Question

find the surface area of the lower prism.
11 cm 3 cm 7 cm
6 cm 12 cm 12 cm
$\text{sa}_{\text{upper}} = 229\\ \text{cm}^2$
$\text{sa}_{\text{lower}} = ?\\ \text{cm}^2$
$\text{sa}_{\text{total}} = \square\\ \text{cm}^2$
remember!
exclude areas where complex shapes touch.

Explanation:

Step1: Recall surface area formula for rectangular prism

The surface area of a rectangular prism is given by \( SA = 2(lw + lh + wh) \), where \( l \) is length, \( w \) is width, and \( h \) is height. For the lower prism, dimensions are \( l = 12 \, \text{cm} \), \( w = 6 \, \text{cm} \), \( h = 12 \, \text{cm} \). But we need to exclude the area where it touches the upper prism. The area of the touching face is the base area of the upper prism, which is \( 3 \times 11 \) (wait, no, wait: the upper prism has dimensions 11 cm, 3 cm, 7 cm? Wait, no, looking at the diagram: the upper prism's base (the face touching the lower prism) should have dimensions equal to the base of the upper prism. Wait, the upper prism: length 11, width 3, height 7? Wait, no, the lower prism has length 12, width 6, height 12. The upper prism is on top, so the area where they touch is the area of the base of the upper prism. Let's find the base area of the upper prism: from the upper prism's dimensions, the base (the face touching the lower prism) is \( 11 \times 3 \)? Wait, no, maybe I misread. Wait, the upper prism: length 11, width 3, height 7? Wait, the lower prism: length 12, width 6, height 12. So first, calculate the surface area of the lower prism as a standalone rectangular prism, then subtract twice the area of the touching face? Wait, no: when two prisms are joined, the total surface area is the sum of their surface areas minus twice the area of the touching face (because each prism had that face as part of their surface area, but now it's internal). But for the lower prism's surface area (SA_lower), we need to calculate its surface area as a rectangular prism, then subtract the area of the face where it touches the upper prism (because that face is no longer exposed).

First, calculate the surface area of the lower prism as a standalone rectangular prism: \( SA_{\text{lower standalone}} = 2(lw + lh + wh) \), where \( l = 12 \), \( w = 6 \), \( h = 12 \).

So \( lw = 12 \times 6 = 72 \), \( lh = 12 \times 12 = 144 \), \( wh = 6 \times 12 = 72 \). Then \( SA_{\text{lower standalone}} = 2(72 + 144 + 72) = 2(288) = 576 \, \text{cm}^2 \).

Now, the area where the upper prism touches the lower prism is the area of the base of the upper prism. The upper prism's base (the face touching the lower prism) has dimensions: looking at the upper prism's dimensions, the base is \( 11 \times 3 \)? Wait, no, the upper prism: length 11, width 3, height 7? Wait, the upper prism's length is 11, width 3, height 7. So the area of the touching face is \( 11 \times 3 = 33 \, \text{cm}^2 \). So we need to subtract this area from the lower prism's surface area (because that face is no longer exposed). Wait, no: when the upper prism is placed on the lower prism, the lower prism's top face has a portion covered (the area of the upper prism's base), so the exposed area of the lower prism's top face is its original top face area minus the area of the upper prism's base. The original top face area of the lower prism is \( 12 \times 6 = 72 \, \text{cm}^2 \). The area covered by the upper prism is \( 11 \times 3 = 33 \, \text{cm}^2 \). Wait, but that can't be, because 11*3=33, and 12*6=72, so 33 is less than 72, so it fits. Wait, maybe I got the dimensions wrong. Wait, the upper prism: length 11, width 3, height 7. The lower prism: length 12, width 6, height 12. So the base of the upper prism (touching the lower prism) is 11 (length) and 3 (width), so area 11*3=33. Now, the surface area of the lower prism as a standalone is \( 2(12*6 + 12*12 + 6*12) \). Let's calculate that:

\( 12*6 = 72 \), \( 12*12 = 144 \), \( 6*12 = 72 \). Sum: 72 + 144 + 72 = 288. Multiply by 2: 576. Now, the lower prism has a face (the top face) that is covered by the upper prism. So we need to subtract the area of that covered face from the surface area. But wait, the upper prism also has that face as part of its surface area, but for SA_lower, we just need the lower prism's surface area with the covered face excluded. So the covered area is 11*3=33. So SA_lower = 576 - 2*33? Wait, no: when the lower prism is alone, its top face is 12*6=72. When the upper prism is placed on it, the exposed area of the lower prism's top face is 72 - 33 = 39. But the surface area of the lower prism includes the top face (72) and the bottom face (72), front, back, left, right. Wait, no, the formula \( 2(lw + lh + wh) \) includes all six faces: top (lw), bottom (lw), front (lh), back (lh), left (wh), right (wh). So when we place the upper prism on top, the top face of the lower prism (lw = 12*6=72) is now covered by the upper prism's base (11*3=33). So the exposed area of the lower prism's top face is 72 - 33 = 39. But the upper prism's base (33) is now internal, so we need to subtract the area of the covered part from the lower prism's surface area. Wait, actually, the correct way is: the surface area of the lower prism is its total surface area minus twice the area of the touching face? No, no. Wait, when two solids are joined, the total surface area is SA_upper + SA_lower - 2*A, where A is the area of the touching face (because each solid had A as part of their surface area, but now it's internal, so we subtract 2A). But for SA_lower, it's the surface area of the lower prism minus A (because the lower prism's top face had A as part of its surface area, but now it's covered, so we subtract A). Wait, let's confirm:

Standalone lower prism SA: 576 (as calculated). Now, the top face of the lower prism has an area of 12*6=72. But part of it (A=11*3=33) is covered by the upper prism, so the exposed area of the top face is 72 - 33 = 39. The bottom face is still 72 (exposed). The other faces (front, back, left, right) are still fully exposed. So the surface area of the lower prism (SA_lower) is: bottom face (72) + front (12*12=144) + back (144) + left (6*12=72) + right (72) + top exposed (39). Let's calculate that: 72 + 144 + 144 + 72 + 72 + 39. Let's add step by step: 72+144=216; 216+144=360; 360+72=432; 432+72=504; 504+39=543. Wait, but let's check with the formula: SA_lower = SA_standalone_lower - A (the covered face). SA_standalone_lower is 576. The covered face is A=33, so 576 - 33 = 543. Yes, that matches. Because the standalone lower prism had the top face as 72, but now it's 39, so the difference is 72 - 39 = 33, so subtracting 33 from 576 (which had 72 as part of the top face) gives 576 - 33 = 543.

Wait, let's verify the standalone surface area again: \( 2(lw + lh + wh) = 2(12*6 + 12*12 + 6*12) = 2(72 + 144 + 72) = 2(288) = 576 \). Correct. Then, the top face area is 12*6=72. We need to subtract the area covered by the upper prism, which is 11*3=33 (since the upper prism's base is 11x3). So the exposed top face area is 72 - 33 = 39. So the total surface area of the lower prism is: bottom face (72) + front (12*12=144) + back (144) + left (6*12=72) + right (72) + top (39). Sum: 72 + 144 + 144 + 72 + 72 + 39 = 543. So SA_lower = 543.

Step2: Confirm the calculation

Alternatively, think of it as: the lower prism's surface area is the surface area of a rectangular prism with length 12, width 6, height 12, minus the area of the face where it touches the upper prism (since that face is no longer exposed). The area of the touching face is the area of the base of the upper prism, which is 11*3=33. So 576 - 33 = 543.

Answer:

543