Surface Area of Non-Overlapping Rectangular Prisms Worksheets - Free Printable

Explanation:
We are asked to find the surface area of each solid figure. These are all rectangular prisms (or combinations of them), and surface area means the total area of *all* the outer faces.

For a rectangular prism, the surface area formula is:
SA = 2(lw + lh + wh)
where *l* = length, *w* = width, *h* = height.

But many of these figures are L-shaped or stepped, meaning they’re made of two or more rectangular prisms joined together. In such cases, we can’t just use the simple formula — instead, we’ll break the shape into visible rectangles, find the area of each face, and add them up. Or better yet: find the area of *all* exposed faces by identifying each rectangle’s dimensions.

A reliable method for these composite solids:
1. Identify all individual rectangular faces (front/back, left/right, top/bottom).
2. For each face, determine its dimensions (length × width).
3. Add up all face areas.
4. Be careful: where two parts are glued together, those touching faces are *not* part of the surface area — so don’t double-count or include hidden faces.

Let’s solve each one carefully.

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Problem 1:
Shape looks like a 2-step L-block. Dimensions given:
- Bottom base: 12 ft (length) × 4 ft (depth)
- Height of lower part: 5 ft
- Upper step: 7 ft (length) × 4 ft (depth), height = 2 ft (so total height = 5 + 2 = 7 ft)

Better to compute by listing all exposed faces:

Let’s define orientation:
- Front face: consists of two rectangles:
- Lower rectangle: 12 ft wide × 5 ft tall = 60 ft²
- Upper rectangle: 7 ft wide × 2 ft tall = 14 ft²
→ Front = 60 + 14 = 74 ft²
- Back face: same as front = 74 ft²

- Top face: two rectangles:
- Lower part top (exposed): (12 − 7) = 5 ft long × 4 ft deep = 20 ft²
- Upper part top: 7 ft × 4 ft = 28 ft²
→ Top = 20 + 28 = 48 ft²
- Bottom face: full base: 12 ft × 4 ft = 48 ft²

- Left side (view from left): rectangle 5 ft (height) × 4 ft (depth) = 20 ft²
Plus on top, a vertical face of upper block: 2 ft × 4 ft = 8 ft²
→ Left = 20 + 8 = 28 ft²
- Right side: full height 7 ft × 4 ft = 28 ft²

Wait — need to be consistent. Let's draw mentally:

The shape is like:
- A base: 12 ft (x) × 4 ft (y) × 5 ft (z)
- On top of the *left* part of the base (7 ft long), a smaller block: 7 ft × 4 ft × 2 ft

So coordinates:
- Base occupies x = 0 to 12, y = 0 to 4, z = 0 to 5
- Upper block occupies x = 0 to 7, y = 0 to 4, z = 5 to 7

Now list all external faces:

1. Front (y = 0 plane):
- From x=0–12, z=0–5 → 12×5 = 60
- From x=0–7, z=5–7 → 7×2 = 14
Total front = 74
Back (y=4) same = 74

2. Top (z = 7 plane): only upper block top: x=0–7, y=0–4 → 7×4 = 28
Also, the part of base not covered: x=7–12, y=0–4, z=5 → that’s a horizontal rectangle at height 5, but it's *exposed* on top? Yes — because upper block only covers x=0–7. So top of base from x=7–12 is exposed: 5×4 = 20
So total top = 28 + 20 = 48
Bottom (z=0): full base: 12×4 = 48

3. Left side (x = 0): full height 7 ft, depth 4 ft → 7×4 = 28
4. Right side (x = 12): only base height (since upper block doesn’t extend to x=12): 5 ft × 4 ft = 20
5. Inner vertical face (at x = 7, between upper and lower blocks): this is exposed! Because upper block sits only on left part, the “step” face at x=7, y=0–4, z=5–7 is vertical and exposed. Its size: height = 2 ft, depth = 4 ft → 8 ft²

Now sum all:
- Front: 74
- Back: 74
- Top: 48
- Bottom: 48
- Left: 28
- Right: 20
- Step face (x=7): 8

Total = 74+74 = 148
148 + 48 + 48 = 244
244 + 28 = 272
272 + 20 = 292
292 + 8 = 300 ft²

✔ So Problem 1: 300 ft²

Let’s verify with alternative method: total surface area = surface area of both prisms minus 2×area of contact face (since when glued, two faces disappear).

