Cylinders - High School Math

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Question

This figure is a right cylinder with radius of 2 m and a height of 10 m.

What is the surface area of the right cylinder (m2)?

Answer

In order to find the surface area of a right cylinder you must find the area of both bases (the circles on either end) and add them to the lateral surface area. The area of the two circles is easy to find with $\pi r^{2}$ but remember to multiply by 2 for both bases

$2\cdot2^{2}\cdot\pi= 25.14$ .

Next find the lateral area. The lateral area if unrounded would be a rectangle with height of 10 m and length equal to the circumference of the base circles. Thus the lateral area is

$4\cdot\pi\cdot10= 125.66$

Now add the lateral area to the area of the two bases:

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Question

Find the surface area of a cylinder given that its radius is 2 and its height is 3.2.

Answer

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

$Surface\ Area=2\pi r^2+2\pi rh$

where denotes the radius and denotes the height.

Plug in the given values for and to find the area of the cylinder:

$Area=2\pi (2)^2+2\pi (2)(3.2)=8\pi +12.8\pi =20.8\pi$

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Question

The base of a cylinder has an area of $9\pi$ and the cylinder has a height of . What is the surface area of this cylinder?

Answer

The standard equation for the surface area of a cylinder is

$SA=2\pi r^2+2\pi rh$

where denotes radius and denotes height. We've been given the height in the question, so all we're missing is the radius. However, we are able to find the radius from the area of the circle:

$Area=\pi r^2$

We know the area is $9\pi$

$9\pi =\pi r^2$ so

Now that we have both and , we can plug them into the standard equation for the surface area of a cylinder:

$SA=2\pi (3)^2+2\pi (3)(12)=18\pi +72\pi =90\pi$

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Question

Find the surface area of the following cylinder.

Answer

The formula for the surface area of a cylinder is:

$SA = lateral\ area + 2\cdot (base)$

$SA = 2 \pi r (h) + 2 (\pi r^2)$

where is the radius of the base and is the length of the height.

Plugging in our values, we get:

$SA = 2 \pi (7m) (11m) + 2 \pi (7m)^2$

$SA = 154 \pi m^2 + 98 \pi m^2 = 252 \pi m^2$

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Question

Find the surface area of the following cylinder.

Answer

The formula for the surface area of a cylinder is:

$SA = 2(\pi r^2)+(2 \pi r h)$

Where is the radius of the cylinder and is the height of the cylinder

Plugging in our values, we get:

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

$SA = 104 \pi m^2$

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Question

Find the surface area of the following partial cylinder.

Answer

The formula for the surface area of this partial cylinder is:

$SA = 2(\pi r^2)(part)+2(r)(h)+(2 \pi r)(part)(h)$

Where is the radius of the cylinder, is the height of the cylinder, and is the sector of the cylinder.

Plugging in our values, we get:

$SA = 2(\pi (8m)^2)\left(\frac{60}{360}\right)+2(8m)(5m)+(2 \pi (8m))\left(\frac{60}{360}\right)(5m)$

$SA = \frac{64 \pi m^2}{3} + 80m^2 + \frac{40 \pi m^2}{3}$

$SA = \frac{104 \pi m^2}{3} + 80m^2$

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Question

Find the surface area of the following partial cylinder.

Answer

The formula for the surface area of this partial cylinder is:

$SA = 2(\pi r^2)(part)+2(r)(h)+(2 \pi r)(part)(h)$

where is the radius of the cylinder, is the height of the cylinder, and is the sector of the cylinder.

Plugging in our values, we get:

$SA = 2(\pi r^2)(part)+2(r)(h)+(2 \pi r)(part)(h)$

$SA = 2(\pi (6m)^2)\left(\frac{270}{360}\right)+2(6m)(20m)+(2\pi (6m))\left(\frac{270}{360}\right)(20m)$

$SA = 54 \pi m^2 + 240m^2 +180 \pi m^2$

$SA = 234 \pi m^2 + 240m^2$

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Question

What is the surface area of cylinder with a radius of 3 and height of 7?

