Area of a Frustum of a Cone Calculator

Find the surface area of a frustum — the side plus the two circular ends.

How the Math Works

A frustum of a cone is formed when the top of a cone is cut off by a plane parallel to its base, creating two circular faces of different radii. The lateral surface area is calculated using the formula π(R + r)·slant, where R is the radius of the larger base, r is the radius of the smaller top base, and slant is the slant height (the diagonal distance between the edges of the two bases). The total surface area adds the areas of both circular bases (πR² and πr²) to the lateral area, giving the complete surface measurement of the frustum shape.

Practical Applications

To use this calculator effectively, you need three measurements: the radius of the bottom base (R), the radius of the top base (r), and the slant height of the frustum. You can find the slant height using the Pythagorean theorem if you have the vertical height (h) of the frustum: slant = √((R-r)² + h²). Simply input these values into the calculator to instantly obtain both the lateral surface area (the curved part only) and the total surface area (including the top and bottom circles) for any conical frustum shape.

Day-to-Day Use

This calculation has practical applications in everyday scenarios like determining how much material you need for construction projects involving truncated cone shapes, such as decorative columns, flowerpot volumes, or architectural elements. Homeowners might use it to calculate paint or wallpaper needs for conical architectural features, while craft enthusiasts can determine fabric requirements for making items like party hats or conical storage containers. It's also useful in manufacturing when working with materials like metal, plastic, or fabric that need to be formed into frustum shapes.

Worked example

R = 5, r = 3, h = 8 → lateral ≈ 207.4.

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