# cutting circle into triangles

i am confused in one problem and please help me,i will give picture from problem below and question says : $AB$ is a diameter of the circle. All triangles above the diameter in the diagram are equal in area. All triangles below the diameter are equal in area.compare total area of triangles above $AB$ and total are below $AB$

i have chosen that this can't be determined by given information,because in spite of we have fact that below $AB$ we have more triangles then above $AB$,we dont know length of bases of each triangles,all sides except bases are radius so equal to each other,but in question answer is different,and explanation is following:

The total area of the upper triangles is less than the area of the lower triangles. The more triangles that you cut the semicircle into, the more of the circle that is occupied. is this right?it is test taken from GRE test,i am preparing for passing it

• Is the joining point the centre of the circle? – Mark Bennet Oct 21 '12 at 11:54
• yes it is,it is center of circle – dato datuashvili Oct 21 '12 at 12:08

## 3 Answers

GRE is correct; they in fact check if you know a famous proof of formula for circle area. The proof takes area of circle as limit of total area of triangles, which become smaller and smaller and closer to the circle, making a circle as a limit. Full proof is for example here, visualized: http://faraday.physics.utoronto.ca/GeneralInterest/Harrison/Flash/AreaOfCircle/AreaOfCircle.html

So just by taking more triangles, they will necessarily be closer to circumference and hence their total area is larger. GRE is about intuitiveness, so this reasoning is mainly intuitive but based on some knowledge you need to have.

HINT You need to use the fact that although the triangles in each half of the diagram look to be different sizes, triangles in the same half all have the same area. Express the area of such a triangle in terms of the angle at the centre (knowing that two sides in each triangle have length equal to the radius). You should be able to prove that you have two sets of congruent triangles.

Then you could use the fact that the pentagon is constructible to compute the sine of the central angles in the bottom half. The sine of the angles in the top half should be known to you. And that should be enough information to give the ratio explicitly (if this is what is requited) - else use a calculator or tables to give a numerical value.

Upper 4 triangles area = 4 ( 1/2 * b*h) = 2bh Lower 5 triangles area = 5 ( 1/2 * b * h) = 2.5 (bh)

which is clear that lower triangles area is larger

Why we need complex theory for simple though a tricky question ... Unless they specify the all sides of triangles.