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PSAT Math : How to find the length of the diameter

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Example Questions

Example Question #132 : Plane Geometry

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

Sat_math_picture

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

Possible Answers:

$\displaystyle 81$

162 $\displaystyle 162$

211 $\displaystyle 211$

100 $\displaystyle 100$

144 $\displaystyle 144$

Correct answer:

162 $\displaystyle 162$

Explanation:

Obtain the radius of the circle from the area.

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

r^2=81 $\displaystyle r^2=81$

r=9 $\displaystyle r=9$

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 $\displaystyle x$ , $\displaystyle x$ , and $x\sqrt{2}$ $\displaystyle x\sqrt{2}$ . 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 $9\sqrt{2}$ $\displaystyle 9\sqrt{2}$ .

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

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Example Question #522 : Plane Geometry

Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?

Possible Answers:

$25\pi$ $\displaystyle 25\pi$

$12.5\pi$ $\displaystyle 12.5\pi$

$6\pi$ $\displaystyle 6\pi$

$5\pi$ $\displaystyle 5\pi$

$6.25\pi$ $\displaystyle 6.25\pi$

Correct answer:

$6.25\pi$ $\displaystyle 6.25\pi$

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.

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

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Example Question #3 : Diameter And Chords

Sector

Note: Figure NOT drawn to scale.

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

Possible Answers:

$50 \frac{2}{5 }$ $\displaystyle 50 \frac{2}{5 }$

$\displaystyle 35$

$\displaystyle 42$

$\displaystyle 36$

$\displaystyle 49$

Correct answer:

$50 \frac{2}{5 }$ $\displaystyle 50 \frac{2}{5 }$

Explanation:

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

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

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

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

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

Possible Answers:

19.1 $\displaystyle 19.1$

9.5 $\displaystyle 9.5$

Insufficient information exists to answer the question.

15.9 $\displaystyle 15.9$

8.0 $\displaystyle 8.0$

Correct answer:

15.9 $\displaystyle 15.9$

Explanation:

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

$\pi d = C$ $\displaystyle \pi d = C$

$\pi d = 50$ $\displaystyle \pi d = 50$

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

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PSAT Math : How to find the length of the diameter

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #132 : Plane Geometry

Example Question #2 : How To Find The Length Of The Diameter

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

Example Question #522 : Plane Geometry

Example Question #3 : Diameter And Chords

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

All PSAT Math Resources

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