Surface Area of a Cone Formula

The surface area of a cone includes the area of its curved surface and, if it has a base, the area of the circular base. Calculating the surface area of a cone requires knowledge of its radius, slant height, and sometimes its height, depending on the given measurements.

Here are all the formulas for calculating the surface area of a cone, with detailed explanations and examples for each method.

Formulas for the Surface Area of a Cone

Using Radius and Slant Height:

If the radius \( r \) and the slant height \( s \) of the cone are known,

The total surface area \( S \) (including the base) can be calculated with the formula:

\( S = \pi r (r + s) \)

Using Diameter and Slant Height:

If the diameter \( d \) and the slant height \( s \) are known,

The total surface area \( S \) can be calculated as:

\( S = \dfrac{\pi d (d/2 + s)}{2} \)

Using Radius and Height:

If the radius \( r \) and height \( h \) of the cone are known,

The total surface area \( S \) can be calculated by first finding the slant height \( s = \sqrt{r^2 + h^2} \) and then using the formula:

\( S = \pi r (r + \sqrt{r^2 + h^2}) \)

In these formulas:

  • \( r \) is the radius of the base of the cone
  • \( d \) is the diameter of the base of the cone, where \( d = 2r \)
  • \( s \) is the slant height of the cone
  • \( h \) is the vertical height of the cone
  • \( \pi \) (Pi) is approximately equal to 3.14159

Detailed Explanation of Each Formula

1. Formula for Surface Area Using Radius and Slant Height

The formula \( S = \pi r (r + s) \) calculates the surface area of a cone by adding the area of the base \( \pi r^2 \) to the area of the curved surface \( \pi r s \). This method is commonly used when both the radius and slant height of the cone are known.

Example 1: Calculating Surface Area with Radius and Slant Height

Problem: Find the surface area of a cone with a radius of \( r = 3 \, \text{cm} \) and a slant height of \( s = 5 \, \text{cm} \).

Solution:

  1. Write down the formula: \( S = \pi r (r + s) \).
  2. Substitute \( r = 3 \) and \( s = 5 \): \( S = \pi \times 3 \times (3 + 5) \).
  3. Calculate inside the parentheses: \( S = \pi \times 3 \times 8 \).
  4. Multiply: \( S = 24\pi \approx 75.4 \, \text{cm}^2 \) (using \( \pi \approx 3.14159 \)).

The surface area of the cone is approximately \( 75.4 \, \text{cm}^2 \).

2. Formula for Surface Area Using Diameter and Slant Height

The formula \( S = \dfrac{\pi d (d/2 + s)}{2} \) calculates the surface area by using the diameter of the base instead of the radius. This formula is convenient if only the diameter and slant height are given.

Example 2: Calculating Surface Area with Diameter and Slant Height

Problem: A cone has a diameter of \( d = 6 \, \text{cm} \) and a slant height of \( s = 8 \, \text{cm} \). Find its surface area.

Solution:

  1. Write down the formula: \( S = \dfrac{\pi d (d/2 + s)}{2} \).
  2. Substitute \( d = 6 \) and \( s = 8 \): \( S = \dfrac{\pi \times 6 \times (3 + 8)}{2} \).
  3. Calculate inside the parentheses: \( S = \dfrac{\pi \times 6 \times 11}{2} \).
  4. Multiply and divide: \( S = \dfrac{66\pi}{2} = 33\pi \approx 103.67 \, \text{cm}^2 \).

The surface area of the cone is approximately \( 103.67 \, \text{cm}^2 \).

3. Formula for Surface Area Using Radius and Height

The formula \( S = \pi r (r + \sqrt{r^2 + h^2}) \) calculates the surface area by finding the slant height \( s = \sqrt{r^2 + h^2} \) using the radius and height. This approach is useful if the radius and height are given, but the slant height is not.

Example 3: Calculating Surface Area with Radius and Height

Problem: A cone has a radius of \( r = 4 \, \text{cm} \) and a height of \( h = 6 \, \text{cm} \). Find its surface area.

Solution:

  1. Write down the formula: \( S = \pi r (r + \sqrt{r^2 + h^2}) \).
  2. Substitute \( r = 4 \) and \( h = 6 \): \( S = \pi \times 4 \times \left(4 + \sqrt{4^2 + 6^2}\right) \).
  3. Calculate \( r^2 \) and \( h^2 \): \( S = \pi \times 4 \times (4 + \sqrt{16 + 36}) \).
  4. Simplify inside the square root: \( S = \pi \times 4 \times (4 + \sqrt{52}) \).
  5. Approximate \( \sqrt{52} \approx 7.21 \): \( S = \pi \times 4 \times (4 + 7.21) = \pi \times 4 \times 11.21 \).
  6. Multiply: \( S = 44.84\pi \approx 140.83 \, \text{cm}^2 \).

The surface area of the cone is approximately \( 140.83 \, \text{cm}^2 \).

These examples demonstrate how to calculate the surface area of a cone using different known measurements, whether it’s the radius and slant height, diameter and slant height, or radius and height.