Understand the pyramid's dimensions.

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Here's the solution to the pyramid problem.

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.
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:
$A_b = s^2$ $A_b = (12 \text{ cm})^2$ $A_b = 144 \text{ cm}^2$
The base area of the pyramid is $\boxed{\text{144 cm}^2}$ .
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 $\frac{\sqrt{3}}{2}s$ .
$l_s = \frac{\sqrt{3}}{2} \times 12 \text{ cm}$ $l_s = 6\sqrt{3} \text{ cm}$
Next, consider a right-angled triangle formed by the pyramid's height ( $h$ ), the slant height ( $l_s$ ), and half the base side length ( $\frac{s}{2}$ ). The base of this right triangle is $\frac{12}{2} = 6$ cm.
Using the Pythagorean theorem:
$h^2 + \left(\frac{s}{2}\right)^2 = l_s^2$ $h^2 + (6 \text{ cm})^2 = (6\sqrt{3} \text{ cm})^2$ $h^2 + 36 \text{ cm}^2 = (36 \times 3) \text{ cm}^2$ $h^2 + 36 \text{ cm}^2 = 108 \text{ cm}^2$ $h^2 = 108 \text{ cm}^2 - 36 \text{ cm}^2$ $h^2 = 72 \text{ cm}^2$ $h = \sqrt{72} \text{ cm}$ $h = \sqrt{36 \times 2} \text{ cm}$ $h = 6\sqrt{2} \text{ cm}$
The height of the pyramid is $\boxed{6\sqrt{2} \text{ cm}}$ .
Calculate the volume of the pyramid (c).
The volume $V$ of a pyramid is given by the formula:
$V = \frac{1}{3} \times A_b \times h$ $V = \frac{1}{3} \times 144 \text{ cm}^2 \times 6\sqrt{2} \text{ cm}$ $V = (48 \times 6\sqrt{2}) \text{ cm}^3$ $V = 288\sqrt{2} \text{ cm}^3$
The volume of the pyramid is $\boxed{288\sqrt{2} \text{ cm}^3}$ .
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 $\frac{\sqrt{3}}{4}s^2$ .
$A_{\text{face}} = \frac{\sqrt{3}}{4} \times (12 \text{ cm})^2$ $A_{\text{face}} = \frac{\sqrt{3}}{4} \times 144 \text{ cm}^2$ $A_{\text{face}} = 36\sqrt{3} \text{ cm}^2$
The total surface area excluding the base is the sum of the areas of the 4 faces:
$A_L = 4 \times A_{\text{face}}$ $A_L = 4 \times 36\sqrt{3} \text{ cm}^2$ $A_L = 144\sqrt{3} \text{ cm}^2$
The total surface area of the pyramid excluding the base is $\boxed{144\sqrt{3} \text{ cm}^2}$ .

Understand the pyramid's dimensions.

Step-by-step solution

Understand the pyramid's dimensions.

Calculate the base area (a).

Calculate the height of the pyramid (b).

Calculate the volume of the pyramid (c).

Calculate the total surface area excluding the base (d).

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