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High School Math : Cones

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Example Questions

Example Question #1 : Cones

What is the volume of a cone with a height of $\displaystyle 9$ and a base with a radius of $\displaystyle 13$?

Possible Answers:

$117\pi$ $\displaystyle 117\pi$

$396\pi$ $\displaystyle 396\pi$

$507\pi$ $\displaystyle 507\pi$

$1521\pi$ $\displaystyle 1521\pi$

Correct answer:

$507\pi$ $\displaystyle 507\pi$

Explanation:

To find the volume of a cone we must use the equation $V=\frac{1}{3}(A_{base})(H)$ $\displaystyle V=\frac{1}{3}(A_{base})(H)$. In this formula, $A_{base}$ $\displaystyle A_{base}$ is the area of the circular base of the cone, and $\displaystyle H$ is the height of the cone.

We must first solve for the area of the base using r=13 $\displaystyle r=13$.

The equation for the area of a circle is $A=\pi(r^{2})$ $\displaystyle A=\pi(r^{2})$. Using this, we can adjust our formula and plug in the value of our radius.

$V=\frac{1}{3}(A_{base})(H)=\frac{1}{3}(\pi r^2)(H)=\frac{1}{3}(\pi (13)^2)(H)$ $\displaystyle V=\frac{1}{3}(A_{base})(H)=\frac{1}{3}(\pi r^2)(H)=\frac{1}{3}(\pi (13)^2)(H)$

$\displaystyle 13^2=169$

$V=\frac{1}{3}(169\pi)(H)$ $\displaystyle V=\frac{1}{3}(169\pi)(H)$

Now we can plug in our given height, H=9 $\displaystyle H=9$.

$V=\frac{1}{3}(169\pi)(9)$ $\displaystyle V=\frac{1}{3}(169\pi)(9)$

Multiply everything out to solve for the volume.

$V=\frac{1}{3}(169\pi)(9)=(169\pi)(3)=507\pi$ $\displaystyle V=\frac{1}{3}(169\pi)(9)=(169\pi)(3)=507\pi$

The volume of the cone is $507\pi$ $\displaystyle 507\pi$.

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Example Question #1 : Solid Geometry

What is the equation of a circle with a center of (5,15) and a radius of 7?

Possible Answers:

$\displaystyle (x-5)^2+(y-15)^2=7$

$\displaystyle (x-5)^2+(y-15)^2=14$

$\displaystyle (x+5)^2+(y+15)^2=49$

$\displaystyle (x-5)^2+(y-15)^2=49$

Correct answer:

$\displaystyle (x-5)^2+(y-15)^2=49$

Explanation:

To find the equation of a circle we must first know the standard form of the equation of a circle which is

The letters $\displaystyle h$ and $\displaystyle k$ represent the $\displaystyle x$-value and $\displaystyle y$-value of the center of the circle respectively.

In this case $\displaystyle h$ is 5 and k is 15 so plugging the values into the equation yields $\displaystyle (x-5)^2+(y-15)^2=r^2$

We then plug the radius into the equation to get $\displaystyle (x-5)^2+(y-15)^2=7^2$

Square it to yield $\displaystyle (x-5)^2+(y-15)^2=49$

The equation with a center of (5,15) and a radius of 7 is $\displaystyle (x-5)^2+(y-15)^2=49$.

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Example Question #1 : How To Find The Volume Of A Cone

$\textup{Find the volume, to the nearest cubic inch, of a cone with a radius of 8 inches}$ $\displaystyle \textup{Find the volume, to the nearest cubic inch, of a cone with a radius of 8 inches}$

$\textup{at its base and a height of one foot.}$ $\displaystyle \textup{at its base and a height of one foot.}$

Possible Answers:

2413 $\displaystyle 2413$

$\displaystyle 67$

1206 $\displaystyle 1206$

302 $\displaystyle 302$

804 $\displaystyle 804$

Correct answer:

804 $\displaystyle 804$

Explanation:

$V= \frac{1}{3} \pi r^{2}h\;\;\textup{Given: }r=8\textup{ inches}, h=12\textup{ inches}$ $\displaystyle V= \frac{1}{3} \pi r^{2}h\;\;\textup{Given: }r=8\textup{ inches}, h=12\textup{ inches}$

$V = \frac{1}{3} \pi\times8^{2}\times12\;\;\;\;\;V = 256\pi\;\;\;V=804.2$ $\displaystyle V = \frac{1}{3} \pi\times8^{2}\times12\;\;\;\;\;V = 256\pi\;\;\;V=804.2$

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Example Question #1 : How To Find The Volume Of A Cone

What is the volume of a cone that has a radius of 3 and a height of 4?

