Learn how to find the area of a cone using the surface area of a cone formula. Calculate the volume of the cone section and the volume of a truncated cone.
Updated: 10/15/2021
Surface Area and Volume of a Cone
Cone and Truncated Cone
There are many three-dimensional geometrical shapes that are commonly used in the real world, such as cubes, cylinders, and rectangular prisms. These shapes are seen in everyday objects, e.g., boxes, cans, and suitcases. Another very common three-dimensional shape is the cone.Cones are seen in ice cream cones, traffic cones, or disposable water cups often used in waiting rooms. Cones are fan-like shapes that are wrap around so that their sides meet. If a cone were to be sliced down the side and then spread onto a two-dimensional plane, a fan-like shape would be observed.
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All cones have a radius, slant height, and height.
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The illustration above shows a cone with a radius, slant height, and height. The radius of the cone is the radius of the circle located at the bottom. The slant height is the distance alone the side of the cone from the bottom to the top, while the height is the distance from the center of the circle to the bottom of the cone.
Now picture that same shape that was spread out from a cone and imagine the tip of it being cut off, then wrap the sides to touch again as they were before when it formed the cone. This resulting shape is called a truncated cone. It is a special type of cone that has two circles: one on the top and one on the bottom. Similar to regular cones, truncated cones have a radius, slant height, and height. However, truncated cones have two radii, one for the top circle and another for the bottom.
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Truncated cones are the shapes used for funnels, where it has a cone shape but an opening on the top and on the bottom to let matter flow through it. However, these do not necessarily have to be 'openings' for it to be considered a truncated cone. Picture a popcorn container for example. It has an opening on the top, but the bottom is closed to keep the popcorn contained. This shape is still considered a truncated cone.
Truncated Cone Definition
What was the last really great film you went to see at a movie theater? Going to the movies is always a great time! Of course, the experience of going to a movie at the theater isn't complete without popcorn! Most likely, you've never taken note of the shape of a popcorn tub at the theater, but here's an interesting fact! There is a mathematical name for this shape, and it's called a truncated cone.
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A truncated cone is a three-dimensional object that looks like a cone with the top pointy end chopped off.
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The two circular ends of a truncated cone are called the bases. We will call the radius of the smaller circular base the little radius and represent it with r, and we will call the radius of the larger circular base the big radius and represent it with R. The distance between the centers of the two bases is called the height of the truncated cone, and we represent it with h. Lastly, the shortest distance between the outer edges of the bases is called the slant height, and we represent it with s.
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Do you now see that a popcorn tub has this shape? Pretty interesting, huh? Well, it's about to get even more interesting! Let's talk a bit about the volume and surface area of a truncated cone!
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How to Find the Area of a Cone?
Recall that area describes the amount of space contained in a two-dimensional shape, and it is always expressed in units squared. Calculating an area usually involves multiplying the length of the shape by the height. For triangles, the area is found by multiplying the base by the height of the triangle.
For three dimensional shapes, there are two types of areas: Lateral surface area and total surface area. The lateral surface area of a shape is the area around the sides of the shape. For a cone, the lateral surface area would be the area of the fan shape that wraps around to form the cone. Lateral surface area can be thought of as the surface area without the area of the base(s).
What about the area of the base? What if that were to be included? That is where total surface area comes in. The total surface area of a shape includes both the lateral surface area as well as the area of the base(s). It is, as the name suggests, the total surface area of the entire shape and includes every side of the shape. Total surface area can be found by adding the lateral surface area to the area of the base(s).
Surface Area of a Cone Formula
As previously mentioned, the lateral surface area for a cone is found by finding the area of the fan-like shape that forms the cone. For the case of a cone, the height of the fan shape would be the slant height of the cone, while the base of the fan shape comes from the circle that is formed at the base of the cone. The formula for the lateral area of a cone is the following:
{eq}\text{A}_{l} = \pi r s {/eq}
where {eq}\text{A}_{l} {/eq} is the lateral surface area of a cone, {eq}r {/eq} is the circle radius of the cone while {eq}s {/eq} is the slant height of the cone.
If the slant height of a cone is unknown but the radius and height of the cone are provided, then the slant height can be solved for using the following formula:
{eq}s = \sqrt{ r^2 + h^2 } {/eq}
where {eq}r {/eq} is the circle radius of a cone while {eq}h {/eq} is the height of a cone.
Using this formula for the slant height of the cone and plugging it in for the lateral area, the result is the following formula:
{eq}\text{A}_{l} = \pi r \sqrt{r^2 + h^2} {/eq}
Using this formula, the lateral surface area of a cone can be solved for if the slant height is not provided.
