Cone Surface Area Calculator including Base

The surface area of a cone consists of two parts: the base area and the lateral (side) surface area. The total surface area ($�$) of a cone is the sum of the area of its base (${�}_{����}$) and the lateral surface area (${�}_{�������}$).

Formula

The total surface area of a cone is given by:

$�={�}_{����}+{�}_{�������}$

Where:

• ${�}_{����}=�{�}^2$ (Area of the base circle, with $�$ being the radius of the base)
• ${�}_{�������}=���$ (Lateral surface area, with $�$ being the slant height of the cone)

So, the formula for the total surface area of a cone can be expressed as:

$�=�{�}^2+���$

Or simplified to:

$�=��(�+�)$

Components of the Formula

• $�$: Radius of the base of the cone.
• $�$: Slant height of the cone, which is the distance from the top of the cone (vertex) to any point on the edge of the base circle. The slant height can be found using the Pythagorean theorem if you know the height of the cone ($ℎ$) and the radius ($�$):

$�=\sqrt{{�}^2+{ℎ}^2}$

• $ℎ$: Height of the cone, which is the perpendicular distance from the base to the vertex.

Example Calculation

If you have a cone with a base radius of 3 cm and a slant height of 5 cm, you can calculate its surface area as follows:

$�=�\times 3\times (3+5)$

$�=�\times 3\times 8$

$�=24�$

$�\approx 75.4{\text{cm}}^2$

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