How to find the length of the diameter - PSAT Math

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

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Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).

Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.

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

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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$

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

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The perimeter of a circle = 2 πr = πd

Therefore d = 36

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?

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Obtain the radius of the circle from the area.

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

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 .

Note: Figure NOT drawn to scale.

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

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Call the diameter . Since , $\widehat{MN}$ 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 }$$

Note: Figure NOT drawn to scale.

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

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Call the diameter . Since , $\widehat{MN}$ 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 in the circumference formula:

$\pi d = C$

$\pi d = 50$

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

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

Tap to see back →

Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).

Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.

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

Tap to see back →

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$

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

Tap to see back →

The perimeter of a circle = 2 πr = πd

Therefore d = 36

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?

Tap to see back →

Obtain the radius of the circle from the area.

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

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 .

Note: Figure NOT drawn to scale.

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

Tap to see back →

Call the diameter . Since , $\widehat{MN}$ 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 }$$

Note: Figure NOT drawn to scale.

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

Tap to see back →

Call the diameter . Since , $\widehat{MN}$ 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 in the circumference formula:

$\pi d = C$

$\pi d = 50$

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

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

Tap to see back →

Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).

Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.

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

Tap to see back →

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$

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

Tap to see back →

The perimeter of a circle = 2 πr = πd

Therefore d = 36

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?

Tap to see back →

Obtain the radius of the circle from the area.

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

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 .

Note: Figure NOT drawn to scale.

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

Tap to see back →

Call the diameter . Since , $\widehat{MN}$ 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 }$$

Note: Figure NOT drawn to scale.

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

Tap to see back →

Call the diameter . Since , $\widehat{MN}$ 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 in the circumference formula:

$\pi d = C$

$\pi d = 50$

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

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

Tap to see back →

Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).

Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.

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

Tap to see back →

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$