Answer:
the surface area of the given pyramid is approximately 261.24 cm².
Step-by-step explanation:
To find the surface area of a square-based pyramid, we need to calculate the area of each of its faces and add them together.
1. Base: The base of the pyramid is a square. Given that the base has a side length of 6 cm, we can find its area by multiplying the length of one side by itself: 6 cm × 6 cm = 36 cm².
2. Triangular faces: The pyramid has four identical triangular faces. To find the area of one triangular face, we need to calculate the area of a triangle. Since the triangle is isosceles, we can use the formula: area = (base × height) ÷ 2. In this case, the base of the triangle is 6 cm, and to find the height, we can use the Pythagorean theorem. The height forms a right triangle with the base and one of the sides, and the hypotenuse (slant height) of the triangle is given as 19 cm. Using the Pythagorean theorem, we can calculate the height as √(19 cm² - 3 cm²) = √(361 cm² - 9 cm²) = √(352 cm²) ≈ 18.77 cm. Thus, the area of one triangular face is (6 cm × 18.77 cm) ÷ 2 = 56.31 cm².
3. Total surface area: To find the total surface area of the pyramid, we add the area of the base to the combined areas of the four triangular faces: 36 cm² + (4 × 56.31 cm²) = 36 cm² + 225.24 cm² = 261.24 cm².
