Surface Area of a Cone: Formula Derivation and Application

Deriving the Lateral Surface Area Formula

• 💡 To understand the lateral surface area of a cone, imagine flattening its side into a sector of a circle, resembling a "Pac-Man" shape.
• 🎯 The arc length of this sector corresponds to the circumference of the cone's base, which is 2πr.
• 🔑 The radius of this sector is the slant height (L) of the cone, which can be calculated using the Pythagorean theorem: L = √(r² + h²).
• 🧠 The area of the sector is a fraction of the area of a full circle with radius L (πL²), determined by the ratio of the base circumference (2πr) to the sector's full circumference (2πL).
• ⚡ This calculation simplifies to the formula for lateral surface area: πrL.

Calculating Total Surface Area

• ➕ The total surface area of a cone is the sum of its lateral surface area and the area of its circular base.
• 📐 The area of the base is given by the standard formula πr².
• 🧮 Therefore, the total surface area of a cone is πr² + πrL.

Applying Formulas to a Specific Cone

• 📌 For a cone with a base radius (r) of 7 cm and a height (h) of 8 cm, the slant height (L) is calculated as √(7² + 8²) = √(49 + 64) = √113 cm.
• 📊 The lateral surface area is π * 7 * √113 cm².
• 📈 The total surface area is 49π cm² + 7π√113 cm².