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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.
Use this quiz to check your grade 1, 5 and 6 students’ understanding of 3D shape. 10+ questions with answers covering a range of 1st, 5th and 6th grade 3D shape topics to identify areas of strength and support!
DOWNLOAD FREEUse this quiz to check your grade 1, 5 and 6 students’ understanding of 3D shape. 10+ questions with answers covering a range of 1st, 5th and 6th grade 3D shape topics to identify areas of strength and support!
DOWNLOAD FREEThe 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:
Look 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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