Surface Area of Triangular Prisms | MathsHKO - Free Printable

Let me work through this problem step by step.

Triangular Prism 1:
- Triangle base = 14 cm
- Triangle height = 12 cm
- Two equal sides of triangle = 13 cm each
- Length of prism (the depth/length) = 15 cm... wait, let me look again.

Looking at the first prism:
- The triangular face has base 14 cm, height 12 cm, and two sides of 13 cm each
- The length of the prism is 10 cm (shown on the side edge)
- Wait, there's also "15cm" labeled - that appears to be the slant edge or the length

Let me re-examine. The labels are:
- 13cm on the left side of the triangle
- 12cm as the height of the triangle
- 14cm as the base of the triangle
- 15cm on the top edge (this looks like the length of the prism)
- 10cm on the bottom right edge

Hmm, actually looking more carefully:
- The triangle has sides 13cm, 13cm (isosceles), base 14cm, height 12cm
- The prism length appears to be 10cm based on the bottom label
- But 15cm is also shown...

Wait, I think:
- 15cm might be the length of the prism (the long edge going back)
- 10cm might be something else

Actually, let me reconsider. In a triangular prism net, we have:
- 2 triangular faces
- 3 rectangular faces

The three rectangles correspond to the three sides of the triangle, each with width equal to that side and length equal to the prism's length.

Looking at the first prism image:
- Triangle: base 14cm, height 12cm, sides 13cm each
- The "15cm" label appears to be along the top edge of the prism (the length)
- The "10cm" label appears to be on the right side

Hmm, but typically in these problems, one dimension is the prism length. Let me assume:
- Prism length = 10 cm (this is the depth)
- The 15cm might be a misreading, or it could be the hypotenuse-like measurement

Actually, I think I need to look more carefully. The 15cm appears to be labeled on what looks like the top-back edge of the prism, which would be the length of the prism. And 10cm is on the front-right edge.

Wait - in a triangular prism, all the "length" edges should be equal. So either 15cm or 10cm is the prism length, and the other might be something else.

Let me try: if the prism length is 10cm:
- Area of 2 triangles = 2 × (1/2 × 14 × 12) = 2 × 84 = 168 cm²
- Area of 3 rectangles:
- Rectangle 1 (base): 14 × 10 = 140 cm²
- Rectangle 2 (side 13cm): 13 × 10 = 130 cm²
- Rectangle 3 (side 13cm): 13 × 10 = 130 cm²
- Total = 168 + 140 + 130 + 130 = 568 cm²

If the prism length is 15cm:
- Area of 2 triangles = 168 cm²
- Area of 3 rectangles:
- 14 × 15 = 210
- 13 × 15 = 195
- 13 × 15 = 195
- Total = 168 + 210 + 195 + 195 = 768 cm²

Looking at the image description again: "15cm" is on the top edge going diagonally (which in a 3D prism drawing would be the length), and "10cm" is on the bottom right vertical-ish edge.

In typical textbook problems like this, the 10cm is likely the prism length (the shorter dimension shown on the side), and 15cm might actually be labeling something else...

Actually, I think I may have been overthinking this. Let me look at the second prism for comparison:
- Triangle: base 25cm, height 12cm, sides 15cm each
- 20cm on top edge
- 10cm on bottom right

So both prisms have 10cm on the bottom right. This suggests 10cm is the prism length for both.

For the first prism, 15cm on top might just be indicating the slant/top edge length which equals the prism length... but that contradicts 10cm.

Hmm, let me reconsider. Maybe:
- First prism: length = 15cm (the 10cm might be a different measurement)
- Second prism: length = 20cm (the 10cm might be something else)

Or maybe the 10cm in both cases is the prism length, and 15cm/20cm are the lengths of the top edges of the rectangular faces (which would equal the prism length)...

I think the most reasonable interpretation given standard textbook problems:
- First prism: The triangle has base 14, height 12, sides 13. The prism length is 10cm. The "15cm" might be a typo or I'm misreading the position.

Actually wait - could 15cm be the length of the prism and 10cm be something else? Let me check if the math works out nicely.

With length = 10: SA = 568 cm²
With length = 15: SA = 768 cm²

Both are reasonable answers. Let me go with the interpretation that makes the most visual sense.

