ACT Math - How to find the length of the diameter

In a group of students, it was decided that a pizza would be divided according to its crust size. Every student wanted 3 inches of crust (measured from the outermost point of the pizza). If the pizza in question had a diameter of 14 inches, what percentage of the pizza was wasted by this manner of cutting the pizza? Round to the nearest hundredth.

Explanation

What we are looking at is a way of dividing the pizza according to arc lengths of the crust. Thus, we need to know the total circumference first. Since the diameter is , we know that the circumference is $14\pi$ . Now, we want to ask how many ways we can divide up the pizza into pieces of inch crust. This is:

$\frac{14\pi}{3}$ or approximately pieces.

What you need to do is take this amount and subtract off . This is the amount of crust that is wasted. You can then merely divide it by the original amount of divisions:

$\frac{0.66076571675237...}{14.66076571675237....}=0.04507034144863....$

(You do not need to work in exact area or length. These relative values work fine.)

This is about of the pizza that is wasted.

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

Find the length of the diameter of a circle given the area is $4\pi$ .

Explanation

To solve, simply use the formula for the area of a circle, solve for r, and multiply by 2 to get the diameter. Thus,

$A=\pi{r^2}\Rightarrow r=\sqrt{\frac{A}{\pi}}$

$r=\sqrt{\frac{4\pi}{\pi}}=\sqrt{4}=2$

A circle has an area of $\textup{in}^{2}$ . What is its diameter?

$\textup{in}$

Explanation

To solve a question like this, first remember that the area of a circle is defined as:

$A=\pi r^2$

For your data, this is:

$45 = \pi r^2$

To solve for , first divide both sides by $\pi$ . Then take the square root of both sides. Thus you get:

The diameter of the circle is just double that:

Rounding to the nearest hundredth, you get .

Find the diameter given the radius is .

$\frac{3}{2}$

Explanation

Diameter is simply twice the radius. Therefore, .

If a circle has an area of $49\pi$ , what is the diameter of the circle?

Explanation

1. Use the area to find the radius:

$Area=\pi r^{2}$

$49\pi=\pi r^{2}$

$49=r^{2}$

2. Use the radius to find the diameter:

$d=2r=2\cdot 7=14$

A circle has an area of $39\pi$ . What is the diameter of the circle?

$2\sqrt{39}$

$2\sqrt{39\pi }$

$2\sqrt{39}\pi$

$\sqrt{39\pi }$

$\sqrt{39}$

Explanation

The equation for the area of a circle is $\pi r^2$ , which in this case equals $39\pi$ . Therefore, The only thing squared that equals an integer (which is not a perfect root) is that number under a square root. Therefore, $r=\sqrt{39}$ . Since diameter is twice the radius, $d=2\sqrt{39}.$

What is the diameter of a semi-circle that has an area of $25\pi$ ?

$10\sqrt{2}$

$5\sqrt{2}$

$25\sqrt{3}$

Explanation

To begin, be very careful to note that the question asks about a semi-circle—not a complete circle! This means that a complete circle composed out of two of these semi-circles would have an area of $50\pi$ . Now, from this, we can use our area formula, which is: