Diameter - ACT Math

Card 0 of 12

Question

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

Answer

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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Question

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

Answer

The perimeter of a circle = 2 πr = πd

Therefore d = 36

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Question

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

Answer

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:

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

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Question

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.

Answer

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.

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Question

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

Answer

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 .

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Question

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

Answer

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:

$A=\pi r^2$

For our data, this is:

$50\pi=\pi r^2$

Solving for , we get:

$r=\sqrt{50}$

This can be simplified to:

$r = 5\sqrt{2}$

The diameter is , which is $2 * 5\sqrt{2}$ or $10\sqrt{2}$ .

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Question

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

Answer

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, $d=2\sqrt{39}.$

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Question

Find the diameter given the radius is .

Answer

Diameter is simply twice the radius. Therefore, .

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Question

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

Answer

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

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

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

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Question

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

Answer

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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Question

A circle has a radius of . What is the ratio of its circumference to its area?

Answer

1. Find the circumference:

$C=\pi d=2\pi r$

$C=2\cdot \pi \cdot 2=4\pi$

2. Find the area:

$A=\pi r^{2}$

$A=\pi \cdot (2^{2})=4\pi$

3. Divide the circumference by the area:

$\frac{C}{A}=\frac{4\pi }{4\pi }=1$

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Question

How far must a racehorse run in a lap race where the course is a circle with an area of $36\pi$ ?

Answer

To find the answer, we need to first find the circumference of the circle then mulitply that by 4 because the racehorse is running 4 laps.

We know that the area of the circle is $36\pi$ . We can then find the radius.

$A=\pi r^2$

$36\pi=\pi r^2$

Because , our diameter is 12.

We can now find the circumference.

$C=d\pi$

$C=12 \pi$

Multiply the circumference by 4 (our racehorse is running 4 laps) to find the answer.

$12 \pi \cdot 4=48\pi$

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