Prism A (base): 12×4×5 → SA = 2(12·4 + 12·5 + 4·5) = 2(48 + 60 + 20) = 2(128) = 256
Prism B (top): 7×4×2 → SA = 2(7·4 + 7·2 + 4·2) = 2(28 + 14 + 8) = 2(50) = 100
Contact face: where they join = 7×4 = 28
When glued, we lose *two* faces of area 28 (one from each prism) → subtract 2×28 = 56
Total SA = 256 + 100 − 56 = 300 ✔

Great — method confirmed.

Now do the rest quickly using the prism-sum minus 2×contact method when shapes are two prisms joined.

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Problem 2:
Looks like an L-shape: bottom part 5 ft × 2 ft × 2 ft? Wait labels:
From diagram:
- Overall width: 5 ft
- Depth: 2 ft
- Left vertical part: height 2 ft, width 2 ft
- Right horizontal part: extends 3 ft right, height 1 ft? Actually labeled:
Left column: 2 ft (width) × 2 ft (height)
Right arm: 3 ft (length) × 1 ft (height)
Depth = 2 ft (same for all)

So it's two prisms:
- Prism A (vertical left): 2 ft (x) × 2 ft (y) × 2 ft (z)
- Prism B (horizontal right): 3 ft (x) × 2 ft (y) × 1 ft (z)
They join along a 2 ft × 2 ft face? Wait: the vertical part is 2 ft wide (x), 2 ft deep (y), 2 ft high (z). The horizontal part sits on top of the base? No — looking: the shape is like a backwards L on its side: the tall part is on left, short arm goes right at bottom.

Actually, from labels:
Left side height = 2 ft
Bottom horizontal length = 5 ft
The step is at 2 ft from left, height drops to 1 ft for remaining 3 ft.

So:
- Left block: 2 ft (width) × 2 ft (depth) × 2 ft (height)
- Right block: 3 ft (width) × 2 ft (depth) × 1 ft (height)
They sit side-by-side on same base (z=0), so contact face is vertical: where they meet at x=2, it's a rectangle 2 ft (depth) × 1 ft (height) — because right block is only 1 ft tall, so only lower 1 ft touches.

Thus contact area = 2 × 1 = 2 ft²
But careful: the left block has a face 2 (depth) × 1 (height) that is covered; the right block has same face exposed inward — so we subtract 2 × 2 = 4 ft² total.

Compute SA:

Prism A (2×2×2): SA = 2(2·2 + 2·2 + 2·2) = 2(4+4+4)=24
Prism B (3×2×1): SA = 2(3·2 + 3·1 + 2·1) = 2(6+3+2)=2(11)=22
Total before gluing: 24+22 = 46
Subtract 2×contact = 2×(2×1)=4
→ SA = 46 − 4 = 42 ft²

Double-check by face counting:
- Front: left 2×2 = 4, right 3×1 = 3 → total front = 7
- Back: same = 7
- Top: left top 2×2 = 4, right top 3×2 = 6 → total top = 10
- Bottom: full base 5×2 = 10
- Left side: 2×2 = 4
- Right side: 3×2? No — right side is just the end of right block: 2 (depth) × 1 (height) = 2
- Inner step face (between them, vertical): height difference = 1 ft, depth = 2 ft → 2 ft²
Sum: 7+7+10+10+4+2+2 = 42 ✔

Problem 2: 42 ft²

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Problem 3:
Dimensions:
- Base: 10 in × 4 in × 3 in
- Upper block on left: 3 in × 4 in × 2 in (since height total = 3+2 = 5 in, and upper part is 2 in tall)

So:
Prism A (base): 10×4×3 → SA = 2(10·4 + 10·3 + 4·3) = 2(40+30+12)=2(82)=164
Prism B (upper): 3×4×2 → SA = 2(3·4 + 3·2 + 4·2)=2(12+6+8)=2(26)=52
Contact face = 3×4 = 12
Subtract 2×12 = 24
Total SA = 164 + 52 − 24 = 192 in²