The total surface area of a cone can be found by adding the lateral surface area to the base surface area of the cone. The area of a circle can be found from the following formula:
{eq}\text{Circle area} = \pi r^2 {/eq}
where {eq}r {/eq} is the radius of the circle. Adding this formula to the lateral surface area of a cone formula yields the total surface area of a cone:
{eq}\text{A}_{t} = \text{S}_{l} + \pi r^2 {/eq}
or
{eq}\text{A}_{t} = \pi r \sqrt{r^2 + h^2} + \pi r^2 {/eq}
Using these formulas, one could solve for the lateral and total surface area of a cone.
How to Calculate the Surface Area of a Cone?
When calculating the surface area of a cone, there are specifics that must be taken into consideration. Take the following example:
- Example
- A cone has a radius of {eq}4\text{in} {/eq} and a height of {eq}7\text{in} {/eq}, what is the lateral and total surface area of this cone?
→ Solution:
The lateral surface area can be found by plugging in the values for radius and height into the formula for lateral surface area of a cone:
{eq}\text{A}_{l}= \pi r \sqrt{ r^2 + h^2 } \\ \text{A}_{l}= \pi 4 \sqrt{16 + 49} \\ \text{A}_{l}= 12.57\sqrt{65} \\ \text{A}_{l} \approx 101.34\text{in}^{2} \\ {/eq}
Now that the lateral surface area was solved for, this value can be used to calculate the total surface area of the cone:
{eq}\text{A}_{t} = \text{A}_{l} + \pi r^2 \\ \text{A}_{t} = 101.34 + \pi r^2 \\ \text{A}_{t} = 101.34 + \pi 4^2 \\ \text{A}_{t} = 101.34 + 16\pi \\ \text{A}_{t} \approx 151.61\text{in}^{2} {/eq}
Therefore, the lateral surface area of the cone is about {eq}101.34 \text{in}^2 {/eq} while the total surface area of the cone is about {eq}151.61 \text{in}^2 {/eq}
Surface Area of Truncated Cone
The surface areas for truncated cones are slightly different than the surface areas of regular cones due to the differences within the shapes. To find the lateral surface area of a truncated cone, the following formula can be used:
{eq}\text{A}_{l\text{ }t} = \pi (R + r) s {/eq}
where {eq}\text{A}_{l\text{ }t} {/eq} is the lateral surface area for a truncated cone, {eq}R {/eq} is the radius of the larger circle, {eq}r {/eq} is the radius of the smaller circle, and {eq}s {/eq} is the slant height of the truncated cone. Similar to the slant height of a regular cone, the truncated cone slant height can be calculated using the values for the height and radii of the truncated cone:
{eq}s = \sqrt{h^2 + (R - r )^2} {/eq}
where {eq}s {/eq} is the slant height of a truncated cone and {eq}h {/eq} is the height of the truncated cone.
To find the total surface area of a truncated cone, simply add the lateral surface area to the areas of both bases. Unlike regular cones, truncated cones have two bases since they have two circles. The total surface area formula for truncated cones is the following:
{eq}\text{A}_{t\text{ }t} = \pi [ s (R + r) + R^2 + r^2] {/eq}
where {eq}\text{A}_{t\text{ }t} {/eq} is the total surface area of a truncated cone. Plugging in the formula for {eq}s {/eq}, the formula for the total surface area of a truncated cone can also be written as the following:
{eq}\text{A}_{t\text{ }t} = \pi \sqrt{h^2 + (R - r )^2} \text{ }(R + r) + R^2 + r^2 {/eq}
Using these formulas, the total surface area or the lateral surface area of a truncated cone can be solved for.
How to Calculate the Surface Area of a Truncated Cone?
Calculating the surface area of truncated cone is a little different. Take the following example:
- Example
- A truncated cone has a larger radius of {eq}5\text{in} {/eq} and a smaller radius of {eq}3\text{in} {/eq}. The height of the truncated cone is {eq}10\text{in} {/eq}. What is the lateral and total surface area of this truncated cone?
