PSAT Math - 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?

$6.25\pi$

$6\pi$

$5\pi$

$25\pi$

$12.5\pi$

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.

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

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

Explanation

The perimeter of a circle = 2 πr = πd

Therefore d = 36

Sector

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?

$50 \frac{2}{5 }$

Explanation

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

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

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

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?

Explanation

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 $(9\sqrt{2})^2=162$ .

Sector

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).

Insufficient information exists to answer the question.

Explanation

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: