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Here you will learn about cones, including how to classify and identify a cone, how to find the volume of a cone and how to find the surface area of a cone.
Students will first learn about a cone as part of geometry in 1 st grade. They will expand their learning in middle school and high school when they learn how to find the volume and surface area of a cone.
A cone is a three dimensional object that tapers from a circular base to a point. The term cone comes from the Greek word, “konos”, meaning a wedge or peak.
There is more than one type of cone, and the cone most commonly used is referred to as a “right circular cone”.
Examples of cones:
Right circular cone  Oblique cone 


Reallife examples of cone like shapes include traffic cones, ice cream cones, volcano shapes, and party hats.
Parts of a cone:
Base
The base of the cone is a circle.
Vertex or apex
The vertex or apex of the cone is the point where all lateral sides meet.
Dimensions of a cone
The radius of the base of a cone is r .
The perpendicular height of a cone is h . The height of the cone is a line segment that connects the apex to the center of the circular base. It is perpendicular to the base of the cone.
The slant height of a cone is l . The slant height of the cone is the distance from any point on the base to the apex, along the curved surface of the cone.
The volume of a cone is how much space there is inside a cone.
The formula for the volume of a cone is:
\text{Volume}=\cfrac{1}{3} \, \pi r^2 h
For example, find the volume of the cone, rounded to the nearest tenth.
\begin{aligned} \text { Volume of cone }&=\cfrac{1}{3} \, \pi r^2 h \\\\ & =\cfrac{1}{3} \times \pi \times 3^2 \times 4 \\\\ & =12 \pi \\\\ & =37.7 \mathrm{~cm}^3 \end{aligned}
The surface area of a cone is the area which covers the outer surface of a cone.
The surface area is made up of two parts, a curved surface area and a circular base.
The formula for calculating the curved surface area of a cone is:
\text{Curved surface area}=\pi rl
The formula for calculating the area of a circle:
\text{Area of circle}=\pi r^2
For the TOTAL surface area, you can add the two parts together:
\text{TOTAL surface area}=\pi rl+\pi r^2
For example, find the surface area rounded to the nearest tenth.
\text{Curved surface area}=\pi rl=\pi \times 3\times 5=15\pi
\text{Area of circle}=\pi r^2=\pi \times 3^2=9\pi
\text{TOTAL surface area}= 15\pi + 9\pi = 24\pi = 75.4 {~cm}^2
How does this relate to 1 st, 7 th and 8 th grade math?
In order to identify a cone, you will:
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DOWNLOAD FREELook at the image below and determine if it is a cone or not.
A cone has a circular base with a curved surface area that meets at a vertex that is directly above the center of the circular base.
This shape also has a circular base and a curved surface area that meets at a vertex.
2State whether or not the shape is a cone.
This shape is a cone because it has a circular base with a curved surface that meets at a vertex.
Look at the image below and determine if it is a cone or not.
Look for the characteristics of a cone.
A cone has a circular base with a curved surface area that meets at a vertex that is directly above the center of the circular base.
This shape has a rectangular base with 4 triangular faces that meet at a vertex.
State whether or not the shape is a cone.
This shape is a pyramid, not a cone.
If the shape is not a cone, explain what characteristics are different.
This shape has a rectangular base with 4 triangular faces that meet at a vertex.
In order to calculate the volume of a cone:
Find the volume of the cone with radius 5.3{~cm} and perpendicular height 7.8{~cm} .
Give your answer to the nearest centimeter.
Write down the formula.
\text {Volume }=\cfrac{1}{3} \, \pi r^{2} h
Substitute the given values.
\begin{aligned} & r=5.3 \\\\ & h=7.8 \end{aligned}
\begin{aligned} \text { Volume } &=\cfrac{1}{3} \, \pi r^{2} h \\\\ &=\cfrac{1}{3} \times \pi \times 5.3^{2} \times 7.8 \end{aligned}
Calculate the volume of the cone.
\begin{aligned} &=\cfrac{1}{3} \times \, \pi \times 5.3^{2} \times 7.8 \\\\ &=229.443 \ldots \end{aligned}
Write the final answer, including the units.
The answer rounded to the nearest centimeter is V = 229 {~cm}^3 .
Find the volume of the cone with radius 9{~cm} and perpendicular height 11{~cm} .
Leave your answer in terms of \pi .
Write down the formula.
\text {Volume }=\cfrac{1}{3} \, \pi r^{2} h
Substitute the given values.
\begin{aligned} & r=9 \\\\ & h=11 \end{aligned}
\begin{aligned} \text { Volume } &=\cfrac{1}{3} \, \pi r^{2} h \\\\ &=\cfrac{1}{3} \times \pi \times 9^{2} \times 11 \end{aligned}
Calculate the volume of the cone.
\begin{aligned} &=\cfrac{1}{3} \times \, \pi \times 9^{2} \times 11 \\\\ &=297 \pi \end{aligned}
Write the final answer, including the units.
The question asks for the answer in terms of \pi so the final answer is
=297 \pi {~cm}^3
In order to calculate the surface area of a cone:
Find the curved surface area of the cone with radius 4.3{~cm} and slant height 9.6{~cm} .
Give your answer to the nearest centimeter.
Calculate the area of each face.
\begin{aligned} \text {Curved surface area} &= \, \pi r l \\\\ &=\pi \times 4.3 \times 9.6 \\\\ &=129.6849 \ldots \\\\ &=129.7 \text { (round to the nearest tenth) } \end{aligned}
\begin{aligned} \text {Area of circle } &= \, \pi r^2 \\\\ &=\pi \times 4.3^2 \\\\ &=58.0880 \ldots \\\\ &=58.1 \text { (rounded to the nearest tenth) } \end{aligned}
Add the area of each face together.
Total surface area: 129.7+58.1=187.8
Include units.
Surface area =188 \; cm^2
Find the curved surface area (lateral area) of the cone with radius 8{~cm} and slant height 13{~cm} .
Leave your answer in terms of \pi .
Calculate the area of each face.
\begin{aligned} \text{Curved surface area}&=\, \pi rl\\\\ &=\pi \times 8 \times 13\\\\ &= 104 \pi \end{aligned}
\begin{aligned} \text{Area of circle }&=\, \pi r^2\\\\ &=\pi \times 8^2\\\\ &=64\pi \end{aligned}
Add the area of each face together.
Total surface area: 104\pi +64\pi = 168\pi
Include units.
=168 \pi \mathrm{~cm}^{2}
1) Which shape is an example of a cone?
This shape has a circular base with a curved surface area that meets at a vertex that is directly above the center of the circular base.
This shape is a cone.
2) Which shape is an example of a cone?
This shape has a circular base with a curved surface area that meets at a vertex that is directly above the center of the circular base.
This shape is a cone.
3) Find the volume of a cone to the nearest whole cubic centimeter with a radius of 9.4 {~cm} and perpendicular height of 8.7 {~cm} .
Find the volume of a cone so you can substitute the values of r and h into the formula.
\begin{aligned} & r=9.4 \\\\ & h=8.7 \end{aligned}
\begin{aligned} & V=\cfrac{1}{3} \, \pi r^2 h \\\\ & V=\cfrac{1}{3} \times \pi \times 9.4^2 \times 8.7 \end{aligned}
V=805.014 \ldots
( 805.014 rounded to the nearest whole number would give you…)
V=805 \mathrm{~cm}^3
4) Find the volume of a cone of radius 8 {~cm} and perpendicular height 6 {~cm} . Leave your answer in terms of \pi .
Find the volume of a cone so you can substitute the values of r and h into the formula.
\begin{aligned} & r=8 \\\\ & h=6 \\\\ & V=\cfrac{1}{3} \, \pi r^2 h \\\\ & V=\cfrac{1}{3} \times \pi \times 8^2 \times 6 \\\\ & V=128 \pi \\\\ & V=128 \pi \mathrm{~cm}^3 \end{aligned}
5) Find the curved surface area (lateral area) of a cone of radius 4.3 {~cm} and slant height 6.2 {~cm} .
You will find the curved surface area of a cone, so you can substitute the values of r and h into the formula.
\begin{aligned} \text{Curved surface area}&= \, \pi r l \\\\ & =\pi \times 4.3 \times 6.2 \\\\ & =83.754 \ldots \text{ (round to the nearest tenth)}\\\\ & =83.8 \mathrm{~cm}^2 \end{aligned}
6) Find the curved surface area (lateral area) of a cone of radius 7 {~cm} and slant height 9 {~cm} . Leave your answer in terms of \pi .
You are finding the curved surface area of a cone, so substitute the values of r and l into the formula.
\begin{aligned} \text{Curved surface area}&= \,\pi rl\\\\ &=\pi \times 7\times 9\\\\ &=63\pi\\\\ &=63\pi \ cm^2\\\\ \end{aligned}
A right circular cone is the most common type of cone. It has a circular base, and the apex is directly above the center of the base. The axis of the cone is perpendicular to the base, creating a right angle. An oblique cone is any cone that is not a right circular cone. In an oblique cone, the axis is not perpendicular to the base, resulting in a slanted shape.
During your math journey, you will encounter right circular cones, oblique cones, acute cones, and obtuse cones.
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