→ Solution:
Using the values for both the large and small radii as well as the value for the height, the surface areas can be solved by using the formulas for the lateral and total surface area of a truncated cone. The first step is to solve for the slant height:
{eq}s =\sqrt{ h^2 + (R - r )^2} \\ s= \sqrt {10^2 + (5 - 3 )^2} \\ s =\sqrt{100 + 4} \\ s =\sqrt{104} \\ s \approx 10.2 {/eq}
Now this value for the slant height can be plugged into the equation for the lateral surface area of a truncated cone:
{eq}\text{A}_{l\text{ }t} = \pi (R + r) s \\ \text{A}_{l\text{ }t} = \pi (5 + 3) 10.2 \\ \text{A}_{l\text{ }t} = \pi (8) 10.2 \\ \text{A}_{l\text{ }t} = 256.35\text{in}^{2} \\ {/eq}
The total surface area can also be calculated using the formula and the value for {eq}s {/eq}:
{eq}\text{A}_{t\text{ }t} = \pi [ s (R + r) + R^2 + r^2] \\ \text{A}_{t\text{ }t} = \pi [ 10.2 (5 + 3) + 5^2 + 3^2] \\ \text{A}_{t\text{ }t} = \pi [ 10.2 (8) + 25+9 ] \\ \text{A}_{t\text{ }t} = \pi [ 10.2 (8) + 34 ] \\ \text{A}_{t\text{ }t} \approx 363.2\text{in}^{2} {/eq}
Therefore, the total surface area of the truncated cone is about {eq}363.2\text{in}^{2} {/eq} and the lateral surface area is about {eq}256.35\text{in}^{2} \\ {/eq}
The Formula for the Volume of a Cone Section
Another important geometrical characteristic of cones is the volume of a cone. The volume of a cone can be found using the following formula:
{eq}\text{V} = \pi r^2 \frac{h}{3} {/eq}
where {eq}\text{V} {/eq} is the volume, {eq}r {/eq} is the radius of the cone, and {eq}h {/eq} is the height of the cone. This formula can be used to calculate the volume of any standard cone.
How to Calculate the Volume of a Cone?
When it comes to calculating the volume of a cone, the answer is going to end with cubic inches. Take for example:
- Example
- A disposable water cup takes the shape of a cone. The cone has a radius of {eq}2\text{in} {/eq} and a height of {eq}5\text{in} {/eq}. How much water can this cone hold?
→ Solution:
The amount of water the cone can hold is found by finding the volume of the cone using the formula for the volume of a cone:
{eq}\text{V} = \pi r^2 \frac{h}{3} \\ \text{V} = \pi 2^2 \frac{5}{3} \\ \text{V} \approx 20.9 \text{in}^{3} {/eq}
The cone can hold about {eq}21\text{in}^{3} {/eq} of water.
The Formula for the Volume of a Truncated Cone
Just like regular cones, truncated cones also have a volume that can be calculated. The formula for the volume of a truncated cone is the following:
{eq}\text{V}_{t} = \frac{1}{3} \pi h(R^{2} + r^{2} + R r) {/eq}
where {eq}\text{V}_{t} {/eq} is the volume of a truncated cone, {eq}h {/eq} is the height of the cone, {eq}R {/eq} is the larger radius, and {eq}r {/eq} is the smaller radius.
How to Calculate the Volume of a Truncated Cone?
As with calculating the surface area, calculating for volume of a truncated cone requires extra steps in the process. Take for example:
- Example
- An extra large popcorn container has a larger radius of {eq}8\text{in} {/eq} and a smaller radius of {eq}5\text{in} {/eq}. The height of the container is {eq}12\text{in} {/eq}. How much popcorn can this container hold?
→ Solution:
The amount of popcorn the container can hold is found by solving for the volume of the truncated cone. Using the formula for the volume of a truncated cone, the values for the height and radii can be plugged in:
{eq}\text{V}_{t} = \frac{1}{3} \pi h(R^{2} + r^{2} + R r) \\ \text{V}_{t} = \frac{1}{3} \pi 12(8^{2} + 5^{2} + (8) (5)) \\ \text{V}_{t} = \frac{1}{3} \pi 12(64+25+40) \\ \text{V}_{t} \approx 1,621.1 \text{in}^{3} {/eq}
Therefore, the truncated cone can hold about {eq}1,621.1\text{in}^{3} {/eq} of popcorn.
