Finding the Volume of a Cone with Diameter and Height: Step-by-Step Guide

To find the volume of a cone when its diameter and height are given, one must use the formula for the volume of a cone, which is a fundamental concept in geometry. The formula for the volume of a cone is given by:

V 1/3 × π × r2 × h

where:

V is the volume of the cone π (pi) is the mathematical constant approximately equal to 3.14159 r is the radius of the circular base of the cone (which is half of the diameter) h is the height of the cone

Given the diameter and height of a cone, the first step is to determine the radius. The formula for the radius, given the diameter, is:

r d/2

where d is the diameter.

Let's solve the problem step-by-step using an example.

Example: A Cone with Diameter 5 cm and Height 13 cm

Step 1: Calculate the Radius

The diameter is 5 cm, so the radius is:

r 5 cm / 2 2.5 cm

Step 2: Substitute the Values into the Volume Formula

With the radius (r 2.5 cm) and height (h 13 cm), substitute these values into the volume formula:

V 1/3 × π × 2.52 × 13 cm

Step 3: Calculate the Radius Squared

First, calculate the radius squared:

2.52 6.25 cm2

Step 4: Calculate the Volume

Now, substitute the squared radius into the formula and calculate the volume:

V 1/3 × π × 6.25 cm2 × 13 cm (1/3) × π × 81.25 cm3

Using an approximate value for π (3.14), we get:

V ≈ (81.25 × 3.14) / 3 255.3125 / 3 ≈ 85.1041667 cm3

Thus, the volume of the cone is approximately 85.1 cm3.

Alternate Method Using Pythagoras Theorem

Another method can be used if the altitude (height) of the cone and the diameter of the base are provided. According to the given information:

The altitude and the radius (diameter / 2) form a right triangle, where the altitude is the height of the triangle. Using the Pythagorean theorem:

h2 132 – 52

Calculate the height:

h2 169 – 25 144

Then,

h √144 12 cm

The volume can be calculated as:

V (1/3) × 22/7 × 25 cm2 × 12 cm (1/3) × 22/7 × 300 cm3 2200 / 7 ≈ 314.2857 cm3

So, the volume of the cone is approximately 314.3 cm3.

Conclusion

Understanding and applying the volume of a cone formula is a valuable skill for solving geometric problems. Whether using basic multiplication or utilizing the Pythagorean theorem, both methods provide a means to accurately calculate the volume of a cone when its diameter and height are known.