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PSAT Math : Diameter and Chords

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Example Questions

Example Question #1 : How To Find The Length Of A Chord

Circle_120_degrees

The circle above has a radius of , and the measure of $\angle AOB$ is $120 ^\circ$ . What is the length of chord ?

Possible Answers:

$6\sqrt{2}$

$3\sqrt{3}$

$6\sqrt{3}$

Correct answer:

$6\sqrt{3}$

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.

Circle_120_degrees_chord

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: Triangle306090-3

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 $\sqrt{3}$ side of the model triangle.

$\frac{x}{6}=\frac{\sqrt{3}}{2}$

Therefore, $x=3\sqrt{3}$

Because x is equal to half of the chord, the answer is $6\sqrt{3}$ .

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Example Question #1 : Diameter And Chords

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

Possible Answers:

C^2

Correct answer:

Explanation:

The formula for the area of a circle is $A=\pi r^2$ , and the formula for circumference is $C=2\pi r$ . If we solve for C in terms of r, we get
$r=C/2\pi$ .

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

$A=\pi r^2$

$=\pi (C/2\pi )^2$

$=C^2\pi /4\pi ^2$

$=C^2/4\pi$

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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?

Possible Answers:

Correct answer:

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 d = 16.

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Example Question #2 : Diameter

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

Possible Answers:

Correct answer:

Explanation:

The perimeter of a circle = 2 πr = πd

Therefore d = 36

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Example Question #31 : Circles

Sat_math_picture

If the area of the circle touching the square in the picture above is $81\pi$ , what is the closest value to the area of the square?

Possible Answers:

211

162

100

144

Correct answer:

162

Explanation:

Obtain the radius of the circle from the area.

$A=\pi r^2=81\pi$

r^2=81

r=9

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 $x\sqrt{2}$ . 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 $9\sqrt{2}$ .

The area of the square is then $(9\sqrt{2})^2=162$ .

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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?

Possible Answers:

$6.25\pi$

$12.5\pi$

$25\pi$

$5\pi$

$6\pi$

Correct answer:

$6.25\pi$

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.

$A=\pi (5/2)^2=6.25\pi$

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Example Question #2 : Diameter And Chords

Sector

Note: Figure NOT drawn to scale.

In the above circle, the length of arc $\widehat{MYN}$ is $42 \pi$ , and x = 60 . What is the diameter of the circle?

Possible Answers:

$50 \frac{2}{5 }$

Correct answer:

$50 \frac{2}{5 }$

Explanation:

Call the diameter . Since x = 60 , $\widehat{MN}$ is $\frac{60}{360} = \frac{1}{6}$ of the circle, and $\widehat{MYN}$ is $\frac{5}{6 }$ of a circle with circumference $\pi d$ .

$\widehat{MYN}$ is $42 \pi$ in length, so

$\frac{5}{6} \pi d = 42 \pi$

$d = 42 \pi \div \frac{5}{6} \pi =\frac{ 42 \pi }{1}\cdot \frac{6}{5 \pi} = \frac{252}{5 } = 50 \frac{2}{5 }$

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

Sector

Note: Figure NOT drawn to scale.

In the above circle, the length of arc $\widehat{M N}$ is 10, and x = 72 . Give the diameter of the circle. (Nearest tenth).

Possible Answers:

19.1

15.9

8.0

9.5

Insufficient information exists to answer the question.

Correct answer:

15.9

Explanation:

Call the diameter . Since x = 72 , $\widehat{MN}$ is $\frac{72}{360} = \frac{1}{5}$ of a circle with circumference $\pi d$ . Since it is of length 10, the circumference of the circle is 5 times this, or 50. Therefore, set C = 50 in the circumference formula:

$\pi d = C$

$\pi d = 50$

$d = \frac{50 }{\pi } \approx \frac{50 }{3.14159} \approx 15.9$

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PSAT Math : Diameter and Chords

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Find The Length Of A Chord

Example Question #1 : Diameter And Chords

Example Question #1 : Diameter

Example Question #2 : Diameter

Example Question #31 : Circles

Example Question #5 : How To Find The Length Of The Diameter

Example Question #2 : Diameter And Chords

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

All PSAT Math Resources

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