Spheres - High School Math

Card 0 of 20

Question

What is the diameter of a sphere with a volume of $36\pi$ ?

Answer

To find the diameter of a sphere we must use the equation for the volume of a sphere to find the radius which is half of the diameter.

The equation is

First we enter the volume into the equation yielding $36\pi=(\frac{4}{3})(\pi)(r^{3})$

We then divide each side by to get $36=(\frac{4}{3})(r^{3})$

We then multiply each side by to get $27=r^{3}$

We then take the cubic root of each side to solve for the radius $\sqrt[3]{27}=\sqrt[3]{r^{3}}$

The radius is

We then multiply the radius by 2 to find the diameter

The answer for the diameter is .

Compare your answer with the correct one above

Question

If the surface area of a sphere is $64\pi$ , find the diameter of this sphere.

Answer

The standard equation to find the surface area of a sphere is

$SA=4\pi r^2$ where denotes the radius. Rearrange this equation in terms of :

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

Substitute the given surface area into this equation and solve for the radius and then double the radius to get the diameter:

$r=\sqrt{\frac{64\Pi }{4\Pi }}=\sqrt{16}=4$

$diameter=2\cdot r=2\cdot 4=8$

Compare your answer with the correct one above

Question

Given that the volume of a sphere is $4\pi$ , find the diameter.

Answer

The standard equation to find the volume of a sphere is

$V=\frac{4}{3}\pi r^3$ where denotes the radius. Rearrange this equation in terms of :

$r=\sqrt[3]{\frac{3V}{4\pi }}$

Substitute the given volume into this equation and solve for the radius. Double the radius to find the diameter:

$r=\sqrt[3]{\frac{3V}{4\pi }}=\sqrt[3]{\frac{3\cdot 4\pi }{4\pi }}=\sqrt[3]{\frac{12\pi }{4\pi }}=\sqrt[3]{3 }$

$diameter=2\cdot r=2\sqrt[3]{3}$

Compare your answer with the correct one above

Question

The volume of a sphere is $2 8 8 \pi$ . What is the diameter?

Answer

To find the diameter of the sphere, we need to find the radius.

The volume of a sphere is $V=\frac{4}{3}\pi r^3$ .

Plug in our given values.

$2 88\pi=\frac{4}{3}\pi r^3$

Notice that the $\pi$ 's cancel out.

$288=\frac{4}{3}r^3$

$\frac{3}{4}*288=r^3$

$\sqrt[3]{216}=\sqrt[3]{r^3}$

The diameter is twice the radius, so .

Compare your answer with the correct one above

Question

What is the diameter of a sphere with a volume of $36\pi$ ?

Answer

The volume of a sphere is determined by the following equation:

$V=\frac{4}{3}\pi r^3$

$36\pi=\frac{4}{3}\pi r^3$

$(\frac{3}{4})36\pi=(\frac{3}{4})(\frac{4}{3})\pi r^3$

$\frac{27\pi}{\pi}=\frac{\pi r^3}{\pi}$

$\sqrt[3]{27}=\sqrt[3]{r^3}$

Compare your answer with the correct one above

Question

Find the radius of a sphere whose surface area is $SA=100cm^{2}$ .

Answer

We know that the surface area of the spere is $SA=100cm^{2}$ .

$SA=4\pi r^{2}$

$100cm^{2}=4\pi r^{2}$

Rearrange and solve for .

$r^{2}=\frac{100cm^{2}}{4\pi}$

$\dpi{100} r=\sqrt{\frac{100cm^{2}}{4\pi}}$

$\dpi{100}\rightarrow 2.82cm$

Compare your answer with the correct one above

Question

What is the radius of a sphere that has a surface area of $16\pi$ ?

Answer

The standard equation to find the area of a sphere is $SA=4\pi r^2$ where denotes the radius. Rearrange this equation in terms of :

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

To find the answer, substitute the given surface area into this equation and solve for the radius:

$r=\sqrt{\frac{16\pi }{4\pi }}=\sqrt{4}=2$

Compare your answer with the correct one above

Question

Given that the volume of a sphere is $12\pi$ , what is the radius?

Answer

The standard equation to find the volume of a sphere is

$V=\frac{4}{3}\pi r^3$ where denotes the radius. Rearrange this equation in terms of :

$r=\sqrt[3]{\frac{3V}{4\pi }}$

Substitute the given volume into this equation and solve for the radius:

$r=\sqrt[3]{\frac{3V}{4\pi }}=\sqrt[3]{\frac{3\cdot 12\pi }{4\pi }}=\sqrt[3]{\frac{36\pi }{4\pi }}=\sqrt[3]{9 }$

Compare your answer with the correct one above

Question

What is the radius of a sphere with a volume of $288\pi$ ?

Answer

$V=\frac{4}{3}\pi r^3$

$288\pi=\frac{4}{3}\pi r^3$

$\frac{288\pi}{\pi}=\frac{4\pi r^3}{3\pi}$

$288=\frac{4r^3}{3}$

$(288)(\frac{3}{4})=\frac{3}{4}(\frac{4r^3}{3})$

$\sqrt[3]{216}=\sqrt[3]{r^3}$

Compare your answer with the correct one above

Question

What is the surface area of a composite figure of a cone and a sphere, both with a radius of 5 cm, if the height of the cone is 12 cm? Consider an ice cream cone as an example of the composite figure, where half of the sphere is above the edge of the cone.

