Surface Area of a Cone

The total surface area of a cone is the sum of the area of its base and the lateral (side) surface.

The lateral surface area of a cone is the area of the lateral or side surface only.

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Since a cone is closely related to a pyramid , the formulas for their surface areas are related.

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Remember, the formulas for the lateral surface area of a pyramid is $\frac{1}{2} p l$ and the total surface area is $\frac{1}{2} p l + B$ .

Since the base of a cone is a circle, we substitute $2 π r$ for $p$ and $π r^{2}$ for $B$ where $r$ is the radius of the base of the cylinder.

So, the formula for the lateral surface area of a right cone is $L . S . A = π r l$ , where $l$ is the slant height of the cone .

Example 1:

Find the lateral surface area of a right cone if the radius is $4$ cm and the slant height is $5$ cm.

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$L . S . A = π (4) (5) = 20 π \approx 62.82 {cm}^{2}$

The formula for the total surface area of a right cone is $T . S . A = π r l + π r^{2}$ .

Example 2:

Find the total surface area of a right cone if the radius is $6$ inches and the slant height is $10$ inches.

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$\begin{array}{l} T . S . A = π (6) (10) + π {(6)}^{2} \\ = 60 π + 36 π \\ = 96 π {inches}^{2} \\ \approx 301.59 {inches}^{2} \end{array}$