Calculating the Surface Area of a Right Circular Cone: A Practical Guide

In geometry, the surface area of a right circular cone is a measure of the total area of its exterior. This includes the area of the base and the lateral surface. The right circular cone in question has a base with a radius of 3.5 cm and a height of 8 cm. To accurately calculate its surface area, we need to apply the formulas for the lateral surface area (LSA) and the base area, and then sum these values.

Lateral Surface Area (LSA)

The Lateral Surface Area (LSA) of a cone, given by the formula πr√(r^2 h^2), is essential to calculate first. Here, r represents the base radius (3.5 cm) and h represents the height (8 cm).

Step-by-Step Calculation of LSA:

LSA π(3.5)√(3.5^2 8^2)

LSA π(3.5)√(12.25 64)

LSA π(3.5)√76.25

LSA ≈ 45.02 cm2

Base Area

The base area of a cone is simply the area of the circular base, given by the formula πr^2. Here, the base radius is 3.5 cm.

Step-by-Step Calculation of Base Area:

Base Area π(3.5)^2

Base Area ≈ 38.48 cm2

Total Surface Area (TSA)

The total surface area (TSA) is the sum of the Lateral Surface Area and the Base Area. Hence, we get:

Total Surface Area (TSA)

TSA LSA Base Area

TSA ≈ 45.02 38.48

TSA ≈ 83.50 cm2

Therefore, the total surface area of the cone is approximately 83.50 cm2 when considering the radius as 3.5 cm.

Alternative Consideration: Base Diameter as 3 cm

If we consider the base diameter to be 3 cm, then the radius will be 1.5 cm. Let's recalculate the surface area based on this radius.

Lateral Surface Area (LSA)

LSA π(1.5)√(1.5^2 8^2)

LSA π(1.5)√(2.25 64)

LSA π(1.5)√66.25

LSA ≈ 25.01 cm2

Base Area

Base Area π(1.5)^2

Base Area ≈ 7.07 cm2

Total Surface Area (TSA)

TSA ≈ 25.01 7.07

TSA ≈ 32.08 cm2

In this scenario, the total surface area of the cone is approximately 32.08 cm2.

Note: The brightness of the cone's surface does not affect its surface area. The term "bright" in the original query is irrelevant to the geometric calculation.