To work out the area of the next square which is tilted and following the next square number (1,4) we add the new area for the tilted square that's 1 cm by 1 cm is 1 cm squared (1 cm) so one of the right-angled triangle's area is 2 cm squared.
This is my solution for how to work out tilted squares area's quickly in an exam.
Thanks.