A cone has a base radius of 5 cm and a slant height of 13 cm. What is the total surface area of the cone? (Use \(\pi \approx 3.14\))
["# Total Surface Area of a Cone: A Step-by-Step Calculation with Radius 5 cm and Slant Height 13 cm", "Understanding the total surface area of a cone is essential in geometry, especially when studying 3D shapes in math and engineering. In this article, we’ll calculate the total surface area of a cone with a base radius of 5 cm and a slant height of 13 cm using the approximate value (\pi \approx 3.14).", "## What Is a Cone?", "A cone is a 3D geometric figure with a circular base and a single vertex (apex) connected by a slant surface. It has two key measurements involved in surface area calculation:", "- Base radius ((r)): The radius of the circular base\n- Slant height ((l)): The distance from the base edge to the apex along the side surface", "## Given Values", "- Radius (r = 5) cm\n- Slant height (l = 13) cm", "## Formula for Total Surface Area of a Cone", "The total surface area (TSA) of a cone consists of two parts:", "1. Base area – the area of the circular base\n2. Lateral (side) surface area – the curved surface area", "Mathematically, the total surface area is:", "[\n\ ext{TSA} = \pi r^2 + \pi r l\n]", "Or factored:", "[\n\ ext{TSA} = \pi r (r + l)\n]", "## Step-by-Step Calculation", "### Step 1: Calculate base area", "[\n\ ext{Base area} = \pi r^2 = 3.14 \ imes (5)^2 = 3.14 \ imes 25 = 78.5 , \ ext{cm}^2\n]", "### Step 2: Calculate lateral surface area", "[\n\ ext{Lateral area} = \pi r l = 3.14 \ imes 5 \ imes 13 = 3.14 \ imes 65 = 204.1 , \ ext{cm}^2\n]", "### Step 3: Add both areas to get total surface area", "[\n\ ext{TSA} = 78.5 + 204.1 = 282.6 , \ ext{cm}^2\n]", "## Summary", "For a cone with a base radius of 5 cm and a slant height of 13 cm:", "- Base area = 78.5 cm²\n- Lateral surface area = 204.1 cm²\n- Total surface area = 282.6 cm²", "This calculation highlights how geometry helps us determine shaping components in real-world applications, from construction to design.", "---", "Key takeaway: Total surface area of a cone with radius 5 cm and slant height 13 cm is approximately 282.6 cm² using (\pi \approx 3.14). Understanding this formula enables better comprehension of 3D surfaces in academic and practical contexts."]