Answer

The surface area of a cylinder can be determined by the following equation:

$SA=2\pi rh+\pi r^2$

$SA=(2\pi)(3)(7)+\pi(3^2)$

$SA=42\pi+9\pi=51\pi$

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Question

What is the surface area of a cylinder with a base diameter of and a height of ?

Answer

Area of a circle $A=\pi r^{2}=\pi(\frac{d}{2})^{2}=\pi(\frac{6}{2})^{2}=\pi(3)^{2}=9\pi in^{2}$

Circumference of a circle $C=\pi d=6\pi in^{2}$

Surface area of a cylinder $SA=2A+hC=2(9\pi)+8(6\pi)=18\pi+48\pi=66\pi in^{2}$

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Question

The volume of a cylinder is $81\pi$ . If the radius of the cylinder is , what is the surface area of the cylinder?

Answer

The volume of a cylinder is equal to:

$V=\pi r^2h$

Use this formula and the given radius to solve for the height.

$81\pi=\pi(3)^2h$

$81\pi=9\pi(h)$

Now that we know the height, we can solve for the surface area. The surface area of a cylinder is equal to the area of the two bases plus the area of the outer surface. The outer surface can be "unwrapped" to form a rectangle with a height equal to the cylinder height and a base equal to the circumference of the cylinder base. Add the areas of the two bases and this rectangle to find the total area.

$A=2A_{base}+A_{rec}$

$A=2\pi r^2+(2\pi r)(h)$

Use the radius and height to solve.

$A=2\pi (3)^2+2\pi (3)(9)$

$A=18\pi+54\pi$

$A=72\pi$

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Question

What is the surface area of a cylinder with a radius of 2 cm and a height of 10 cm?

Answer

SAcylinder = 2πrh + 2πr2 = 2π(2)(10) + 2π(2)2 = 40π + 8π = 48π cm2

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Question

A circle has a circumference of $4\pi$ and it is used as the base of a cylinder. The cylinder has a surface area of $16\pi$ . Find the volume of the cylinder.

Answer

Using the circumference, we can find the radius of the circle. The equation for the circumference is $2\pi r$ ; therefore, the radius is 2.

Now we can find the area of the circle using $\pi r^{2}$ . The area is $4\pi$ .

Finally, the surface area consists of the area of two circles and the area of the mid-section of the cylinder: $2\cdot 4\pi +4\pi h=16\pi$ , where $h$ is the height of the cylinder.

Thus, $h=2$ and the volume of the cylinder is $4\pi h=4\pi \cdot 2=8\pi$ .

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Question

If a cylinder has a radius, $\small r$ , of 2 inches and a height, $\small h$ , of 5 inches, what is the total surface area of the cylinder?

Answer

The total surface area will be equal to the area of the two bases added to the area of the outer surface of the cylinder. If "unwrapped" the area of the outer surface is simply a rectangle with the height of the cylinder and a base equal to the circumference of the cylinder base. We can use these relationships to find a formula for the total area of the cylinder.

$A=2A_{base}+A_{rectangle}$

$A=2(\pi r^2)+(2\pi r)(h)$

Use the given radius and height to solve for the final area.

$\small 2\pi(2)^{2} + 2\pi (2)(5)$

$\small 8\pi + 20\pi$

$\small 28\pi$

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Question

A water glass has the shape of a right cylinder. The glass has an interior radius of 2 inches, and a height of 6 inches. The glass is 75% full. What is the volume of the water in the glass (in cubic inches)?

Answer

The volume of a right cylinder with radius and height is:

$\pi r^2 h = \pi (2)^2 (6) = 24 \pi$

Since the glass is only 75% full, only 75% of the interior volume of the glass is occupied by water. Therefore the volume of the water is:

$24 \pi \times ( 75 / 100 ) = 24 \pi (0.75) = 18 \pi$

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Question

What is the volume of a cylinder with a radius of 2 and a length that is three times as long as its diameter?