Check quickly:
Front: base 10×3=30, upper 3×2=6 → 36
Back: same 36
Top: base exposed: (10−3)×4 = 7×4=28, upper top: 3×4=12 → 40
Bottom: 10×4=40
Left: full height 5×4=20
Right: base only 3×4=12
Step face (x=3, z=3 to 5): 2×4=8
Sum: 36+36=72; +40+40=152; +20=172; +12=184; +8=192 ✔

Problem 3: 192 in²

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Problem 4:
Looks like an L:
- Vertical part: 4 in (width) × 2 in (depth) × 10 in (height)
- Horizontal arm: extends 6 in (length) × 2 in (depth) × 3 in (height)
They join at bottom: contact face = 2 in (depth) × 3 in (height) = 6 in²

Prism A (vertical): 4×2×10 → SA = 2(4·2 + 4·10 + 2·10) = 2(8+40+20)=2(68)=136
Prism B (horizontal): 6×2×3 → SA = 2(6·2 + 6·3 + 2·3)=2(12+18+6)=2(36)=72
Subtract 2×6 = 12
Total = 136+72−12 = 196 in²

Verify quickly:
Front: vertical 4×10=40, horizontal 6×3=18 → 58
Back: same 58
Top: vertical top 4×2=8, horizontal top 6×2=12 → 20
Bottom: full base = (4+6)×2 = 10×2=20
Left side: vertical left 2×10=20
Right side: horizontal end 2×3=6
Step face (inner vertical at junction): height diff = 10−3=7, depth=2 → 14
Sum: 58+58=116; +20+20=156; +20=176; +6=182; +14=196 ✔

Problem 4: 196 in²

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Problem 5:
Similar to #4 but mirrored. Dimensions:
- Base horizontal: 7 yd × 2 yd × 3 yd
- Vertical part on right: 4 yd (width) × 2 yd (depth) × 5 yd (height)
Wait labels:
From diagram: total length = 7 yd, vertical part is 4 yd tall, horizontal part height = 3 yd, depth = 2 yd everywhere.

So:
- Horizontal prism: 7×2×3
- Vertical prism sits on right end, width = ? The vertical part width is 4 yd? No — the vertical part protrudes upward, and its base is part of the horizontal: likely the vertical part is 4 yd (depth?) Let's read:
Labels: left horizontal segment: 3 yd (length), then vertical part width = 4 yd? Actually diagram shows:
- Bottom: total length 7 yd
- At right end, a block rises: its width (into page) = 2 yd, depth same, height = 5 yd, and its base length = 4 yd? But 3 + 4 = 7, so yes:
- Left arm: 3 yd long × 2 yd deep × 3 yd high
- Right arm (taller): 4 yd long × 2 yd deep × 5 yd high
They sit on same base, so contact face is where they meet: vertical face of size 2 (depth) × 3 (height) — because left is only 3 ft tall, so only lower 3 ft of right block contacts.

Contact area = 2 × 3 = 6
Prism A (left): 3×2×3 → SA = 2(3·2 + 3·3 + 2·3)=2(6+9+6)=2(21)=42
Prism B (right): 4×2×5 → SA = 2(4·2 + 4·5 + 2·5)=2(8+20+10)=2(38)=76
Subtract 2×6 = 12
Total = 42+76−12 = 106 yd²

Check faces:
Front: left 3×3=9, right 4×5=20 → 29
Back: same 29
Top: left top 3×2=6, right top 4×2=8 → 14
Bottom: 7×2=14
Left side: 2×3=6
Right side: 2×5=10
Step face (at junction, vertical): height diff = 5−3=2, depth=2 → 4
Sum: 29+29=58; +14+14=86; +6=92; +10=102; +4=106 ✔

Problem 5: 106 yd²

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Problem 6:
Dimensions:
- Base: 11 in × 2 in × 2 in
- Upper block on left: 3 in × 2 in × 10 in? Wait labels:
Left vertical part height = 10 in, width = 3 in, depth = 2 in
Right horizontal part: length = 11−3 = 8 in? But label says 11 in total, and a 3 in segment on left with height 10 in, rest is 2 in high.

