Understand the pyramid's dimensions.

Here's the solution to the pyramid problem. Step 1: Understand the pyramid's dimensions. The pyramid has a square base. Each sloping face is an equilateral triangle with a side length of 12 cm. This means the side length of the square base is also s = 12 cm. Step 2: Calculate the base area (a). The base is a square with side length s = 12 cm. The area of the base A_b is given by: A_b = s^2 A_b = (12 cm)^2 A_b = 144 cm^2 The base area of the pyramid is 144 cm^2. Step 3: Calculate the height of the pyramid (b). First, we find the slant height (l_s) of one of the equilateral triangular faces. For an equilateral triangle with side s, the height is sqrt(3)2s. l_s = sqrt(3)2 × 12 cm l_s = 6sqrt(3) cm Next, consider a right-angled triangle formed by the pyramid's height (h), the slant height (l_s), and half the base side length ((s)/(2)). The base of this right triangle is (12)/(2) = 6 cm. Using the Pythagorean theorem: h^2 + ((s)/(2))^2 = l_s^2 h^2 + (6 cm)^2 = (6sqrt(3) cm)^2 h^2 + 36 cm^2 = (36 × 3) cm^2 h^2 + 36 cm^2 = 108 cm^2 h^2 = 108 cm^2 - 36 cm^2 h^2 = 72 cm^2 h = sqrt(72) cm h = sqrt(36 × 2) cm h = 6sqrt(2) cm The height of the pyramid is 6sqrt(2) cm. Step 4: Calculate the volume of the pyramid (c). The volume V of a pyramid is given by the formula: V = (1)/(3) × A_b × h V = (1)/(3) × 144 cm^2 × 6sqrt(2) cm V = (48 × 6sqrt(2)) cm^3 V = 288sqrt(2) cm^3 The volume of the pyramid is 288sqrt(2) cm^3. Step 5: Calculate the total surface area excluding the base (d). This is the lateral surface area (A_L). The pyramid has 4 identical equilateral triangular faces. The area of one equilateral triangular face with side s is given by sqrt(3)4s^2. A_face = sqrt(3)4 × (12 cm)^2 A_face = sqrt(3)4 × 144 cm^2 A_face = 36sqrt(3) cm^2 The total surface area excluding the base is the sum of the areas of the 4 faces: A_L = 4 × A_face A_L = 4 × 36sqrt(3) cm^2 A_L = 144sqrt(3) cm^2 The total surface area of the pyramid excluding the base is 144sqrt(3) cm^2. Send me the next one 📸