Have you ever tried to figure out the covering of a cone? Take a look at this dilemma.
Jessica came to buy paper at the card store. She is decorating conical party hats for her party by wrapping them in colored tissue paper. Each hat has a radius of 4 centimeters and a slant height of 8 centimeters.
If she wants to wrap 6 party hats, how much paper will she need?
This Concept will teach you how to find the surface area of cones. Then you will know how to figure out this dilemma.
In the last Concept, you learned about pyramids and surface area. Let's look at cones.
Cones have different nets. Imagine you could unroll a cone.
The shaded circle is the base. Remember, cones always have circular bases. The unshaded portion of the cone represents its side. Technically we don’t call this a face because it has a round edge.
To find the surface area of a cone, we need to calculate the area of the circular base and the side and add them together.
To find the area of the sector, we take the area of the portion of the circle.
To find the area of the cone’s side, then, we multiply the radius, the slant height, and pi.
This may seem a little tricky, but as you work through a few examples you will see that this becomes easier as you go along.
What is the surface area of the figure below?
Now that we have the measurements of the sides of the cone, let’s calculate the area of each. Remember to use the correct area formula.
We know the area of each side of the cone when we approximate pi as 3.14. Now we can add these together to find the surface area of the entire cone.
Cones have a different formula because they have a circular base. But the general idea is the same. The formula is a short cut to help us combine the area of the circular base and the area of the cone’s side. Here’s what the formula looks like.
What is the surface area of the cone?
This cone has a surface area of 113.04 square inches.
A cone has a radius of 2.5 meters and a slant height of 7.5 meters. What is its surface area?
This cone has a surface area of 78.5 square inches.
Take a few minutes to write this formula down in your notebooks.
Now try a few of these on your own.
Find the surface area of each cone.
Radius = 4 in, slant height = 6 in
Radius = 3.5 in, slant height = 5 in
Radius = 5 m, slant height = 7 m
You’re right! Just be sure to put the numbers in the correct place in the formula!
Now back to Jessica and the hats.
Jessica is decorating conical party hats for her party by wrapping them in colored tissue paper. Each hat has a radius of 4 centimeters and a slant height of 8 centimeters. If she wants to wrap 6 party hats, how much paper will she need?
Jessica will need 150.72 square centimeters of tissue paper to cover one hat.
Jessica will need 904.32 square centimeters of paper to cover all 6 hats.
Here are the vocabulary words in this Concept.
a three-dimensional object with a circle as a base and a side that wraps around the base connecting at one vertex at the top.
the measurement of the outer covering or surface of a three-dimensional figure.
Here is one for you to try on your own.
Figure out the surface area of a cone with a radius of 4.5 inches and a slant height of 8 inches.
To figure this out, we can use the formula for finding the surface area of a cone.
This is our answer.
Here is a video for review.
- This is a video on surface area of cones.