Actually:
- Total length = 11 in
- Left portion: 3 in long, height = 10 in
- Right portion: 8 in long, height = 2 in
Depth = 2 in everywhere.

So:
Prism A (left tall): 3×2×10 → SA = 2(3·2 + 3·10 + 2·10)=2(6+30+20)=2(56)=112
Prism B (right short): 8×2×2 → SA = 2(8·2 + 8·2 + 2·2)=2(16+16+4)=2(36)=72
Contact face = 2 (depth) × 2 (height of short part) = 4
Subtract 2×4 = 8
Total = 112+72−8 = 176 in²

Check:
Front: left 3×10=30, right 8×2=16 → 46
Back: 46
Top: left top 3×2=6, right top 8×2=16 → 22
Bottom: 11×2=22
Left side: 2×10=20
Right side: 2×2=4
Step face: height diff = 10−2=8, depth=2 → 16
Sum: 46+46=92; +22+22=136; +20=156; +4=160; +16=176 ✔

Problem 6: 176 in²

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Problem 7:
Dimensions:
- Base: 13 yd × 2 yd × 9 yd? Labels:
Front: width 13 yd, depth 2 yd, height 9 yd for most, but a step: left part height = 9 yd, right part height = 3 yd, and the drop happens after 10 yd? Wait labels:
From diagram: total length = 13 yd
At left, height = 9 yd for 10 yd length
Then right 3 yd length, height = 3 yd
Depth = 2 yd

So:
- Left prism: 10×2×9
- Right prism: 3×2×3
Contact face = 2 × 3 = 6 (since right is only 3 ft tall)

SA left: 2(10·2 + 10·9 + 2·9) = 2(20+90+18)=2(128)=256
SA right: 2(3·2 + 3·3 + 2·3)=2(6+9+6)=2(21)=42
Subtract 2×6 = 12
Total = 256+42−12 = 286 yd²

Check:
Front: 10×9=90, 3×3=9 → 99
Back: 99
Top: left top 10×2=20, right top 3×2=6 → 26
Bottom: 13×2=26
Left side: 2×9=18
Right side: 2×3=6
Step face: height diff = 9−3=6, depth=2 → 12
Sum: 99+99=198; +26+26=250; +18=268; +6=274; +12=286 ✔

Problem 7: 286 yd²

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Problem 8:
This is a single rectangular prism? Labels:
Length = 10 ft, width = 2 ft, height = 4 ft? Wait:
Diagram shows: front face 4 ft tall, depth 2 ft, length 10 ft, and a slant? No — it's a rectangular prism, possibly with a cut? Actually looks like a full box: 10 ft (length) × 2 ft (depth) × 4 ft (height)

Yes — all edges labeled: 10 ft, 2 ft, 4 ft.

So SA = 2(lw + lh + wh) = 2(10·2 + 10·4 + 2·4) = 2(20 + 40 + 8) = 2(68) = 136 ft²

Problem 8: 136 ft²

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Problem 9:
Looks like a rectangular prism tilted? But labels:
- Base: 5 m × 3 m
- Height: 3 m
- Also a slanted top? Wait — it's a parallelepiped? No, in elementary worksheets, this is still a rectangular prism; the angle is just perspective.

Dimensions given: 5 m (length), 3 m (depth), 3 m (height) — all right angles.

So SA = 2(5·3 + 5·3 + 3·3) = 2(15 + 15 + 9) = 2(39) = 78 m²

But check labels: left vertical edge = 3 m, front bottom = 5 m, depth = 3 m, top edge also 5 m — yes, it's a box 5×3×3.

Problem 9: 78 m²

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Final answers:
1) 300
2) 42
3) 192
4) 196
5) 106
6) 176
7) 286
8) 136
9) 78

Make sure units match: ft², ft², in², in², yd², in², yd², ft², m² — as per problem.

Final Answer:
300, 42, 192, 196, 106, 176, 286, 136, 78