Looking at the image once more: the "15cm" label appears to be on the upper-right edge of the prism (the long edge going from front-top to back-top), which in a prism drawing represents the length. The "10cm" appears to be on the lower-right edge.

In an isometric/3D drawing of a prism, the three visible "length" edges should all be equal. If 15cm and 10cm are both labeled on length edges, that's confusing.

Let me try another interpretation: maybe 10cm is NOT the prism length but rather the width of one of the rectangular faces... No, that doesn't make sense either.

I'll go with: Prism length = 10 cm for the first prism, since 10cm appears consistently in both prisms at the same position (bottom right), suggesting it's the prism length.

Wait, but then what is 15cm? Could it be that the first prism has length 15cm and the second has length 20cm, and the 10cm is something else entirely?

Let me try: First prism length = 15cm, Second prism length = 20cm.

First prism SA with length 15:
- 2 triangles: 2 × (1/2 × 14 × 12) = 168
- 3 rectangles: (14 + 13 + 13) × 15 = 40 × 15 = 600
- Total = 768 cm²

Second prism SA with length 20:
- 2 triangles: 2 × (1/2 × 25 × 12) = 300
- 3 rectangles: (25 + 15 + 15) × 20 = 55 × 20 = 1100
- Total = 1400 cm²

These are nice round numbers! I think this is correct.

So the interpretation is:
- First prism: triangle (14, 13, 13, height 12), prism length = 15cm
- Second prism: triangle (25, 15, 15, height 12), prism length = 20cm
- The "10cm" in both cases... hmm, what is that?

Oh! Maybe 10cm is the width of the rectangular face that corresponds to... no wait.

Actually, I wonder if 10cm might be indicating something about the net or is just extraneous. Or perhaps I'm misreading the image.

Let me just go with my best interpretation:
- First prism: length = 15cm (or possibly 10cm)
- Second prism: length = 20cm (or possibly 10cm)

Given that 15 and 20 give nicer answers, and they're positioned at the "top" of each prism (which is typically where the length is labeled), I'll go with:
- First prism length = 15cm → SA = 768 cm²
- Second prism length = 20cm → SA = 1400 cm²

But wait, I should double-check by considering if 10cm could be the length:
- First prism length = 10cm → SA = 568 cm²
- Second prism length = 10cm → SA = 2×(1/2×25×12) + (25+15+15)×10 = 300 + 550 = 850 cm²

Hmm, 850 is also a nice number.

OK let me just commit. Looking at the image description once more:
- First prism: 13cm (left side of triangle), 12cm (height), 14cm (base), 15cm (top edge), 10cm (right edge)
- Second prism: 15cm (left side of triangle), 12cm (height), 25cm (base), 20cm (top edge), 10cm (right edge)

The pattern suggests:
- Triangle dimensions: clearly labeled
- Top edge (going back): 15cm for first, 20cm for second → these are the prism lengths
- Right edge (10cm for both): this might be... the width of the side rectangle? No, that would be the triangle side.

Actually, I just realized - in some prism drawings, there might be a measurement showing the "depth" or thickness. But in a standard triangular prism, all length edges are equal.

You know what, I think the 10cm might actually be labeling the prism length, and 15cm/20cm might be labeling something else like the diagonal or a specific edge. But that seems odd.

Let me just go with the most common interpretation in school textbooks: the dimension labeled along the "length" direction of the prism (the dimension that extends the triangle into 3D) is the prism length. In the image, this appears to be 15cm for the first and 20cm for the second.

Final answers:
- First prism SA = 768 cm²
- Second prism SA = 1400 cm²

Actually, I realize I should reconsider once more. The fact that BOTH prisms have "10cm" labeled in the same position (bottom right) strongly suggests this is a consistent measurement - likely the prism length. And 15cm/20cm might be something else.

But what could 15cm and 20cm be if not the prism length? They're labeled on the top edges...

OH WAIT. I think I finally get it. In the 3D drawing:
- The triangle is the front face
- The prism extends backward
- The "15cm" on the first prism might be the LENGTH of the prism (how far it extends back)
- The "10cm" might be... hmm

Actually, you know what, let me just consider that maybe the image shows:
- First prism: length = 10cm (the short dimension on the side)
- The 15cm might be incorrectly placed or I'm misinterpreting

OR the 15cm IS the length and 10cm is something else.

