Calculating the Area of a Regular Pentagon Using Radius and Trigonometry

Area Formula for Regular Polygons

• 🎯 The area of a regular polygon is calculated as 1/2 * apothem length * perimeter.
• 💡 This formula is derived by dividing the polygon into congruent triangles, where the area of each triangle is 1/2 * height * base.

Finding the Apothem Length

• 📐 To find the apothem, we first determine the central angle of one of the triangles formed by connecting the center to two adjacent vertices. For a pentagon, this angle is 360° / 5 = 72°, and half of this angle, used in a right triangle, is 36°.
• 🔑 Using trigonometry, the apothem (adjacent side) can be found with the cosine function: apothem = radius * cos(36°), given the radius.

Calculating the Perimeter

• 📏 The side length of the polygon can be found using the sine function: half_side_length = radius * sin(36°).
• 🌐 The full perimeter is then 10 * half_side_length, which equals 20 * radius * sin(36°).

Final Area Calculation

• 🧮 With the apothem and perimeter calculated, the area of the regular pentagon is 1/2 * (radius * cos(36°)) * (20 * radius * sin(36°)).
• 📈 For a radius of 20 cm, the apothem is approximately 16.18 cm and the perimeter is approximately 117.56 cm, resulting in an area of approximately 951 square cm.