Lesson Summary
Cones are used in everyday objects such as ice cream cones or paper cups. They are a three-dimensional object formed from a two-dimensional triangle folded around so the sides of the triangle meet. Another type of cone is a truncated cone which is similar to a regular cone except with an opening at the bottom. Truncated cones are used in objects such as funnels or popcorn containers. All cones have a radius, slant height, and height. The radius of a cone is the radius of the circle. Truncated cones have two radii since they have two circles, so labeling which radius belongs to what circle is important. The slant height of a cone is the distance from the bottom to the top of the cone along the side, while the height is the distance from the bottom to the top through the center of the cone. Like all three-dimensional objects, both cones and truncated cones have a surface area and a volume that can be found. There are two types of surface area: Lateral surface area and total surface area. The lateral surface area is the side area of the shape while total surface area includes both the lateral surface area and the base surface area. The surface areas and volumetric formulas for both cones and truncated cones are the following:
- Lateral surface area of a cone: {eq}\text{A}_{l} = \pi r \sqrt{r^2 + h^2} {/eq}
- Total surface area of a cone: {eq}\text{A}_{t} = \pi r \sqrt{r^2 + h^2} + \pi r^2 {/eq}
- Lateral surface area of a truncated cone: {eq}\text{A}_{l\text{ }t} = \pi (R + r) (\sqrt{h^2 + (R - r )^2}) {/eq}
- Total surface area of a truncated cone: {eq}\text{A}_{t\text{ }t} = \pi \sqrt{h^2 + (R - r )^2} \text{ }(R + r) + R^2 + r^2 {/eq}
- Volume of a cone: {eq}\text{V} = \pi r^2 \frac{h}{3} {/eq}
- Volume of a truncated cone: {eq}\text{V}_{t} = \frac{1}{3} \pi h(R^{2} + r^{2} + R r) {/eq}
Using these formulas, volumes and surface areas of both cones and truncated cones can be solved for given enough information.
Truncated Cone Volume
Suppose you go to the movies, and you get a large popcorn. You can't wait to sink your teeth into the salty, buttery goodness, but before doing so, you begin to wonder just how much popcorn fits inside the large popcorn tub. I've got news for you! You're actually wondering what the volume of the popcorn tub is!
The volume of a truncated cone is the amount of space inside the truncated cone. The good news is that we have a nice formula that we can use to find the volume of a truncated cone.
Volume = (1/3) ⋅ π ⋅ h(R 2 + r 2 + R ⋅ r)
Your curiosity gets the best of you, so you ask the worker at the concessions counter to see an empty large popcorn tub. When he gives it to you, you take some quick measurements and find that the big radius is 5 inches, the small radius is 4 inches, the height is 10 inches, and the slant height is 11 inches. You give the empty tub back to the concessions worker and pull out a pencil and some paper to calculate the volume of the large popcorn tub using your formula.
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So, we see that the volume of the large popcorn tub is approximately 638.79 cubic inches. Wow! That's a lot of popcorn! Dig in!
Truncated Cone Surface Area
As you begin to gobble down your popcorn, you begin to have a bit of buyer's remorse. After all, you paid $8 for this large popcorn, and that's not cheap! This gets you to wondering just how much the movie theater has to pay for a popcorn tub. You have a bit more time until your movie starts, so you ask a worker how much these tubs cost them. The worker says that the manufacturer charges based on how much material was used to make the tub and a lid for the tub, and they charge 0.00125 cents per square inch of material used.
Hmmm, to answer your question, you would need some way to find the total amount of material used to make a large tub and a lid. Guess what? You're actually thinking mathematically again, because this total amount of area that the surface of a truncated cone takes up is called the surface area of the truncated cone, and this is exactly what you're looking for! Thankfully, once again, we have a nice formula we can use to calculate this surface area.
Surface Area = π (s(R + r) +R 2 + r 2)
Great! You can find the surface area of the large tub with a lid and then multiply it by 0.00125 cents to find out how much the theater actually pays for these tubs. First, let's find the surface area. You already have the measurements, since you took them when finding the volume, so we just plug in s = 11, R = 5, and r = 4 to the formula and then simplify.
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So we can now see that the surface area of the large tub is approximately 439.82 square inches. Okay, let's multiply this by the amount the manufacturer charges per square inch of material.
439.82 â‹… 0.00125 = 0.549775
We see that the movie theater pays approximately $0.55 per tub of popcorn. Whoa! You paid $8! Talk about a major markup. But, as you shove another handful of popcorn in your mouth, you think it was probably worth it.
Lesson Summary
Let's take a few moments to review what we've learned about truncated cones and measuring them. A truncated cone is a three-dimensional object that has the shape of a cone with the top pointy end chopped off. The volume of a truncated cone is the amount of space inside the truncated cone. The surface area of a truncated cone is the total amount of area that the entire surface of the truncated cone takes up.
We can calculate the volume and surface area of a truncated cone using the following formulas.
Volume = (1/3) ⋅ π ⋅ h(R 2 + r 2 + R ⋅ r)
Surface Area = π (s(R + r) +R 2 + r 2)
You may never look at a tub of popcorn in the same way again! Speaking of which, you've been so enthralled and amused by all this mathematics that you are missing the previews at your movie - better get in there! Enjoy the show!
