PSAT Math : Diameter and Chords
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Example Question #1 : How To Find The Length Of A Chord
The circle above has a radius of 
To solve a chord problem, draw right triangles using the chord, the radii, and a line connecting the center of the circle to the chord at a right angle.
Now, the chord is split into two equal pieces, and angle AOB is bisected. Instead of one 120 degree angle, you now have two 30-60-90 triangles. 30-60-90 triangles are characterized by having sides in the following ratio:
So, to find the length of the chord, first find the length of each half. Because the triangles in your circle are similar to the 30-60-90 triangle above, you can set up a proportion. The hypotenuse of our triangle is 6 (the radius of the circle) so it is set over 2 (the hypotenuse of our model 30-60-90 triangle). Half of the chord of the circle is the leg of the triangle that is across from the 60 degree angle (120/2), so it corresponds to the 
Therefore,
Because x is equal to half of the chord, the answer is 
Example Question #1 : Diameter And Chords
Let 
The formula for the area of a circle is 
We can then substitute this value of r into the formula for the area:
Example Question #1 : Diameter
If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?
2
4
32
16
8
16
Set the area of the circle equal to four times the circumference πr2 = 4(2πr).
Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.
Example Question #2 : Diameter
The perimeter of a circle is 36 π. What is the diameter of the circle?
18
72
6
3
36
36
The perimeter of a circle = 2 πr = πd
Therefore d = 36
Example Question #31 : Circles
If the area of the circle touching the square in the picture above is 
Obtain the radius of the circle from the area.
Split the square up into 4 triangles by connecting opposite corners. These triangles will have a right angle at the center of the square, formed by two radii of the circle, and two 45-degree angles at the square's corners. Because you have a 45-45-90 triangle, you can calculate the sides of the triangles to be 
The area of the square is then 
Example Question #5 : How To Find The Length Of The Diameter
Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
Example Question #2 : Diameter And Chords
Note: Figure NOT drawn to scale.
In the above circle, the length of arc 
Call the diameter 
Example Question #1 : How To Find The Length Of The Diameter
Note: Figure NOT drawn to scale.
In the above circle, the length of arc 
Insufficient information exists to answer the question.
Call the diameter 
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