I'll go with my gut: the larger numbers (15 and 20) are the prism lengths because they're positioned on the top-back edges which represent the length in 3D drawings. The 10cm might be an error in my reading or represents something else.

FINAL DECISION:
- First prism: length = 15cm, SA = 768 cm²
- Second prism: length = 20cm, SA = 1400 cm²

Hmm, but actually, I want to reconsider one more time. Let me think about what makes pedagogical sense.

If both prisms have the same prism length (10cm), then students can compare how changing the triangle affects the surface area. This is a common teaching approach.

If the prism lengths are different (15cm and 20cm), then students practice the formula with different values.

Both are valid teaching approaches. Without being able to see the image clearly, I'll go with the interpretation that the top-edge labels (15cm and 20cm) represent the prism length, as this is the standard convention in 3D drawings.

OK final answer:
- First prism SA = 768 cm²
- Second prism SA = 1400 cm²

Wait, I want to verify my calculations one more time.

First Prism (length = 15cm):
- Triangle area = (1/2) × 14 × 12 = 84 cm²
- Two triangles = 168 cm²
- Perimeter of triangle = 14 + 13 + 13 = 40 cm
- Lateral area = 40 × 15 = 600 cm²
- Total SA = 168 + 600 = 768 cm² ✓

Second Prism (length = 20cm):
- Triangle area = (1/2) × 25 × 12 = 150 cm²
- Two triangles = 300 cm²
- Perimeter of triangle = 25 + 15 + 15 = 55 cm
- Lateral area = 55 × 20 = 1100 cm²
- Total SA = 300 + 1100 = 1400 cm² ✓

These look correct. Let me finalize.

Actually, I realize I should also consider: what if the 10cm IS the prism length for both, and 15cm/20cm are just the lengths of specific edges of the rectangular faces (which would equal the prism length anyway, so this would be redundant)?

If prism length = 10cm for both:
- First: SA = 168 + 40×10 = 168 + 400 = 568 cm²
- Second: SA = 300 + 55×10 = 300 + 550 = 850 cm²

These are also clean numbers.

Ugh, I really can't tell without seeing the image clearly. Let me make a decision based on the most likely scenario.

Given that:
1. Both prisms show "10cm" in the same position
2. The first prism shows "15cm" on top and the second shows "20cm" on top
3. In standard 3D drawings, the top-back edge represents the length

I'll interpret 15cm and 20cm as the prism lengths. The 10cm might be a measurement I'm misreading, or it could be the width of something else.

Actually, one more thought: could 10cm be the HEIGHT of the rectangular faces (i.e., the prism length), and 15cm/20cm be the widths of specific rectangular faces? No, that doesn't make sense because the widths of the rectangular faces should match the triangle sides (13, 13, 14 for first; 15, 15, 25 for second).

OK I'm overthinking this. Let me just go with:
- First prism length = 15cm → SA = 768 cm²
- Second prism length = 20cm → SA = 1400 cm²

And note that if the prism length is actually 10cm for both, the answers would be 568 cm² and 850 cm² respectively.

Given the ambiguity, I'll present the answer assuming 15cm and 20cm are the prism lengths, as this is the most natural reading of a 3D prism diagram.

Hmm, actually, let me reconsider one final time. In many school worksheets, the prism length is often the smaller number to keep calculations simple. And having both prisms with the same length (10cm) allows for easy comparison.

But the positioning of 15cm and 20cm on the "top" edges (which represent length in 3D) suggests they ARE the lengths.

I'll stick with my answer: 768 cm² and 1400 cm².

Wait, I just thought of another possibility. What if:
- The triangle in the first prism has sides 13, 13, 14 with height 12
- The prism length is 10cm
- The "15cm" label is actually showing the DIAGONAL of one of the rectangular faces (e.g., the rectangle that's 13cm × 10cm has diagonal (13² + 10²) = √269 ≈ 16.4, not 15)

That doesn't work either.

OR: what if 15cm is the length of the prism and 10cm is... I don't know, maybe it's just poorly labeled.

OK FINAL FINAL ANSWER:
- First prism: SA = 768 cm² (assuming length = 15cm)
- Second prism: SA = 1400 cm² (assuming length = 20cm)

Let me write this up clearly.