Answer

The volume of a cylinder is the base multiplied by the height or length. The base is the area of a circle, which is $\pi r^{2}$ . Here, the radius is 2. The diameter is 4. Three times the diameter is 12. The height or length is 12. So, the answer is $48\pi$ .

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Question

What is the volume of a cylinder that has a base with a radius of 5 and a height of 52?

Answer

To find the volume of a cylinder we must know the equation for the volume of a cylinder which is $Volume:of:cylinder= r^{2}\piheight:of:cylinder$

In this example the height is 52 and the radius is 5 which we plug into our equation which will look like this $Volume:of:cylinder= 5^{2}\pi52$

We then square the 5 to get $Volume=25\pi52$

Then perform multiplication to get $Volume=1300\pi$

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Question

A balloon, in the shape of a sphere, is filled completely with water. The surface area of the filled balloon is $144\pi$ . If all of the balloon's water was emptied into a cylindrical cup of base radius 4, how high would the water level be?

Answer

The surface area of a sphere is given by the formula

$S (sphere) = 4\pir^2$ , which here equals $144\pi$ . So

$4\pir^2 = 144\pi$

To find out what the water level would be, we need to know how much water is in the balloon. So we need to find the volume of the balloon.

The amount of water that is in the balloon is

$V(sphere) = \frac{4\pir^3}{3}$

= $\frac{4\pi6^3}{3}$

$288\pi$

The cylindrical column of water will have this volume, and will have base radius 4. Using the formula for volume of a cylinder, we can solve for the height of the column of water.

$V(cylinder) = \pir^2h$ , where is the base radius 4 in this case.

$288\pi = \pi(4^2)h$

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Question

A cylinder has a radius of and a height of . What is its volume?

Answer

In order to calculate the volume of a cylinder, we must utilize the formula $V=\pi r^{2}h$ . We were given the radius, $\dpi{100} r=3cm$ , and the height, $\dpi{100} h=27cm$ .

Insert the known variables into the formula and solve for volume $\dpi{100} V$ .

$V=\pi (3cm)^{2}27cm$

$V=\pi9cm^{2}27cm$

$V=243\pi cm^{3}$

In essence, we find the area of the cylinder's circular base, $\dpi{100} A=\pi r^{2}$ , and multiply it by the height.

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Question

A cylinder has a radius of $\dpi{100} r=\pi$ and a height of $\dpi{100} h=10$ . What is its volume

Answer

In order to calculate the volume of a cylinder, we must utilize the formula $V=\pi r^{2}h$ . We were given the radius, $\dpi{100} \dpi{100} r=\pi$ , and the height, $\dpi{100} \dpi{100} h=10$ .

Insert the known variables into the formula and solve for volume $\dpi{100} V$ .

$\dpi{100} V=\pi (\pi)^{2}10$

$\dpi{100} V=\pi\pi^{2}10$

$\dpi{100} V=10\pi^{3}$

In essence, we find the area of the cylinder's circular base, $\dpi{100} A=\pi r^{2}$ , and multiply it by the height.

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Question

A sphere with a radius of is circumscribed in a cylinder. What is the cylinder's volume?

Answer

In order to solve this problem, one key fact needs to be understood. A sphere will take up exactly $\frac{2}{3}$ of the volume of a cylinder in which it is circumscribed. Therefore, if we find the volume of the sphere we can then solve for the volume of the cylinder.

First, we need to find the volume of the sphere.

$\dpi{100} V=\frac{4}{3}\pi (2cm)^{3}$

$\dpi{100} \dpi{100} V_{sphere}=10\frac{2}{3}\pi cm^{3}$

This equals $\frac{2}{3}$ of the volume of the cylinder. Therefore,

$\dpi{100} V_{cylinder}=\frac{3}{2}(10\frac{2}{3}\pi cm^{3})$

$\rightarrow 50.27cm^{3}$

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