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Surface area of a cone

Surface area of a cone

Here you will learn about the surface area of a cone, including how to find the surface area given the radius of a cone and the slant height.

Students will first learn about the surface area of a cone as a part of geometry in katex is not definedth grade.

What is the surface area of a cone?

The surface area of a cone is the area which covers the outer surface of the cone.

Surface area of a cone 1 US

For any cone, katex is not defined is the radius of the base of the cone, katex is not defined is the slant height of the cone, and katex is not defined is the perpendicular height of the cone.

In order to calculate the total surface area of a cone (tsa), find the area of the two parts of the surface of the cone and add them together.

There is a curved surface area (CSA) and a circular base. The curved surface area of a cone is sometimes referred to as the lateral surface area of a cone (lsa).

Surface area of a cone 2 US

The surface area of a cone is the area of the two parts added together:

katex is not defined

Surface area of a cone 3 US

The perpendicular height of a cone katex is not defined is not required to calculate the surface area of a cone, although it is used to calculate the volume of the cone.

Step by step guide: Volume of a cone

Note: You cannot use the formula for the surface area of a cone written above for an oblique cone. This is because the curved surface area does not have a constant slant height katex is not defined from the apex (a vertex) to the circular base.

Surface area of a cone 4 US

Example

Calculate the surface area of the cone below to katex is not defined decimal place.

Surface area of a cone 5 US

katex is not defined

katex is not defined

Add these together to find the total surface area of the cone:

katex is not defined

What is the surface area of a cone?

What is the surface area of a cone?

Common Core State Standards

How does this relate to katex is not definedth grade math?

  • Grade 8: Geometry (8.G.C.9)
    Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

[FREE] Surface Area Worksheet (Grade 6)

[FREE] Surface Area Worksheet (Grade 6)

[FREE] Surface Area Worksheet (Grade 6)

Use this quiz to check your grade 6 students’ understanding of surface area. 10+ questions with answers covering a range of 6th grade surface area topics to identify areas of strength and support!

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[FREE] Surface Area Worksheet (Grade 6)

[FREE] Surface Area Worksheet (Grade 6)

[FREE] Surface Area Worksheet (Grade 6)

Use this quiz to check your grade 6 students’ understanding of surface area. 10+ questions with answers covering a range of 6th grade surface area topics to identify areas of strength and support!

DOWNLOAD FREE

How to calculate the surface area of a cone

In order to calculate the surface area of a cone:

  1. Find the area of each face.
  2. Add the areas together.
  3. Write the answer, including the units.

Surface area of a cone examples

Example 1: curved surface area

Find the curved surface area of the cone below. The radius of the cone is katex is not defined and the slant height is katex is not defined Write your answer to katex is not defined decimal place.

Surface area of a cone 6 US

  1. Find the area of each face.

You will only calculate the area of the curved surface for this question.

katex is not defined

2Add the areas together.

Because you are calculating the area of the curved surface, you can move on to step 3.

3Write the answer, including the units.

Round the answer to katex is not defined decimal place and include the units.

katex is not defined

The curved surface area of the cone is katex is not defined

Note: If the question asks for katex is not defined decimal place, the answer must have katex is not defined decimal place, even if the digit is katex is not defined

Example 2: total surface area

Find the total surface area of the cone below to katex is not defined decimal place.

Surface area of a cone 7 US

Find the area of each face.

Show step

Add the areas together.

Show step

Write the answer, including the units.

Show step

Example 3: calculate the surface area in terms of π

A chip shop sells mushy peas in a conical pot. The lid has a radius of katex is not defined and the pot has a slant height of katex is not defined Calculate the outside surface area of the closed pot, including the lid. Write your answer in terms of katex is not defined

Surface area of a cone 8 US

Find the area of each face.

Show step

Add the areas together.

Show step

Write the answer, including the units.

Show step

Example 4: surface area in terms of π

The roof of a tower is a solid cone. The radius of the circular base is katex is not defined and the roof has a slope length of katex is not defined Calculate the surface area of the roof. Write your answer in the form katex is not defined where katex is not defined and katex is not defined are integers with no common factors.

Surface area of a cone 9 US

Find the area of each face.

Show step

Add the areas together.

Show step

Write the answer, including the units.

Show step

Example 5: calculate the surface area given the diameter

Calculate the surface area of a cone with a base diameter of katex is not defined and a slant height of katex is not defined Give your answer to katex is not defined significant figures.

