Diameter, Radius, and Circumference - SAT Math

Card 0 of 64

Question

A barn manager has designed a circular enclosure with a circumference of 75.4 feet. She would like to order sand to fill the enclosure. If one unit of sand covers 27 square feet. How many units of sand will she need to order? Please round your answer up to the nearest whole unit.

Answer

First, you will need to work backward from the circumference to find the radius of the circular enclosure.

$Circumference=2r\pi$

$\small \frac{75.36}{3.14}=\frac{2r(3.14)}{3.14}$

$\small \frac{24}{2}=\frac{2r}{2}$

$\small \small r=12ft$

Now we know what the radius is, we can calculate the surface area of the floor of the enclosure.

$Surface Area=\pi r^{^{2}}$

$\small \small (3.14)(12^{^{2}})=452.16 ft^{^{2}}$

Finally, we need to find the number of units of sand needed to cover the floor of the enclosure.

$\small 452.15ft^{^{2}} \left (\frac{1unit}{27ft^{2}} \right )=16.75 units$

Because we need to round up to the nearest unit, the final is 17 units of sand.

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Question

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

Answer

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.

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Question

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

Answer

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

$C=10\pi cm$

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Question

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

Answer

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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Question

Determine the circumference of the circle with an area of $5\pi$ .

Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the area.

$5\pi=\pi r^2$

Square root both sides to solve for radius.

$\sqrt{5}=\sqrt{r^2}$

$r=\sqrt5$

Write the formula for circumference.

$C=2\pi r$

Substitute the radius.

The answer is: $C=2\pi\sqrt{5}$

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Question

Find the diameter of a circle if the circumference is .

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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:

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

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Question

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

Answer

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

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Question

If the circumference of a circle is $2\pi +1$ , what must be the diameter?

Answer

Write the circumference formula for the circle.

$C=2\pi r$

Substitute the circumference into the equation.

$2\pi +1 = 2\pi r$

Divide by $2\pi$ on both sides.

$\frac{2\pi +1}{ 2\pi } =\frac{ 2\pi r}{ 2\pi }$

$r=\frac{2\pi +1}{ 2\pi }$

The diameter is twice the radius.

$d=2r=2\cdot \frac{2\pi +1}{ 2\pi } = \frac{2\pi+1}{\pi}$

The answer is: $\frac{2\pi+1}{\pi}$

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Question

What is the circumference of a circle with a diameter of $\frac{\pi }{4}$ ?

Answer

Write the formula for the circumference of a circle.

$C= \pi D$

Substitute the diameter into the equation.

$C= \pi (\frac{\pi}{4}) = \frac{\pi^2}{4}$

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

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Question

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

Answer

The diameter of a circle is twice the radius.

Substitute the radius.

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

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

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Question

Determine the radius if the circumference of a circle is $2\sqrt2$ .

Answer

Write the formula for the circumference of a circle.

$C=2\pi r$

Substitute the circumference.

$2\sqrt2=2\pi r$

Divide by $2\pi$ on both sides.

$\frac{2\sqrt2}{2\pi }=\frac{2\pi r}{2\pi }$

Reduce both sides.

The answer is: $r= \frac{\sqrt2}{\pi}$

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Question

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

Answer

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: $\frac{1}{3}$

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Question

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

Answer

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

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

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

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Question

Determine the circumference of a circle with a radius of $\frac{2}{\pi}$ .

Answer

The circumference of a circle is: $C=2\pi r$

Substitute the radius.

$C=2\pi (\frac{2}{\pi})$

The answer is:

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Question

A barn manager has designed a circular enclosure with a circumference of 75.4 feet. She would like to order sand to fill the enclosure. If one unit of sand covers 27 square feet. How many units of sand will she need to order? Please round your answer up to the nearest whole unit.

Answer

First, you will need to work backward from the circumference to find the radius of the circular enclosure.

$Circumference=2r\pi$

$\small \frac{75.36}{3.14}=\frac{2r(3.14)}{3.14}$

$\small \frac{24}{2}=\frac{2r}{2}$

$\small \small r=12ft$

Now we know what the radius is, we can calculate the surface area of the floor of the enclosure.

$Surface Area=\pi r^{^{2}}$

$\small \small (3.14)(12^{^{2}})=452.16 ft^{^{2}}$

Finally, we need to find the number of units of sand needed to cover the floor of the enclosure.

$\small 452.15ft^{^{2}} \left (\frac{1unit}{27ft^{2}} \right )=16.75 units$

Because we need to round up to the nearest unit, the final is 17 units of sand.

Compare your answer with the correct one above

Question

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

Answer

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.

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Question

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

Answer

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

$C=10\pi cm$

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Question

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

Answer

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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