# PSAT Math : Diameter and Chords

## Example Questions

### Example Question #1 : Diameter And Chords

The circle above has a radius of , and the measure of  is . What is the length of chord ?

Explanation:

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  side of the model triangle.

Therefore,

Because x is equal to half of the chord, the answer is .

### Example Question #1 : How To Find The Ratio Of Diameter And Circumference

Let  represent the area of a circle and  represent its circumference. Which of the following equations expresses  in terms of

Explanation:

The formula for the area of a circle is , and the formula for circumference is . If we solve for C in terms of r, we get
.

We can then substitute this value of r into the formula for the area:

### Example Question #1 : Diameter And Chords

If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?

8

32

16

4

2

16

Explanation:

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 = 16.

### Example Question #1 : How To Find The Length Of The Diameter

The perimeter of a circle is 36 π.  What is the diameter of the circle?

3

72

36

18

6

36

Explanation:

The perimeter of a circle = 2 πr = πd

Therefore d = 36

### Example Question #3 : How To Find The Length Of The Diameter

If the area of the circle touching the square in the picture above is , what is the closest value to the area of the square?

Explanation:

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 , , and . The radii of the circle (from the center to the corners of the square) will be 9. The hypotenuse (side of the square) must be .

The area of the square is then .

### Example Question #1 : 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?

Explanation:

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  is , and . What is the diameter of the circle?

Explanation:

Call the diameter . Since  is  of the circle, and  is  of a circle with circumference .

is  in length, so

### Example Question #3 : Diameter And Chords

Note: Figure NOT drawn to scale.

In the above circle, the length of arc  is 10, and . Give the diameter of the circle. (Nearest tenth).