A square-based pyramid has a base side of 20 cm and a slant edge of 26 cm. How many total surfaces are there, and what is the total surface area?

You're on a roll — here are the solutions to problem 5: Given: Length of a side of the base of a square-based pyramid (a) = 20 cm Length of slant edge (e) = 26 cm a) How many total surfaces are there in a square base pyramid? Write it. A square-based pyramid has one square base and four triangular lateral faces. Total number of surfaces = 1 (base) + 4 (lateral faces) = 5. The final answer is 5 surfaces. b) Find the total surface area of the pyramid? Step 1: Calculate the area of the base. The base is a square with side length a = 20 cm. Area of base = a^2 = (20 cm)^2 = 400 cm^2 Step 2: Calculate the slant height (l) of the pyramid. Consider a right-angled triangle formed by the slant edge (e), half the base side ((a)/(2)), and the slant height (l). The slant edge is the hypotenuse. e^2 = l^2 + ((a)/(2))^2 (26 cm)^2 = l^2 + (20 cm2)^2 676 cm^2 = l^2 + (10 cm)^2 676 cm^2 = l^2 + 100 cm^2 l^2 = 676 cm^2 - 100 cm^2 l^2 = 576 cm^2 l = sqrt(576 cm)^2 = 24 cm Step 3: Calculate the lateral surface area of the pyramid. The lateral surface area is the sum of the areas of the four triangular faces. Lateral surface area = (1)/(2) × Perimeter of base × l Perimeter of base = 4a = 4 × 20 cm = 80 cm Lateral surface area = (1)/(2) × 80 cm × 24 cm Lateral surface area = 40 cm × 24 cm = 960 cm^2 Step 4: Calculate the total surface area. Total surface area = Area of base + Lateral surface area Total surface area = 400 cm^2 + 960 cm^2 = 1360 cm^2 The final answer is 1360 cm^2. c) Compare between the vertical height and slant height of the pyramid. Step 1: Calculate the vertical height (h) of the pyramid. Consider a right-angled triangle formed by the vertical height (h), half the base side ((a)/(2)), and the slant height (l) as the hypotenuse. l^2 = h^2 + ((a)/(2))^2 We found l = 24 cm and (a)/(2) = 10 cm. (24 cm)^2 = h^2 + (10 cm)^2 576 cm^2 = h^2 + 100 cm^2 h^2 = 576 cm^2 - 100 cm^2 h^2 = 476 cm^2 h = sqrt(476 cm)^2 = sqrt(4 × 119) cm = 2sqrt(119) cm To compare, we can approximate 2sqrt(119): sqrt(119) ≈ 10.9087 h ≈ 2 × 10.9087 cm ≈ 21.8174 cm Step 2: Compare the vertical height (h) and slant height (l). Vertical height h = 2sqrt(119) cm ≈ 21.82 cm Slant height l = 24 cm Since 24 > 2sqrt(119), the slant height is greater than the vertical height. The final answer is The slant height (24 cm) is greater than the vertical height ( 2sqrt(119) cm ≈ 21.82 cm).. Drop the next question.