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Last updated on September 13, 2025
Area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of a cone.
A cone is a three-dimensional shape that tapers smoothly from a flat circular base to a point called the apex or vertex. The surface area of a cone is the total area of its lateral (curved) surface and its base.
To find the surface area of a cone, we use the formula: Lateral Area = π × r × l, where r is the radius of the base and l is the slant height.
The total surface area is given by: Total Surface Area = π × r × l + π × r². Now let’s see how the formula is derived.
Derivation of the formula: The lateral surface area is calculated by unwrapping the cone into a sector of a circle, which is part of a larger circle with a radius equal to the slant height l. The arc length of the sector is equal to the circumference of the base of the cone, i.e., 2πr.
Thus, the lateral surface area of the cone = π × r × l. Adding the area of the base, which is a circle of radius r, gives us the total surface area: Total Surface Area = π × r × l + π × r².
We can find the area of a cone using the following method:
Method Using Radius and Slant Height
If the radius r and slant height l are given, we find the surface area of the cone using the formulas: Lateral Area = π × r × l and Total Surface Area = π × r × l + π × r². For example, if r is 3 cm and l is 5 cm.
What will be the total surface area of the cone? Total Surface Area = π × r × l + π × r² = π × 3 × 5 + π × 3² = 15π + 9π = 24π cm² The total surface area of the cone is approximately 75.36 cm² (using π ≈ 3.14).
We measure the area of a cone in square units. The measurement depends on the system used: In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²). In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).
Since a cone has a circular base and a curved surface, there are specific cases for calculating the area:
Case 1: Surface Area Using Radius and Slant Height If the radius and slant height are given, use the formula Total Surface Area = π × r × l + π × r².
Case 2: Use of Height If the height h of the cone is given instead of the slant height, use the Pythagorean theorem to find the slant height: l = √(r² + h²).
To ensure correct results while calculating the area of a cone, here are some tips and tricks: The slant height is different from the perpendicular height of the cone.
The total surface area includes both the lateral surface area and the base area. Use the Pythagorean theorem to find the slant height if only the height and radius are given.
It is common to make mistakes while finding the area of a cone. Let’s take a look at some common mistakes.
The radius of a cone's base is 6 m and the slant height is 10 m. What will be the total surface area?
We will find the total surface area as approximately 301.44 m².
Here, the radius r is 6 m and the slant height l is 10 m.
The total surface area = π × r × l + π × r² = π × 6 × 10 + π × 6² = 60π + 36π = 96π ≈ 301.44 m².
What will be the total surface area of the cone if the radius is 4 cm and the slant height is 9 cm?
We will find the total surface area as approximately 163.28 cm².
Using the formula
Total Surface Area = π × r × l + π × r², with r = 4 cm and l = 9 cm, we get:
Total Surface Area = π × 4 × 9 + π × 4² = 36π + 16π = 52π ≈ 163.28 cm².
The total surface area of a cone is 282.6 cm² and the radius of the base is 6 cm. Find the slant height.
We find the slant height as approximately 8 cm.
Using the formula:
Total Surface Area = π × r × l + π × r², substitute the known values to find the slant height l: 282.6 = π × 6 × l + π × 6².
Solving for l gives us l = (282.6 - 36π) / (6π) ≈ 8 cm.
Find the total surface area of the cone if its radius is 7 cm and the perpendicular height is 24 cm.
We will find the total surface area as approximately 791.28 cm².
First, find the slant height using the Pythagorean theorem: l = √(r² + h²) = √(7² + 24²) = 25 cm.
Then use the formula Total Surface Area = π × r × l + π × r² = π × 7 × 25 + π × 7² = 175π + 49π = 224π ≈ 791.28 cm².
Help Jane find the total surface area of the cone if the base radius is 5 m and the slant height is 12 m.
We will find the total surface area as approximately 267.94 m².
The base radius is 5 m and the slant height is 12 m.
Using the formula Total Surface Area = π × r × l + π × r², we have:
Total Surface Area = π × 5 × 12 + π × 5² = 60π + 25π = 85π ≈ 267.94 m².
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
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