How to Find the Surface Area of Cones
Set up the formula for the surface area of the cone., Plug the value of the radius into the formula., Plug the value of the slant height into the formula., Calculate the lateral surface area of the cone ((π)(r)(s){\displaystyle (\pi )(r)(s)})...
Step-by-Step Guide
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Step 1: Set up the formula for the surface area of the cone.
The formula is SA=(π)(r)(s)+(π)(r2){\displaystyle {\text{SA}}=(\pi )(r)(s)+(\pi )(r^{2})}, where SA{\displaystyle {\text{SA}}} equals the surface area of the cone, r{\displaystyle r} equals the length of the radius of the cone’s base, and s{\displaystyle s} equals the slant height of the cone.The total surface area of a cone is equal to the sum of the lateral surface area ((π)(r)(s){\displaystyle (\pi )(r)(s)}) and the base area ((π)(r2){\displaystyle (\pi )(r^{2})}), since the base of a cone is a circle.
The slant height is the diagonal distance from the top vertex of the cone to the edge of the base.Make sure you don’t confuse the “slant height” with the “height,” which is the perpendicular distance between the top vertex to the base. -
Step 2: Plug the value of the radius into the formula.
This length should be given, or you should be able to measure it.
Make sure you substitute for both r{\displaystyle r} variables in the formula.
For example, if the radius of the base of a cone is 5 cm, your formula will look like this:
SA=(π)(5)(s)+(π)(52){\displaystyle {\text{SA}}=(\pi )(5)(s)+(\pi )(5^{2})}. , This length should be given, or you should be able to measure it.
For example, if the slant height of a cone is 10 cm, your formula will look like this:
SA=(π)(5)(10)+(π)(52){\displaystyle {\text{SA}}=(\pi )(5)(10)+(\pi )(5^{2})}. , To do this, multiply the radius, slant height, and π{\displaystyle \pi }.
If you are not using a calculator, use
3.14 as the value of π{\displaystyle \pi }.
For example:
SA=(π)(5)(10)+(π)(52){\displaystyle {\text{SA}}=(\pi )(5)(10)+(\pi )(5^{2})}SA=(3.14)(5)(10)+(π)(52){\displaystyle {\text{SA}}=(3.14)(5)(10)+(\pi )(5^{2})}SA=157+(π)(52){\displaystyle {\text{SA}}=157+(\pi )(5^{2})} , To do this, square the radius of the base, then multiply by π{\displaystyle \pi }.
If you are not using a calculator, use
3.14 as the value of π{\displaystyle \pi }.
For example:
SA=157+(π)(52){\displaystyle {\text{SA}}=157+(\pi )(5^{2})}SA=157+(3.14)(25){\displaystyle {\text{SA}}=157+(3.14)(25)}SA=157+78.5{\displaystyle {\text{SA}}=157+78.5} , This will give you the total surface area of the cone, in square units.
For example:
SA=157+78.5=235.5{\displaystyle {\text{SA}}=157+78.5=235.5}So, the surface area of a cone with a radius of 5 cm and a slant height of 10 cm is
235.5 square centimeters. -
Step 3: Plug the value of the slant height into the formula.
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Step 4: Calculate the lateral surface area of the cone ((π)(r)(s){\displaystyle (\pi )(r)(s)}).
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Step 5: Calculate the area of the cone’s base ((π)(r2){\displaystyle (\pi )(r^{2})}).
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Step 6: Add the lateral surface area and the base area of the cone.
Detailed Guide
The formula is SA=(π)(r)(s)+(π)(r2){\displaystyle {\text{SA}}=(\pi )(r)(s)+(\pi )(r^{2})}, where SA{\displaystyle {\text{SA}}} equals the surface area of the cone, r{\displaystyle r} equals the length of the radius of the cone’s base, and s{\displaystyle s} equals the slant height of the cone.The total surface area of a cone is equal to the sum of the lateral surface area ((π)(r)(s){\displaystyle (\pi )(r)(s)}) and the base area ((π)(r2){\displaystyle (\pi )(r^{2})}), since the base of a cone is a circle.
The slant height is the diagonal distance from the top vertex of the cone to the edge of the base.Make sure you don’t confuse the “slant height” with the “height,” which is the perpendicular distance between the top vertex to the base.
This length should be given, or you should be able to measure it.
Make sure you substitute for both r{\displaystyle r} variables in the formula.
For example, if the radius of the base of a cone is 5 cm, your formula will look like this:
SA=(π)(5)(s)+(π)(52){\displaystyle {\text{SA}}=(\pi )(5)(s)+(\pi )(5^{2})}. , This length should be given, or you should be able to measure it.
For example, if the slant height of a cone is 10 cm, your formula will look like this:
SA=(π)(5)(10)+(π)(52){\displaystyle {\text{SA}}=(\pi )(5)(10)+(\pi )(5^{2})}. , To do this, multiply the radius, slant height, and π{\displaystyle \pi }.
If you are not using a calculator, use
3.14 as the value of π{\displaystyle \pi }.
For example:
SA=(π)(5)(10)+(π)(52){\displaystyle {\text{SA}}=(\pi )(5)(10)+(\pi )(5^{2})}SA=(3.14)(5)(10)+(π)(52){\displaystyle {\text{SA}}=(3.14)(5)(10)+(\pi )(5^{2})}SA=157+(π)(52){\displaystyle {\text{SA}}=157+(\pi )(5^{2})} , To do this, square the radius of the base, then multiply by π{\displaystyle \pi }.
If you are not using a calculator, use
3.14 as the value of π{\displaystyle \pi }.
For example:
SA=157+(π)(52){\displaystyle {\text{SA}}=157+(\pi )(5^{2})}SA=157+(3.14)(25){\displaystyle {\text{SA}}=157+(3.14)(25)}SA=157+78.5{\displaystyle {\text{SA}}=157+78.5} , This will give you the total surface area of the cone, in square units.
For example:
SA=157+78.5=235.5{\displaystyle {\text{SA}}=157+78.5=235.5}So, the surface area of a cone with a radius of 5 cm and a slant height of 10 cm is
235.5 square centimeters.
About the Author
Samuel Jones
Samuel Jones is an experienced writer with over 12 years of expertise in educational content. Passionate about sharing practical knowledge, Samuel creates easy-to-follow guides that help readers achieve their goals.
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