Answer

Calculate the slant height height of the cone using the Pythagorean Theorem. The height will be the height of the cone, the base will be the radius, and the hypotenuse will be the slant height.

$s=\sqrt{169}=13$

The surface area of the cone (excluding the base) is given by the formula $A=\pi rs$ . Plug in our values to solve.

$A_{cone}=\pi(5)(13)=65\pi\ cm^2$

The surface area of a sphere is given by $A=4\pi r^2$ but we only need half of the sphere, so the area of a hemisphere is $A=2\pi r^2$ .

$A_{hemisphere}=2\pi(5)^2=50\pi\ cm^2$

So the total surface area of the composite figure is $65\pi\ cm^2+50\pi\ cm^2=115\pi\ cm^2$ .

Compare your answer with the correct one above

Question

What is the surface area of a hemisphere with a diameter of $4\ cm$ ?

Answer

A hemisphere is half of a sphere. The surface area is broken into two parts: the spherical part and the circular base.

The surface area of a sphere is given by $SA = 4\pi r^{2}$ .

So the surface area of the spherical part of a hemisphere is $SA = 2\pi r^{2}$ .

The area of the circular base is given by $A = \pi r^{2}$ . The radius to use is half the diameter, or 2 cm.

Compare your answer with the correct one above

Question

What is the surface area of a sphere with a radius of 15?

Answer

To solve for the surface area of a sphere you must use the equation

First, plug in 15 for and square it

$SA=4(15^2)\pi$

Multiply by 4 and to get

$900\pi=225(4)(\pi)$

The answer is $900\pi$ .

Compare your answer with the correct one above

Question

What is the surface area of a sphere whise radius is .

Answer

The surface area of a sphere is found by the formula $SA=4\pi r^{2}$ using the given radius of .

$SA=4\pi (1.5cm)^{2}$

$SA=4\pi2.25cm^{2}$

$\rightarrow 9\pi cm^{2}$

Compare your answer with the correct one above

Question

Find the surface area of a sphere whose diameter is $D=\pi$ .

Answer

The surface area of a sphere is found by the formula $SA=4\pi r^{2}$ . We need to first convert the given diameter of $D=\pi$ to the sphere's radius.

$\pi =2r$

$r=\pi/2$

Now, we can solve for surface area.

$\dpi{100} \dpi{100} SA=4\pi \left ( \frac{\pi }{2}\right)^{2}$

$=4\pi \cdot \frac{\pi^{2} }{4}$

$\dpi{100} \rightarrow \pi^{3}$

Compare your answer with the correct one above

Question

What is the surface area of a sphere with a radius of ?

Answer

To solve for the surface area of a sphere you must remember the formula:

First, plug the radius into the equation for :

$SA=(4)(12^{2})(\pi)$

Since $(12^{2})=144$ , the surface area is $(4)(144)(\pi)=576\pi$ .

The answer is therefore $576\pi$ .

Compare your answer with the correct one above

Question

To the nearest tenth of a square centimeter, give the surface area of a sphere with volume 1,000 cubic centimeters.

Answer

The volume of a sphere in terms of its radius is

$V = \frac{4\pi r^3}{3}$

Substitute and solve for :

$\frac{4\pi r^3}{3} = V$

$\frac{4\pi r^3}{3} = 1,000$

$\frac{4\pi r^3}{3}\cdot \frac{3}{4\pi }= 1,000\cdot \frac{3}{4\pi }$

$r^3= \frac{3,000}{4\pi }$

$r=\sqrt[3]{ \frac{3,000}{4\pi }} \approx 6.20 \textrm{ cm }$

Substitute for in the formula for the surface area of a sphere:

$A = 4\pi r^2 \approx 4 \cdot \pi \cdot 6.20 ^2 \approx 483.6 \textrm{ cm}^{2}$

Compare your answer with the correct one above

Question

Find the surface area of a sphere with a radius of .

Answer

The standard equation to find the area of a sphere is $4\Pi r^2$ .

Substitute the given radius into the standard equation to get the answer:

$4\Pi (4)^2=4\Pi (16)=64\Pi$

Compare your answer with the correct one above

Question

Given that the radius of a sphere is 3, find the surface area.

Answer

The standard equation to find the area of a sphere is

$SA=4\pi r^2$

where denotes the radius. Plug in the given radius to find the surface area.

$SA=4\pi (3)^2=4\cdot 9\cdot \pi =36\pi$

Compare your answer with the correct one above

Question

Find the surface area of the following sphere.

Answer

The formula for the surface area of a sphere is:

$SA = 4 \pi r^2$

where is the radius of the sphere.

Plugging in our values, we get:

$SA = 4 \pi (6m)^2$

$SA = 4 \pi (36m) = 144 \pi m^2$

Compare your answer with the correct one above

Question

Find the surface area of the following sphere.

Answer

The formula for the surface area of a sphere is:

$SA = 4 \pi (r^2)$

Where is the radius of the sphere

Plugging in our values, we get:

$SA = 4 \pi (9m)^2$

$SA = 324 \pi m^2$

Compare your answer with the correct one above

Tap the card to reveal the answer