Math Problem Statement

Work out the total surface area of the cone shown below. Give your answer to 1 d.p.

Solution

Step-by-Step Solution:

1. Given Information:

  • Radius of the base r=6mr = 6 \, \text{m}
  • Slant height l=23ml = 23 \, \text{m}

The total surface area of a cone includes:

  1. The curved surface area (CSA):
    CSA=πrl\text{CSA} = \pi r l
  2. The base area:
    Base Area=πr2\text{Base Area} = \pi r^2

The total surface area (TSA) is the sum of the two: TSA=πrl+πr2\text{TSA} = \pi r l + \pi r^2

2. Calculate Each Component:

  1. Curved Surface Area (CSA): CSA=πrl=π623=138πm2\text{CSA} = \pi r l = \pi \cdot 6 \cdot 23 = 138 \pi \, \text{m}^2 Approximating π\pi as 3.14163.1416: CSA=1383.1416=433.54m2\text{CSA} = 138 \cdot 3.1416 = 433.54 \, \text{m}^2

  2. Base Area: Base Area=πr2=π62=36πm2\text{Base Area} = \pi r^2 = \pi \cdot 6^2 = 36 \pi \, \text{m}^2 Approximating π\pi: Base Area=363.1416=113.10m2\text{Base Area} = 36 \cdot 3.1416 = 113.10 \, \text{m}^2

  3. Total Surface Area (TSA): TSA=CSA+Base Area=433.54+113.10=546.64m2\text{TSA} = \text{CSA} + \text{Base Area} = 433.54 + 113.10 = 546.64 \, \text{m}^2

3. Final Answer:

The total surface area of the cone, rounded to 1 decimal place, is: 546.6m2\boxed{546.6 \, \text{m}^2}

Let me know if you’d like further clarification or help!


Related Questions:

  1. What is the formula to find the slant height of a cone if not given directly?
  2. How would the total surface area change if the radius doubled but the slant height remained the same?
  3. How can the curved surface area be derived geometrically?
  4. What is the difference between the surface area and the lateral area of a cone?
  5. How is the volume of a cone related to its surface area?

Tip:

Always double-check unit consistency and rounding, especially when working with real-world measurements!

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Math Problem Analysis

Mathematical Concepts

Geometry
Surface Area of Solids

Formulas

Curved Surface Area of a cone = πrl
Base Area of a cone = πr²
Total Surface Area = πrl + πr²

Theorems

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Suitable Grade Level

Grades 8-10