The formula of total surface area of a cone is__________.

The formula for the total surface area of a cone combines both the base area and the lateral (side) area. To express this mathematically, the total surface area (TSA) can be calculated using the formula:

Total Surface Area of a Cone

The formula is given by:

TSA = πr(r + l)

In this formula:

• r represents the radius of the base of the cone.
• l is the slant height of the cone.
• π (pi) is approximately 3.14.

Breaking Down the Formula

To understand how this formula works, let’s break it down into its components:

1. Base Area

The base of the cone is a circle, and the area of a circle is calculated using the formula:

Area = πr²

This area represents the flat circular surface at the bottom of the cone.

2. Lateral Surface Area

The lateral surface area is the area of the cone's side. This is where the slant height comes into play. The formula for the lateral surface area is:

Lateral Area = πrl

Here, l is the slant height, which is the distance from the base of the cone to the apex along the side.

Putting It All Together

To find the total surface area, you simply add the base area and the lateral area:

TSA = Base Area + Lateral Area

Substituting the formulas we discussed:

TSA = πr² + πrl

Factoring out πr gives us:

TSA = πr(r + l)

Example Calculation

Let’s say we have a cone with a radius of 3 cm and a slant height of 5 cm. To find the total surface area:

• Calculate the base area: Area = π(3)² = 9π cm²
• Calculate the lateral area: Lateral Area = π(3)(5) = 15π cm²
• Add them together: TSA = 9π + 15π = 24π cm²

Thus, the total surface area of the cone is approximately 75.4 cm² when using π ≈ 3.14.

This formula is quite useful in various applications, such as in engineering and architecture, where understanding the surface area of conical shapes is essential. By mastering this concept, you can tackle a variety of problems involving cones with confidence!