Calculating Surface Area of a Regular Octagonal Pyramid

Lateral Surface Area Calculation

• 🎯 The goal is to find the lateral surface area of a right pyramid with a regular octagon base.
• 💡 To calculate this, we need the surface area of each of the eight triangular faces.
• 📐 The base of each triangular face is given as 6 meters, and we need to find its height (slant height of the pyramid face), referred to as 'h'.
• 📐 To find 'h', we use the Pythagorean theorem, which requires knowing the distance 'x' from the center of the octagon to the midpoint of a base side, and the pyramid's height (9 meters).
• 📐 The value of 'x' is found using trigonometry: x = 3 / tan(22.5°), derived from the angle of the octagon's central triangles.
• ⚡ The lateral surface area is then calculated as 8 * (1/2 * base * h), substituting the derived 'h' and base (6m).
• 📊 The approximate lateral surface area is 277 square meters.

Total Surface Area Calculation

• ➕ To find the total surface area, we need to add the area of the octagonal base to the lateral surface area.
• 📐 The area of the octagonal base can be calculated by summing the areas of the eight triangles that form it, each with a base of 6 meters and a height of 'x' (3 / tan(22.5°)).
• 🧮 The area of one such triangle is 1/2 * 6 * (3 / tan(22.5°)), and the total base area is 8 * (1/2 * 6 * (3 / tan(22.5°))).
• 📈 The total surface area is the sum of the lateral surface area and the calculated base area.
• 📍 The approximate total surface area is 451 square meters.