The base of a right pyramid is an equilateral triangle with side 10cm ...
Understanding the Problem
To find the surface area of the right pyramid with a triangular base, we need to calculate both the base area and the lateral surface area.
Step 1: Calculate the Base Area
- The base is an equilateral triangle with a side of 10 cm.
- The area of an equilateral triangle can be calculated using the formula: (√3/4) * side².
- Thus, the area = (√3/4) * (10)² = (√3/4) * 100 = 25√3 cm².
Step 2: Calculate the Lateral Surface Area
- The pyramid has three triangular faces.
- To find the area of one triangular face, we need the slant height (l).
Step 3: Calculate the Slant Height
- The slant height can be found using the Pythagorean theorem.
- The height of the pyramid is 5 cm, and the distance from the centroid of the triangle to a vertex is (side/√3) = (10/√3) cm.
- Therefore, l = √(height² + (side/√3)²) = √(5² + (10/√3)²) = √(25 + (100/3)) = √(75/3 + 100/3) = √(175/3) = (5√7/√3) cm.
Step 4: Area of One Triangular Face
- Area = (1/2) * base * height = (1/2) * 10 * l = (1/2) * 10 * (5√7/√3) = (25√7/√3) cm².
Step 5: Total Lateral Surface Area
- Since there are three triangular faces, the total lateral surface area = 3 * (25√7/√3) = 75√7/√3 cm².
Step 6: Total Surface Area
- Total Surface Area = Base Area + Lateral Surface Area.
- Total Surface Area = 25√3 + 75√7/√3 cm².
The correct option based on the calculations should be verified, but the provided answer suggests option 'C' which means further validation is required. However, the approach outlined is the systematic way to calculate the surface area of a right pyramid with a triangular base.