Possible Answers:

$12\pi$ $\displaystyle 12\pi$

$36\pi$ $\displaystyle 36\pi$

$12\pi$ $\displaystyle 12\pi$

$9\pi$ $\displaystyle 9\pi$

$4\pi$ $\displaystyle 4\pi$

Correct answer:

$12\pi$ $\displaystyle 12\pi$

Explanation:

The standard equation for the volume of a cone is

$v=\frac{1}{3}\pi (r)^2h$ $\displaystyle v=\frac{1}{3}\pi (r)^2h$

where $\displaystyle r$ denotes the radius and $\displaystyle h$ denotes the height.

Plug in the given values for $\displaystyle r$ and $\displaystyle h$ to find the answer:

$v=\frac{1}{3}\pi (3)^2(4)=\frac{1}{3}\pi (9)(4)=12\pi$ $\displaystyle v=\frac{1}{3}\pi (3)^2(4)=\frac{1}{3}\pi (9)(4)=12\pi$

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Example Question #1 : Cones

Find the volume of the following cone.

Cone

Possible Answers:

$70 \pi m^3$ $\displaystyle 70 \pi m^3$

$100 \pi m^3$ $\displaystyle 100 \pi m^3$

$130 \pi m^3$ $\displaystyle 130 \pi m^3$

$115 \pi m^3$ $\displaystyle 115 \pi m^3$

$85 \pi m^3$ $\displaystyle 85 \pi m^3$

Correct answer:

$100 \pi m^3$ $\displaystyle 100 \pi m^3$

Explanation:

The formula for the volume of a cone is:

$V= \frac{\pi r^{2}h}{3}$ $\displaystyle V= \frac{\pi r^{2}h}{3}$

where $\displaystyle r$ is the radius of the cone and $\displaystyle h$ is the height of the cone.

In order to find the height of the cone, use the Pythagorean Theorem:

$\displaystyle A^2 + B^2 = C^2$

$\displaystyle A^2 + (5m)^2 = (13m)^2$

$\displaystyle A^2 = 144m^2$

A = 12m $\displaystyle A = 12m$

Plugging in our values, we get:

$V= \frac{\pi (5m)^2 (12m)}{3}$ $\displaystyle V= \frac{\pi (5m)^2 (12m)}{3}$

$V= \pi (25m^2)(4m)=100 \pi m^3$ $\displaystyle V= \pi (25m^2)(4m)=100 \pi m^3$

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Example Question #1 : Cones

Find the volume of the following cone.

Cone

Possible Answers:

$86 \pi m^3$ $\displaystyle 86 \pi m^3$

$106 \pi m^3$ $\displaystyle 106 \pi m^3$

$76 \pi m^3$ $\displaystyle 76 \pi m^3$

$96 \pi m^3$ $\displaystyle 96 \pi m^3$

$116 \pi m^3$ $\displaystyle 116 \pi m^3$

Correct answer:

$96 \pi m^3$ $\displaystyle 96 \pi m^3$

Explanation:

The formula for the volume of a cone is:

$V = \frac{\pi r^2 h}{3}$ $\displaystyle V = \frac{\pi r^2 h}{3}$

Where $\displaystyle r$ is the radius of the cone and $\displaystyle h$ is the height of the cone

Use the Pythagorean Theorem to find the length of the radius:

$\displaystyle A^2 + B^2 = C^2$

$\displaystyle A^2 + (8m)^2 = (10m)^2$

A = 6m $\displaystyle A = 6m$

Plugging in our values, we get:

$V = \frac{\pi (6m)^2 8m}{3}$ $\displaystyle V = \frac{\pi (6m)^2 8m}{3}$

$V = 96 \pi m^3$ $\displaystyle V = 96 \pi m^3$

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Example Question #1 : Cones

Find the volume of the following half cone.

Half_cone

Possible Answers:

$140 \pi m^3$ $\displaystyle 140 \pi m^3$

$170 \pi m^3$ $\displaystyle 170 \pi m^3$

$150 \pi m^3$ $\displaystyle 150 \pi m^3$

$160 \pi m^3$ $\displaystyle 160 \pi m^3$

$180 \pi m^3$ $\displaystyle 180 \pi m^3$

Correct answer:

$160 \pi m^3$ $\displaystyle 160 \pi m^3$

Explanation:

The formula of the volume of a half cone is:

$V=\frac{1}{3}(base)(height)$ $\displaystyle V=\frac{1}{3}(base)(height)$

$V = \frac{1}{3}\left(\frac{1}{2}\pi r^2\right)(h)$ $\displaystyle V = \frac{1}{3}\left(\frac{1}{2}\pi r^2\right)(h)$

Where $\displaystyle r$ is the radius of the cone and $\displaystyle h$ is the height of the cone.

