What is the area of the curved surface of a right circular cone of radius 3 and height 4?

Verified step by step guidance

1. Step 1: Recall the formula for the curved surface area of a right circular cone, which is given by $A\; =\; \pi rl$, where $r$ is the radius of the base and $l$ is the slant height.
2. Step 2: To find the slant height $l$, use the Pythagorean theorem. The slant height is calculated as $l\; =\; \surd (r²\; +\; h²)$, where $r$ is the radius and $h$ is the height of the cone.
3. Step 3: Substitute the given values of $r\; =\; 3$ and $h\; =\; 4$ into the formula for the slant height: $l\; =\; \surd (3²\; +\; 4²)$. Simplify the expression inside the square root.
4. Step 4: Once the slant height $l$ is determined, substitute $r\; =\; 3$ and the calculated value of $l$ into the curved surface area formula $A\; =\; \pi rl$.
5. Step 5: Simplify the expression for $A$ to find the curved surface area of the cone. Leave the result in terms of $\pi$ if required.

Key Concepts

Surface Area of a Cone

The surface area of a right circular cone consists of two parts: the base area and the lateral (curved) surface area. The formula for the lateral surface area is given by πrl, where r is the radius and l is the slant height. The slant height can be calculated using the Pythagorean theorem, l = √(r² + h²), where h is the height of the cone.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is essential for finding the slant height of the cone in this problem.

Calculating Area

Calculating area involves determining the amount of space within a two-dimensional shape. For a cone, the area of the curved surface is calculated using the formula for lateral surface area. Understanding how to apply this formula correctly, along with the necessary dimensions, is crucial for solving the problem of finding the area of the cone's curved surface.