Surface Area of a Right Circular Cone

The surface area of a right circular cone is the total surface area covered inside it. The right circular cone is one of the most known three-dimension shapes. They can be easily spotted in various places around us such as in ice cream cones, birthday caps, traffic cones, rocket nose caps, etc. Finding the area of a right circular cone is an important skill both in maths and in real-life. This complete guide shows you every formula, derivation, and example you need to learn about this topic.

Table of Contents

• What is a Right Circular Cone
• What is Surface Area of a Right Circular Cone
• Surface Area of a Right Circular Cone Formula
• How to Calculate the Surface Area of a Right Circular Cone
• Solved Examples on Surface Area of a Right Circular Cone

What is a Right Circular Cone

A right circular cone is a solid shape in three dimensions that has a circular base and a pointy top called the apex (or vertex). The cone's axis, which runs from the apex to the center of the base, is perpendicular to the base is called its height ‘h’. This is what makes it a right cone. The line along which the cone can be cut in the slant height is called its slanting height ‘l’. All these dimensions are required to find the surface area of a right circular cone.

What is Surface Area of a Right Circular Cone

The surface area of a right circular cone is the sum of the sector which is the lateral face of the cone and the circle which is its base. As the right circular cone has two surfaces, the curved lateral surface and the flat circular base, so the total surface area is the sum of Curved Surface Area + Base Area.

Surface Area of a Right Circular Cone Formula

The right circular cone has two surfaces: the curved lateral surface and the flat circular base. Therefore, the total surface area of the right circular cone is:
Total Surface Area = Curved Surface Area + Base Area = πrl + πr² = πr(r + l)

Here are some key terms related to the surface area of a right circular cone:

• Radius r: Radius of the circular base of the cone

• Height h: Perpendicular distance from apex to the centre of the base

• Slant height l: Distance from apex to any point on the circumference of the base;
  l = √(r² + h²)

• (π) Pi : Mathematical constant ≈ 22/7 or 3.14159
  
  

How to Calculate the Surface Area of a Right Circular Cone

• To calculate the surface area of a right circular cone first, find its height and radius of the base to find the curved surface area using formula = πrl 

• Then find the base area of the right circular cone using formula: Base Area = πr²

• For example, if h = 7 cm, then r = 4 cm then,
  Surface Area = πrl + πr² = (3.14 × 4 × 7) + 3.14 x 4 x 4 = 87.92 + 50.24 = 138.16 cm²
  
  

Solved Examples on Surface Area of a Right Circular Cone 

Example 1: Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm.

Solution: Curved surface area = πrl

= 22/7 × 7 × 10 cm²

= 220 cm²

Example 2: The height of a cone is 16 cm and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone (Use π = 3.14).

Solution: Here, h = 16 cm and r = 12 cm.

So, from l2=h2+r2 we have l = 162+122cm = 20 cm

So, curved surface area = πrl
= 3.14 × 12 × 20cm2
= 753.6cm2

Further, total surface area = πrl + πr2
= (753.6 + 3.14 × 12 × 12)cm2
= (753.6 + 452.16)cm2
= 1205.76cm2