2010 Primary Math Masterclass Project By: Jia Wei, Tyrone and Jia Yong

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?

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.

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.

Stage 2 Prove that when an isosceles triangle has ONE same angle with another triangle in the circle, (the angle cannot any of the angles that are similar to one another because the remaining two angles might be the same) the isosceles triangle is always the larger triangle

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.

Proof for the main question

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.

Thank you!