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Video Transcript
Truncated Cone Definition
What was the last really great film you went to see at a movie theater? Going to the movies is always a great time! Of course, the experience of going to a movie at the theater isn't complete without popcorn! Most likely, you've never taken note of the shape of a popcorn tub at the theater, but here's an interesting fact! There is a mathematical name for this shape, and it's called a truncated cone.
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A truncated cone is a three-dimensional object that looks like a cone with the top pointy end chopped off.
|
The two circular ends of a truncated cone are called the bases. We will call the radius of the smaller circular base the little radius and represent it with r, and we will call the radius of the larger circular base the big radius and represent it with R. The distance between the centers of the two bases is called the height of the truncated cone, and we represent it with h. Lastly, the shortest distance between the outer edges of the bases is called the slant height, and we represent it with s.
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Do you now see that a popcorn tub has this shape? Pretty interesting, huh? Well, it's about to get even more interesting! Let's talk a bit about the volume and surface area of a truncated cone!
Truncated Cone Volume
Suppose you go to the movies, and you get a large popcorn. You can't wait to sink your teeth into the salty, buttery goodness, but before doing so, you begin to wonder just how much popcorn fits inside the large popcorn tub. I've got news for you! You're actually wondering what the volume of the popcorn tub is!
The volume of a truncated cone is the amount of space inside the truncated cone. The good news is that we have a nice formula that we can use to find the volume of a truncated cone.
Volume = (1/3) ⋅ π ⋅ h(R 2 + r 2 + R ⋅ r)
Your curiosity gets the best of you, so you ask the worker at the concessions counter to see an empty large popcorn tub. When he gives it to you, you take some quick measurements and find that the big radius is 5 inches, the small radius is 4 inches, the height is 10 inches, and the slant height is 11 inches. You give the empty tub back to the concessions worker and pull out a pencil and some paper to calculate the volume of the large popcorn tub using your formula.
|
So, we see that the volume of the large popcorn tub is approximately 638.79 cubic inches. Wow! That's a lot of popcorn! Dig in!
Truncated Cone Surface Area
As you begin to gobble down your popcorn, you begin to have a bit of buyer's remorse. After all, you paid $8 for this large popcorn, and that's not cheap! This gets you to wondering just how much the movie theater has to pay for a popcorn tub. You have a bit more time until your movie starts, so you ask a worker how much these tubs cost them. The worker says that the manufacturer charges based on how much material was used to make the tub and a lid for the tub, and they charge 0.00125 cents per square inch of material used.
Hmmm, to answer your question, you would need some way to find the total amount of material used to make a large tub and a lid. Guess what? You're actually thinking mathematically again, because this total amount of area that the surface of a truncated cone takes up is called the surface area of the truncated cone, and this is exactly what you're looking for! Thankfully, once again, we have a nice formula we can use to calculate this surface area.
Surface Area = π (s(R + r) +R 2 + r 2)
Great! You can find the surface area of the large tub with a lid and then multiply it by 0.00125 cents to find out how much the theater actually pays for these tubs. First, let's find the surface area. You already have the measurements, since you took them when finding the volume, so we just plug in s = 11, R = 5, and r = 4 to the formula and then simplify.
|
So we can now see that the surface area of the large tub is approximately 439.82 square inches. Okay, let's multiply this by the amount the manufacturer charges per square inch of material.
439.82 â‹… 0.00125 = 0.549775
We see that the movie theater pays approximately $0.55 per tub of popcorn. Whoa! You paid $8! Talk about a major markup. But, as you shove another handful of popcorn in your mouth, you think it was probably worth it.
Lesson Summary
Let's take a few moments to review what we've learned about truncated cones and measuring them. A truncated cone is a three-dimensional object that has the shape of a cone with the top pointy end chopped off. The volume of a truncated cone is the amount of space inside the truncated cone. The surface area of a truncated cone is the total amount of area that the entire surface of the truncated cone takes up.
We can calculate the volume and surface area of a truncated cone using the following formulas.
Volume = (1/3) ⋅ π ⋅ h(R 2 + r 2 + R ⋅ r)
Surface Area = π (s(R + r) +R 2 + r 2)
You may never look at a tub of popcorn in the same way again! Speaking of which, you've been so enthralled and amused by all this mathematics that you are missing the previews at your movie - better get in there! Enjoy the show!
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Frequently Asked Questions
What are the properties of a truncated cone?
Truncated cones are similar to regular cones, except they have two bases rather than one. They have a larger circle on one end and a smaller circle on the other.
What is the area and volume of a cone?
The area of a cone is A = πr√(r² + h²) + πr² while the volume of a cone is V = πr² ⋅ h⁄3. Area is always expressed in units squared while volume is expressed in units cubed.
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Surface Area of a Cone Formula | How to Find the Area of a Cone
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