1. 2010 Primary Math Masterclass Project By: Liang Jia Wei Tyrone Yeo Lim Jia Yong

2. Our Question is… Is there a relationship the largest possible triangle that can be drawn in the circle and the circle itself? If there is, what is the relationship?

3. Stage 1 Prove that the right isosceles triangle in the circle's area is larger than any other isosceles triangle's area. The base must be the same as the diameter of the circle. Also, the triangle is most like a square.

4. Proof Since the longest line that can be drawn in the circle is the diameter, so it should be the base of the triangle. The height is the tallest when the triangle is an isosceles and thus, the area will be the largest.

5. Stage 2 Prove that when an isosceles triangle has ONE same angle with another triangle in the circle, (the other triangle cannot be an equilateral triangle as the isosceles triangle cannot have an angle that is 60 degrees) the isosceles triangle is always the larger triangle.

6. Proof When the isosceles triangle and a scalene triangle share a same angle, the isosceles triangle’s area is larger as one or more of the lines will be longer than the lines in the scalene triangle.

7. Proof for the main question When you lengthen the equilateral triangle’s base or height, the triangle will lose area because the three sides of the triangle is not to the maximum – one factor (height or base) of the triangle will be less than when it is an equilateral triangle because it is less like a square, which has the largest area of all the quadrilaterals given that the perimeter stays the same, but when it is exactly half a square, the triangle is not the biggest. The reason is that, if you draw a square around the triangle, you will find a little tip of the triangle sticking out, it is this, that causes the difference. So, equilateral triangles are the largest possible triangles that you can draw in a circle.

8. Area of the equilateral triangle in the circle The height of the triangle is 2/3 of the diameter, so, using trigonometry you can find the side of the triangle by dividing the height by 866/1000. Once you found out one side of the triangle, all the rest of the sides are equal to the base, and thus, by multiplying the base to the height then divide it by 2, the area of the triangle can be found out.

9. Thank you for your kind attention!