Find the area of each face.

Show step

Add the areas together.

Show step

Write the answer, including the units.

Show step

Example 6: curved surface area given the perpendicular height

A cone has a perpendicular height of katex is not defined and a circular base with a radius of katex is not defined The apex of the cone is directly above the center of the circular base. Calculate the area of the curved surface in terms of katex is not defined

Find the area of each face.

Show step

Add the areas together.

Show step

Write the answer, including the units.

Show step

Teaching tips for surface area of a cone

  • Link finding the surface area of cones to finding the area and surface area of other geometric shapes students may be familiar with. The formula for finding the area of the circle of the lateral surface area of a cylinder have parallels with the formula for surface area.

  • Provide multiple opportunities for students to practice finding the surface area of a cone, including worksheets, games or the use of interactive online tools. Allow students to work with small groups and provide frequent feedback to help improve understanding.

  • Use of a problem solving strategy can help students tackle word problems. Model and prompt students to identify and label important information within the word problem before attempting to use the formula.

Easy mistakes to make

  • Forgetting to add the base
    Make sure to include the area of the base when calculating the total surface area. The formula includes the curved surface and the area of the base.

  • Forgetting to simplify radicals
    The formula for the curved surface (lateral surface area of the cone) often involves square roots. Keep an eye out for when you need to do this so you don’t leave the answers unsimplified.

  • Mixing up the slant height with the height
    The slant height is the distance from the tip of the cone to any point on the circumference of the base and the height is the perpendicular distance from the base to the tip. Make sure to use the correct height value when calculating the total surface area of the cone.

Practice surface area of a cone questions

1. Find the curved surface area of a cone of radius katex is not defined and slant height katex is not defined Give your answer to katex is not defined significant figures.

 

Surface area of a cone 11 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

To find the curved surface area of a cone, substitute the value of katex is not defined and katex is not defined into the formula.

 

katex is not defined

2. A cone has a radius of katex is not defined and a slant height of katex is not defined Calculate the curved surface area of the cone, to katex is not defined significant figures.

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

To find the curved surface area of a cone, substitute the value of katex is not defined and katex is not defined into the formula.

 

katex is not defined

3. A candle is made in the shape of a cone. The radius of the base is katex is not defined and the slant height is katex is not defined Calculate the surface area of the candle to katex is not defined decimal place.

 

Surface area of a cone 12 US

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

To find the total surface area of a cone, find the curved surface area and add on the area of the circular base.

 

katex is not defined

 

katex is not defined

 

Total surface area: katex is not defined

 

katex is not defined

4. Find the total surface area of a cone of radius katex is not defined and slant height katex is not defined Write your answer as a fraction in the form katex is not defined where katex is not defined and katex is not defined are integers. State the units in your answer.

 

Surface area of a cone 13 US

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

To find the total surface area of a cone, find the curved surface area and add on the area of the circular base.

 

katex is not defined

 

katex is not defined

 

Total surface area:

 

katex is not defined

 

Surface area is equal to katex is not defined

5. A conical party hat has a diameter of katex is not defined and a slant height of katex is not defined An image is being placed on the entire outside surface of the party hat. What is the area of the image? Write your answer in terms of katex is not defined

 

Surface area of a cone 14 US

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

The diameter is twice the length of the radius and so katex is not defined

 

Substituting the values for katex is not defined and katex is not defined into the formula for the curved surface area you get,

 

katex is not defined

6. A cone has a radius of katex is not defined and a perpendicular height of katex is not defined The apex of the cone is directly above the center of the circular base. Calculate the surface area of the cone to katex is not defined significant figures.

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Use Pythagoras’ theorem to calculate the slant height of the cone, then determine the surface area.

 

Pythagoras’ theorem: katex is not defined

 

katex is not defined

 

The slant height katex is not defined

 

katex is not defined

 

katex is not defined

 

Total surface area: katex is not defined

 

The total surface area of the cone is katex is not defined

Surface area of a cone FAQs

What is a cone?

A cone is a 3D geometric shape that tapers from a flat base to a point called the apex. The base of a cone is typically a circle, but not always.

How do you find the surface area of a right circular cone?

Finding the surface area of a right circular cone is the same as finding the surface area of any cone. You will first find the area of the curved surface using the formula katex is not defined You will add the answer of the area of the curved surface to the area of the base, using the formula katex is not defined

The next lessons are

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