The total surface area of a cone with a diameter of 12 cm and a height of 8 cm is 301.44 cm².
Understanding Cone Geometry
To determine the total surface area of a cone, we need to consider both its base area and its lateral (curved) surface area. A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.
The key dimensions for calculating a cone's surface area are:
- Radius (r): Half of the diameter of the circular base.
- Height (h): The perpendicular distance from the apex to the center of the base.
- Slant Height (l): The distance from any point on the circumference of the base to the apex, along the surface of the cone.
Formula for Total Surface Area of a Cone
The total surface area (TSA) of a cone is calculated using the formula:
TSA = πr² + πrl
Where:
- πr² represents the area of the circular base.
- πrl represents the lateral surface area of the cone.
Alternatively, the formula can be factored as:
TSA = πr(r + l)
Step-by-Step Calculation
Let's break down the calculation for a cone with a diameter of 12 cm and a height of 8 cm.
1. Determine the Radius (r)
The diameter is given as 12 cm. The radius is half of the diameter.
- Diameter = 12 cm
- Radius (r) = Diameter / 2 = 12 cm / 2 = 6 cm
2. Calculate the Slant Height (l)
The height (h), radius (r), and slant height (l) form a right-angled triangle. We can use the Pythagorean theorem (a² + b² = c²) to find the slant height:
- l² = r² + h²
- l² = (6 cm)² + (8 cm)²
- l² = 36 cm² + 64 cm²
- l² = 100 cm²
- Slant Height (l) = √100 cm = 10 cm
3. Compute the Total Surface Area (TSA)
Now that we have the radius (r) and slant height (l), we can plug these values into the TSA formula. We will use the approximation of π ≈ 3.14 for practical calculation.
TSA = πr(r + l)
- TSA = 3.14 × 6 cm × (6 cm + 10 cm)
- TSA = 3.14 × 6 cm × 16 cm
- TSA = 3.14 × 96 cm²
- TSA = 301.44 cm²
Summary of Dimensions and Results
To enhance clarity, here's a table summarizing the input values and the calculated result:
| Parameter | Value | Unit |
|---|---|---|
| Diameter | 12 | cm |
| Radius (r) | 6 | cm |
| Height (h) | 8 | cm |
| Slant Height (l) | 10 | cm |
| Total Surface Area | 301.44 | cm² |
This calculation demonstrates how to find the total surface area by first determining the radius and slant height from the given dimensions, then applying the standard formula. For further exploration of cone properties, you can refer to resources like Math is Fun on Cones.
