How to Find Surface Area of a Cone With Slant Height

Surface Area of Cone

The surface area of a cone is the amount of area occupied by the surface of a cone. A cone is a 3-D shape that has a circular base. This means the base is made up of a radius or diameter. The distance between the center of the base and the topmost part of the cone (of course, in the case of ice cream, this portion is at the bottom) is the height of the cone. Like all three-dimensional shapes, you will learn how to calculate the surface area of a cone in this article.

1.	What is the Surface Area of Cone?
2.	Formula of Surface Area of Cone
3.	Derivation of Surface Area of Cone
4.	FAQs on Surface Area of Cone

What is the Surface Area of Cone?

The area occupied by the surface/boundary of a cone is known as the surface area of a cone. It is always measured in square units. Stacking many triangles and rotating them around an axis gives the shape of a cone. As it has a flat base, thus it has a total surface area as well as a curved surface area. We can classify a cone as a right circular cone or an oblique cone. The vertex in the right circular cone is usually vertically above the center of the base whereas the vertex of the cone in an oblique cone is not vertically above the center of the base.

Surface Area of Cone Formula

As a cone has a curved surface, thus we can express its curved surface area as well as total surface area. A cone has two kinds of surface area:

• Total Surface Area
• Curved Surface Area

If the radius of the base of the cone is "r" and the slant height of the cone is "l", the surface area of a cone is given as:

• Total Surface Area, T = πr(r + l) square units
• Curved Surface Area, S = πrl square units

By applying Pythagoras theorem on the cone, we can find the relation between the surface area of the cone and its height. We know, h² + r² = l² where h is the height of the cone, r is the radius of the base, and l is the slant height of the cone.
⇒ l = √(h² + r²)

Thus,

The total surface area in terms of height can be given as, T = πr(r + l) = T = πr(r + √(h² + r²)).
The curved surface area of the cone in terms of height can be given as S = πrl = πr(√(h² + r²)).

Derivation of Surface Area of Cone

Let us take a cone of height "h", base radius "r", and slant height "l". In order to determine the surface area of cone derivation, we cut the cone open from the center which looks like a sector of a circle (a plane shape).

The curved surface area of the cone can be given by finding the area of the sector by using the formula,
Area of the sector (in terms of length of arc) = (arc length × radius)/ 2 = ((2πr) × l)/2 = πrl.
∴ The curved surface area of a cone, S = πrl units².

The total surface area of cone = area of the base of cone + curved surface area of a cone
⇒ Total surface area of cone = πr² + πrl = πr (r + l).
∴ The total surface area of cone, T = πr (r + l) units²

Example: Find the total surface area and curved surface area of the cone whose radius is 7 inches and slant height is 3 inches. (Use π = 22/7).
We know, the total surface area of the cone is πr (r + l), and the lateral surface area of a cone is πrl. Given that: r = 7 inches, l = 3 inches, and π = 22/7. Thus, total surface area of cone, T = πr (r + l) = (22/7) × 7 × (7 + 3) = (22/7) × 7 × 10 = 22 × 10 = 220 in².
∴ The total surface area of the cone is 220 in².

The curved surface area of the cone, S = πrl = (22/7) × 7 × 3 = 66 in². ∴ The curved surface area of the cone is 66 in².

Let us look at some more examples of the surface area of a cone for a deeper understanding.

Surface Area of Cone Examples

1. Example 1: What is the slant height of the cone if the total surface area of the cone is 616 in² and radius 7 inches?
   Solution: The given dimensions are, the total surface area of cone = 616 in² and the radius of the cone = 7 inches. Let the slant height = x inches.

   Substituting the values in the surface area of the cone formula,
   Total surface area of cone = πr (r + l) = (22/7) × 7 × (7 + x) = 616
   ⇒ 22 × (7 + x) = 616
   ⇒ 7 + x = 28
   ⇒ x = 21 inches
   Answer: The slant height of cone is 21 inches.

2. Example 2: What is the height of cone whose radius is 7 inches and curved surface area is 550 in² . (Use π = 22/7)
   Solution: The given dimensions are, radius of cone = 7 in and curved surface area = 550 in². Let the value of slant height be "l" and height of cone be "h".

   Substituting the values in the curved surface area of the cone formula,
   πrl = (22/7) × 7 × l = 550 in²
   ⇒ 22 × l = 550
   ⇒ l = 550/22 = 25 inches
   l = √(h² + r²)
   ⇒ h = √(l² - r²) = √(25² - 7²) = √576 = 24 inches
   Answer: The height of cone is 24 inches.

3. Example 3: Find the total surface area of a cone with radius 14 units and slant height 8 units. (Use π = 22/7)
   Solution: The given dimensions are, radius of cone (r) = 14 units and slant height (l) = 8 units.

   Substituting the values in the total surface area of the cone formula,
   πr (r + l) = (22/7) × 14 × (14 + 8)
   ⇒ 22 × 2 × 22
   ⇒ 968 square units
   Answer: The total surface area of the cone is 968 square units..

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Practice Questions on Surface Area of Cone

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How to Find Surface Area of a Cone With Slant Height

Source: https://www.cuemath.com/measurement/surface-area-of-cone/

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