Surface area of a cone refers to the total area that covers the outer surface of the cone. A cone consists of two main parts: the curved surface and the base.
The base is a circle, and the curved surface wraps around the base. The surface area of a cone can be calculated by adding the area of the base (a circle) and the area of the curved surface (a sector). The total surface area of cone is important in real-world applications like packaging, construction, and design, as it helps in determining the amount of material required to cover a cone-shaped object. The formula for the surface area combines the radius of the base and the slant height, providing a simple yet effective way to calculate the total area of a cone.TSA=𝜋𝑟(𝑙+𝑟)
Where:Also Check: Area of Rectangle
TSA=𝜋𝑟(𝑙+𝑟)
Where : 𝑟 is the radius of the base of the cone, 𝑙 is the slant height of the cone. Curved Surface Area (CSA) of a Cone The curved surface area (CSA) of the cone, which refers to just the slanted area of the cone (excluding the base), is:CSA=𝜋𝑟𝑙
Where:𝑙 is: 𝑙=𝑟2+ℎ2
Where : 𝑟 is the radius of the base, ℎ is the height of the cone, 𝑙 is the slant height.Example Calculation
Let's say you have a cone with: Radius 𝑟=4 cm, Height ℎ= 3cm. First, calculate the slant height 𝑙:𝑙=𝑟2+ℎ 2=42+32=16+9=25=5cm l=r2+h2=4 2+3 2=16+9=25=5 cm Then, calculate the curved surface area (CSA):CSA=𝜋𝑟𝑙=𝜋×4×5=20𝜋≈62.83cm2
To find the total surface area (TSA), use the formula:TSA=𝜋𝑟(𝑙+𝑟)=𝜋×4×(5+4)=𝜋×4×9=36𝜋≈113.10cm2
Thus, the total surface area of the cone is approximately 113.10 cm².Also Check: Simple Interest
Area of the sector = (Arc length × radius) / 2 = ((2πr) × l) / 2 = πrl
Thus, the curved surface area of a cone is given by the formula S = πrl, where 'r' is the radius and 'l' is the slant height.Surface Area Type | Formula | Units |
Curved Surface Area (S) | πr√(r2 + h2) | square units |
Total Surface Area (T) | πr2 + πr√(r2 + h2) | square unit |
Area of sector= 21×arc length×radius of sector
In this case, the arc length of the sector is the circumference of the cone's base, i.e., 2𝜋𝑟, and the radius of the sector is the slant height 𝑙. Therefore, the curved surface area (CSA) of the cone is:CSA=𝜋𝑟𝑙
Area of the base=𝜋𝑟2
TSA=CSA+Area of the base
Substituting the formulas for CSA and the area of the base: TSA=𝜋𝑟𝑙+𝜋𝑟2 This simplifies to: TSA=𝜋𝑟(𝑙+𝑟) Thus, the total surface area of a cone is: TSA=𝜋𝑟(𝑙+𝑟)Also Check: Diameter of a Circle
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