Category: Construction Calculators, Construction Math

Conical Roof Surface Area Calculator and Math

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Jack Gray is an independent commercial roof consultant with over 25 years of experience in the roofing industry. He's trying to make the roofing information you find on the internet better, one article at a time.
A Studio Ghibli style image of a house that features a conical roof.

Introduction

Conical roofs are distinctive roofing structures shaped like a cone. A cone-shaped roof is circular at its base and terminates in a point or sometimes a finial of some sort. They are often used on towers, turrets, gazebos, and small pavilions. Cone roofs are often a feature on Victorian/Queen Anne-style homes.

The surface area of most roofs can be calculated using basic length, width, and slope values. Those calculations should be familiar to anyone involved in roofing, but cone roofs don’t use the same factors. I thought I’d put together a calculator to help with calculating material requirements for dome roofs.

Conical Roof Calculator

Conical Roof Calculator

Results:

Roof Surface Area (ft²):
Cone Base Radius (ft):
Slant Height (Surface Length) (ft):
Cone Base Circumference (ft):
Footprint Area (ft²):
Cone Volume (ft³):

Conical Roof Math Explanation

Calculating the Base Radius
The cone’s base is a circle. The calculator asks for the diameter of this circle. Since the radius is half of the diameter, it uses the formula: radius = diameter / 2.

Finding the Slant Height
The slant height is the distance along the roof from the edge of the base to the tip of the cone. Imagine a right triangle where one leg is the vertical height of the roof and the other is the base radius. The slant height is the hypotenuse of this triangle, calculated using the Pythagorean theorem: slant height = √(radius2 + height2). This gives the true distance along the roof’s surface, which is longer than the vertical rise.

Computing the Roof Surface Area (Lateral Surface Area)
The area of the roof (the curved surface of the cone) is computed by multiplying the base radius by the slant height and then by the constant pi (or π), approximately 3.14159. The formula is: lateral surface area = π × radius × slant height. This tells you how much material is needed to cover the roof’s curved surface.

Determining the Base Circumference
The circumference is the distance around the edge of the circular base. It is found using the formula: circumference = 2 × π × radius. This measurement is useful if you need to know the length around the base of the roof.

Calculating the Base Area (Footprint)
To determine the area of the circular base (the roof's footprint), the calculator uses the formula for the area of a circle: base area = π × radius2.

Computing the Volume of the Cone
The volume represents the space enclosed by the cone. The formula for the volume of a cone is: volume = (1 / 3) × π × radius2 × height. This calculation takes one third of the product of the base area and the vertical height.

In both the USA (feet) and Metric (meters) versions of the calculator, the same math is used. The only difference is the unit of measurement for the inputs and the results. Each formula is based on the basic geometry of a cone.