the net of a square pyramid has a base of 3cm by 3cm and each slant side is 5cm. Include the flaps.
Find the total surface area of the prism.
To find the total surface area of the square pyramid, we need to calculate the areas of all its individual faces and then add them together.
1. Base: The base of the pyramid is a square with sides measuring 3 cm. The formula for the area of a square is A = s^2, where s is the length of a side. Therefore, the area of the base is A_base = 3 cm * 3 cm = 9 cm^2.
2. Slant Faces: Each slant face of the pyramid forms a triangle. To calculate the area of a triangle, we use the formula A = (1/2) * base * height. In this case, the base is the side length of the square base, which is 3 cm, and the height is the slant height, which is 5 cm. Thus, the area of each slant face is A_slant = (1/2) * 3 cm * 5 cm = 7.5 cm^2.
3. Flaps: The flaps of the net represent the lateral faces of the square pyramid. They are in the shape of rectangles, where the length is the slant height (5 cm) and the width is the perimeter of the base (3 cm + 3 cm + 5 cm + 5 cm = 16 cm). Therefore, the area of each flap is A_flap = length * width = 5 cm * 16 cm = 80 cm^2.
4. Total Surface Area: To calculate the total surface area, we add the areas of the base, the slant faces, and the flaps. Thus, the total surface area of the square pyramid is:
Total Surface Area = A_base + 4 * A_slant + 4 * A_flap
= 9 cm^2 + 4 * 7.5 cm^2 + 4 * 80 cm^2
= 9 cm^2 + 30 cm^2 + 320 cm^2
= 359 cm^2
Therefore, the total surface area of the square pyramid, including the flaps, is 359 cm^2.