Disappearing square
Problem
Do you know how to find the area of a triangle?
You could count the squares...
What happens if we turn the triangle on end? Play the video to see.
Try counting the number of units in the triangle after it has been turned.
Do you have any interesting findings to report?
The area of a triangle can be found by multiplying half the base by the height. What is the area of this whole triangle?
Now find the areas of the shapes marked A, B and C.
If you add these answers together, would they equal the same as the whole triangle? Try and see.
Do you have any interesting findings to report?
Getting Started
The area of a triangle is found by multiplying the base by the height then halving your answer.
(Be careful not to use the length of a side but the height from base to apex.)
Could this be to do with the way the squares are drawn? Look out for the parts of squares and how they are matched up.
Student Solutions
Several teachers didn't like this trick but is it really a trick? When the rotation takes place the grid lines up along the slopping sides but does it exactly fit? When you count the squares the second time what are you counting? The counting may not be accurate. Are the 'half' squares all exactly half?
Many of you were puzzled by this. What is really going on here? We can't 'make' an extra square of area just by chopping a shape up and putting the pieces together differently so what is happening? William from Tattingstone describes it like this:
I think he is trying to tell us that the rectangle C doesn't quite fit in there. The line on the sloping side of the triangle isn't really a straight line is it? Measuring is never absolutely accurate is it?