Use the Pythagorean Theorem to find the height of the cone:

$\displaystyle A^2+B^2 = C^2$

$\displaystyle A^2+(8m)^2=(17m)^2$

A=15m $\displaystyle A=15m$

$V = \frac{1}{3}\left(\frac{1}{2}\pi (8m)^2\right)(15m)$ $\displaystyle V = \frac{1}{3}\left(\frac{1}{2}\pi (8m)^2\right)(15m)$

$V=160 \pi m^3$ $\displaystyle V=160 \pi m^3$

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Example Question #1 : Solid Geometry

What is the volume of a right cone with a diameter of 6 cm and a height of 5 cm?

Possible Answers:

$60\pi \ cm^{3}$ $\displaystyle 60\pi \ cm^{3}$

$15\pi \ cm^{3}$ $\displaystyle 15\pi \ cm^{3}$

$120\pi \ cm^{3}$ $\displaystyle 120\pi \ cm^{3}$

$45\pi \ cm^{3}$ $\displaystyle 45\pi \ cm^{3}$

$180\pi \ cm^{3}$ $\displaystyle 180\pi \ cm^{3}$

Correct answer:

$15\pi \ cm^{3}$ $\displaystyle 15\pi \ cm^{3}$

Explanation:

The general formula is given by $V = 1/3Bh = 1/3\pi r^{2}h$ $\displaystyle V = 1/3Bh = 1/3\pi r^{2}h$, where $\displaystyle r$ = radius and $\displaystyle h$ = height.

The diameter is 6 cm, so the radius is 3 cm.

$V=\frac{\pi 3^2\times 5}{3}=15\pi$ $\displaystyle V=\frac{\pi 3^2\times 5}{3}=15\pi$

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Example Question #1 : Solid Geometry

There is a large cone with a radius of 4 meters and height of 18 meters. You can fill the cone with water at a rate of 3 cubic meters every 25 seconds. How long will it take you to fill the cone?

Possible Answers:

$600\ \textup{minutes}$ $\displaystyle 600\ \textup{minutes}$

$800\ \textup{minutes}$ $\displaystyle 800\ \textup{minutes}$

$3\ \textup{minutes}\ 20\ \textup{seconds}$ $\displaystyle 3\ \textup{minutes}\ 20\ \textup{seconds}$

$60\ \textup{minutes}$ $\displaystyle 60\ \textup{minutes}$

$41\ \textup{minutes}\ 53\ \textup{seconds}$ $\displaystyle 41\ \textup{minutes}\ 53\ \textup{seconds}$

Correct answer:

$41\ \textup{minutes}\ 53\ \textup{seconds}$ $\displaystyle 41\ \textup{minutes}\ 53\ \textup{seconds}$

Explanation:

First we will calculate the volume of the cone

$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$ $\displaystyle V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$

Next we will determine the time it will take to fill that volume

$96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$ $\displaystyle 96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$

We will then convert that into minutes

$800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$ $\displaystyle 800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$

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Example Question #3 : How To Find The Volume Of A Cone

You have an empty cylinder with a base diameter of 6 and a height of 10 and you have a cone full of water with a base radius of 3 and a height of 10. If you empty the cone of water into the cylinder, how much volume is left empty in the cylinder?

Possible Answers:

$30\pi$ $\displaystyle 30\pi$

$45\pi$ $\displaystyle 45\pi$

$90\pi$ $\displaystyle 90\pi$

$60\pi$ $\displaystyle 60\pi$

$120\pi$ $\displaystyle 120\pi$

Correct answer:

$60\pi$ $\displaystyle 60\pi$

Explanation:

Cylinder Volume = $\pi*r^{2}*h$ $\displaystyle \pi*r^{2}*h$

Cone Volume = $\frac{1}{3}*\pi*r^{2}*h$ $\displaystyle \frac{1}{3}*\pi*r^{2}*h$

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

$=\pi*r^{2}*h-\frac{1}{3}*\pi*r^{2}*h$ $\displaystyle =\pi*r^{2}*h-\frac{1}{3}*\pi*r^{2}*h$

$=\pi*3^{2}*10-\frac{1}{3}*\pi*3^{2}*10$ $\displaystyle =\pi*3^{2}*10-\frac{1}{3}*\pi*3^{2}*10$

$=\pi*90-\pi*30$ $\displaystyle =\pi*90-\pi*30$

$=\pi*60$ $\displaystyle =\pi*60$

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High School Math : Cones

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Cones

Example Question #1 : Solid Geometry

Example Question #1 : How To Find The Volume Of A Cone

Example Question #1 : How To Find The Volume Of A Cone

Example Question #1 : Cones

Example Question #1 : Cones

Example Question #1 : Cones

Example Question #1 : Solid Geometry

Example Question #1 : Solid Geometry

Example Question #3 : How To Find The Volume Of A Cone

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