Here is a solid square-based pyramid, VABCD - Edexcel - GCSE Maths - Question 5 - 2018 - Paper 1

To calculate the total surface area of the pyramid, follow these steps:

1. Area of the Base: The base of the pyramid is a square with a side length of 6 cm. The area (A_base) can be calculated as:

   $A_{base} = side^2 = 6^2 = 36 ext{ cm}^2$

2. Area of the Triangular Faces: There are four triangular faces. To calculate the area of one triangular face, use the formula for the area of a triangle:

   $A_{triangle} = \frac{1}{2} \times base \times height$

   The base for each triangle is the side of the square base (6 cm). The height of each triangular face can be found using the Pythagorean theorem:

   For triangle VBC:

   • The vertical height is 4 cm.
   • The horizontal distance from M to B (half the base) is 3 cm.

   Using the Pythagorean theorem:

   $h_{triangle} = \sqrt{(5^2 - 3^2)} = \sqrt{(25 - 9)} = \sqrt{16} = 4 ext{ cm}$

   Therefore, the area of one triangular face is:

   $A_{triangle} = \frac{1}{2} \times 6 \times 4 = 12 ext{ cm}^2$

   The total area for all four triangular faces is:

   $A_{triangles} = 4 \times A_{triangle} = 4 \times 12 = 48 ext{ cm}^2$

3. Total Surface Area: Add the area of the base and the area of the triangular faces:

   $A_{total} = A_{base} + A_{triangles}$

   $A_{total} = 36 + 48 = 84 ext{ cm}^2$

Thus, the total surface area of the pyramid is 84 cm².