Calculate the surface area of cones

Surface area of a right circular cone is the sum of its circular base, $\pi r^2$ , and curved (lateral) surface, $\pi r\ell$ , giving $S=\pi r(r+\ell)$ , where $r$ is the radius and $\ell$ is the slant height, not the perpendicular height. The slant height may be found from the radius and perpendicular height using the Pythagorean theorem; results are expressed in square units, with appropriate rounding or exact forms. Composite and oblique-cone surface areas are not included.

Detailed Explanation: Calculate the surface area of cones

To find the surface area of a right circular cone, add:

the area of the circular base: $\pi r^2$
the curved surface area: $\pi r\ell$

So the formula is

S=\pi r^2+\pi r\ell=\pi r(r+\ell),

where $r$ is the radius and $\ell$ is the slant height. The slant height runs along the side of the cone; it is not the perpendicular height.

Example: A cone has radius $5$ cm and perpendicular height $12$ cm. Find its total surface area.

Step 1: Find the slant height.

The radius, perpendicular height, and slant height form a right triangle. Use the Pythagorean theorem:

\ell^2=r^2+h^2

\ell^2=5^2+12^2=25+144=169

\ell=13\text{ cm}

Step 2: Substitute into the surface area formula.

S=\pi r(r+\ell)

S=\pi(5)(5+13)

S=90\pi\text{ cm}^2

Step 3: Give an approximate answer if needed.

90\pi\approx 282.7\text{ cm}^2

The surface area of the cone is therefore $\boxed{90\pi\text{ cm}^2}$ , or approximately $\boxed{282.7\text{ cm}^2}$ .

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Surface Area - Cone - Image to Decimal

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Calculate the surface area of cones

Detailed Explanation: Calculate the surface area of cones

Learn by doing: Calculate the surface area of cones

Practice with unlimited practice problems

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Surface Area - Cone - Image to Decimal