Diameter, Radius, and Circumference

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SAT Math › Diameter, Radius, and Circumference

Questions 1 - 10

1

If a particle accelerator has a circumference of $18.5 \pi mi$ , what is its radius?

Explanation

If a particle accelerator has a circumference of $18.5 \pi mi$ , what is its radius?

Begin with the formula for the circumference of a circle:

$Circ=2 \pi r$

Now, we know the circumference, so just plug in and solve for r.

$18.5 \pi mi=2 \pi r$

Divide both sides by 2 pi to get out answer:

$18.5 \pi mi\div 2 \pi=2 \pi r\div 2\pi$

2

Example circle

Find the diameter, circumference and area of the circle above.

Diameter= 6 ft

Circumference=18.84 ft

Area= 28.27 ft2

Diameter= 6 ft

Circumference= 19 ft

Area= 30 ft2

Diameter= 9ft

Circumference=37.68 ft

Area= 28 ft2

Diameter= 3ft

Circumference=37.68 ft

Area= 28.7 ft2

Diameter= 6ft

Circumference=37.68 ft

Area= 28.27 ft2

Explanation

To find the diameter, you must know that radius is half the diameter (or the diameter is 2 times the radius.

$\small Diameter= 2(radius)$

$\small \small 2(3)= 6$

The diameter is 6ft.

To find the circumference, you must multiply the diameter (6ft) by pi.

$\small Circumference=\pi (diameter)$

$\small \small \pi(6)= 18.84$

The circumference is 18.84 ft.

To find the surface area, you must aquare the radius (3ft) and multiply by pi.

$\small Surface Area=\pi (radius^{2})$

$\small \pi(3{^2})=\pi(9)=28.27$

The surface area is 28.27 ft2.

The diameter is 6ft, the circumference is 18.84 ft, and the surface area is 28.27 ft2.

3

A circle has a diameter of 10cm. What is the circumference?

$10\pi cm$

$20\pi cm$

$5\pi cm$

$25\pi cm$

$100\pi cm$

Explanation

The circumference of a circle is given by the equation:

$C=2\pi r$

The radius is half the diameter, in this case half of 10cm is 5cm

Plug in 5cm for r

$C=2\pi 5cm$

Simplify to get the final answer

$C=10\pi cm$

4

If a particle accelerator has a circumference of $18.5 \pi mi$ , what is its diameter?

Explanation

If a particle accelerator has a circumference of $18.5 \pi mi$ , what is its radius?

Begin with the formula for circumference of a circle:

$C=2\pi r$

Now, we can see that 2r is really the same as d, right? Our radius will always be half the length of the diameter.

So, we can rewrite the above equation as:

$C=d \pi$

Now, plug in our circumference and solve for d:

$18.5 \pi mi=d \pi$

Divide both sides by pi to get:

So our answer is 18.5 miles

5

Determine the diameter if the radius of a circle is $\frac{7}{3}$ .

$\frac{14}{3}$

$\frac{7}{6}$

$\frac{49}{9}$

$\frac{7}{49}$

$\frac{2}{7}$

Explanation

The diameter is double the radius. Multiply the radius by two.

$D= 2(\frac{7}{3}) = \frac{14}{3}$

The answer is: $\frac{14}{3}$

6

What is the diameter of the circle with a radius of $2\sqrt{10}$ ?

$4\sqrt{10}$

$4\sqrt{5}$

$8\sqrt5$

$8\sqrt{10}$

$4+\sqrt{10}$

Explanation

The diameter of a circle is twice the radius.

Substitute the radius.

$D=2(2\sqrt{10}) = 4\sqrt{10}$

The answer is: $4\sqrt{10}$

7

Determine the radius of a circle if the circumference is $\frac{2}{3}\pi$ .

$\frac{1}{3}$

$\frac{1}{6}$

$\frac{4}{3}$

$\frac{4}{9}$

$\frac{2}{5}$

Explanation

Write the formula for the circumference of a circle.

$C=2\pi r$

Substitute the circumference.

$\frac{2}{3}\pi=2\pi r$

Multiply by $\frac{1}{2\pi}$ on both sides to isolate .

$\frac{2}{3}\pi\cdot \frac{1}{2\pi}=2\pi r \cdot \frac{1}{2\pi}$

$r= \frac{1}{3}$

The radius is: $\frac{1}{3}$

8

Find the area of a circle if the circumference is $5\pi$ .

$\frac{25}{4}\pi$

$\frac{25}{2}\pi$

$\frac{5}{2}\pi$

$\frac{5}{2}\pi^2$

$\frac{5}{4}\pi^2$

Explanation

Write the formula for the circumference of a circle.

$C=2\pi r$

Substitute the circumference.

$5\pi=2\pi r$

Divide by $2\pi$ to isolate the .

$\frac{5\pi}{2\pi}=\frac{2\pi r}{2\pi}$

The radius is: $r= \frac{5}{2}$

Write the formula for the area of the circle.

$A=\pi r^2$

Substitute the radius.

$A=\pi (\frac{5}{2})^2 = \frac{25}{4}\pi$

The answer is: $\frac{25}{4}\pi$

9

Find the diameter of a circle if the circumference is .

$\frac{4x^2}{\pi}$

$\frac{2x}{\pi}$

$\frac{4x}{\pi}$

$\frac{8x}{\pi}$

$2\pi x^2$

Explanation

Write the formula for the circumference of the circle.

$C=2\pi r = \pi D$

Substitute the circumference into the equation.

$4x^2 = \pi D$

Divide by pi on both sides to get the diameter.

$\frac{4x^2 }{ \pi }=\frac{ \pi D}{ \pi }$

The answer is: $\frac{4x^2}{\pi}$

10

If the diameter of a circle is , what is the area of the circle?

$25\pi$

$15\pi$

$20\pi$

$10\pi$

Explanation

Step 1: Recall the formula for an area of a circle...

$Area=\pi r^2$ .

Step 2: Given the diameter, find the radius..

We know that the diameter is twice the length of the radius...

Plug in for :

Divide by 2:

$\frac {10}{2}=5=r$

Step 3: Now that we know the radius, plug the radius into the area formula..

$Area=\pi (5)^2$

Simplify:

$Area=25(\pi)=25\pi$

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