Surface area calculation worksheet for composite figures.

Let’s solve each problem one by one. We’ll use π = 3.14 as instructed, and round all answers to two decimal places.

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Problem 1: Cube with a hemisphere on top

The figure is a cube (7 ft × 7 ft × 7 ft) with a hemisphere sitting on top. The hemisphere has the same diameter as the cube’s face — so radius = 7/2 = 3.5 ft.

We need surface area of the *composite* figure. That means:

- Surface area of the cube MINUS the top face (because it’s covered by the hemisphere)
- PLUS the curved surface area of the hemisphere (we don’t include the flat base of the hemisphere because it’s glued to the cube)

Cube surface area normally = 6 × side² = 6 × 49 = 294 ft²
But we remove the top face → subtract 49 → 294 - 49 = 245 ft²

Hemisphere curved surface area = ½ × 4πr² = 2πr²
= 2 × 3.14 × (3.5)²
= 2 × 3.14 × 12.25
= 6.28 × 12.25 = let’s calculate:
6 × 12.25 = 73.5
0.28 × 12.25 = 3.43
Total = 73.5 + 3.43 = 76.93 ft²

Now add to cube part: 245 + 76.93 = 321.93 ft²

✔ Check: Yes, we didn’t double-count or miss anything. Top face removed, only curved hemisphere added.

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Problem 2: Cylinder with a cone on top

Cylinder: height = 12 in, diameter = 2.1 ft? Wait — units are mixed! Look again.

Actually, looking at the diagram: cylinder height is labeled “12” and diameter “2.1 ft”. But then cone height is “4” — probably also in feet? But that would be inconsistent. Wait — maybe all are in feet? Let me check labels.

Actually, re-examining: In problem 2, the cylinder says “12” and “2.1 ft”, and cone says “4”. Probably “12” and “4” are in inches? But that doesn’t make sense with “ft”. Hmm.

Wait — perhaps it’s a typo in my reading. Let me assume all dimensions are in the same unit. Since “2.1 ft” is written, and others are numbers without units, likely they’re all in feet. So:

Cylinder: height = 12 ft? That seems too big compared to diameter 2.1 ft. Maybe “12” is inches? But then “2.1 ft” is 25.2 inches — possible, but messy.

Alternatively — perhaps “12” and “4” are in the same unit as “2.1 ft”? Let’s assume all are in feet for consistency, even if unrealistic.

So:

Cylinder: radius r = 2.1 / 2 = 1.05 ft, height h_cyl = 12 ft
Cone: same radius r = 1.05 ft, height h_cone = 4 ft

Surface area of composite figure:

- Lateral surface area of cylinder (no top or bottom? Wait — does it have a bottom? Usually yes, unless specified. But since cone is on top, we don’t include top circle of cylinder.

Standard approach:

→ Bottom circle of cylinder: πr²
→ Lateral surface of cylinder: 2πrh
→ Lateral surface of cone: πrl, where l = slant height

Slant height of cone: l = √(r² + h²) = √(1.05² + 4²) = √(1.1025 + 16) = √17.1025 ≈ 4.1355 ft

Now compute:

Bottom circle: πr² = 3.14 × (1.05)^2 = 3.14 × 1.1025 ≈ 3.46185

Lateral cylinder: 2πrh = 2 × 3.14 × 1.05 × 12
First: 2 × 3.14 = 6.28
6.28 × 1.05 = 6.594
6.594 × 12 = 79.128

Lateral cone: πrl = 3.14 × 1.05 × 4.1355
First: 3.14 × 1.05 = 3.297
3.297 × 4.1355 ≈ let’s compute:
3.297 × 4 = 13.188
3.297 × 0.1355 ≈ 0.4468
Total ≈ 13.6348

Now sum:
Bottom: 3.46185
Cylinder lateral: 79.128
Cone lateral: 13.6348
Total ≈ 3.46185 + 79.128 = 82.58985 + 13.6348 ≈ 96.22465 → 96.22 ft²

Wait — but is the bottom included? In many such problems, if it's sitting on ground, sometimes bottom is excluded. But the problem doesn't specify. Looking back at problem 1, we included all except hidden faces. Here, likely we include bottom unless told otherwise.

But let me double-check standard practice. For composite figures like this, usually we include all external surfaces. So bottom should be included.

However, I recall that in some textbooks, for a cylinder with cone on top, they might not include the bottom if it's open, but here no indication. To be safe, let’s see if the answer makes sense.

Alternative thought: Perhaps the "12" and "4" are in inches, and "2.1 ft" is a mistake? Or vice versa? This is confusing.

Wait — looking at other problems: Problem 3 uses cm, problem 4 uses m, problem 5 uses cm, problem 6 uses ft. So units vary per problem. In problem 2, it says "2.1 ft", and the other numbers are likely also in feet. I'll proceed with that.

But 12 ft tall cylinder with 2.1 ft diameter is very skinny, but mathematically ok.

Another way: perhaps "12" is the total height? No, diagram shows cylinder height 12, cone height 4 separately.

I think my calculation is correct. Total ≈ 96.22 ft²

But let me recalculate slant height more accurately.

r = 1.05, h = 4
l = √(1.05² + 4²) = √(1.1025 + 16) = √17.1025

√17.1025: 4.135^2 = 17.098225, close. 4.136^2 = 4.136*4.136.

4.135^2 = (4+0.135)^2 = 16 + 2*4*0.135 + 0.135^2 = 16 + 1.08 + 0.018225 = 17.098225
17.1025 - 17.098225 = 0.004275, so approx 4.135 + 0.004275/(2*4.135) ≈ 4.135 + 0.000516 ≈ 4.1355, as before.

Now πrl = 3.14 * 1.05 * 4.1355

Compute step by step:

1.05 * 4.1355 = 1.05 * 4 = 4.2, 1.05 * 0.1355 ≈ 0.142275, total 4.342275

Then 3.14 * 4.342275 ≈ 3.14 * 4.342 = 3.14*4 = 12.56, 3.14*0.342 = approximately 1.07388, total 13.63388

Earlier I had 13.6348, close.

Now bottom: πr² = 3.14 * 1.1025 = let's calculate: 3.14*1.1 = 3.454, 3.14*0.0025=0.00785, total 3.46185

Cylinder lateral: 2*3.14*1.05*12 = 2*3.14=6.28, 6.28*1.05=6.594, 6.594*12=79.128

Sum: 3.46185 + 79.128 = 82.58985; +13.63388 = 96.22373 → rounds to 96.22 ft²

Okay.

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Problem 3: Two rectangular prisms stacked

Bottom prism: 12 cm long, 6 cm wide, 4 cm high
Top prism: 6 cm long, 6 cm wide, 4 cm high (since it sits on half the length? Diagram shows top is centered, so length 6, width 6, height 4)

When stacked, the bottom face of top prism covers part of top face of bottom prism. Specifically, it covers a 6x6 area.

So total surface area = SA of bottom prism + SA of top prism - 2 * area of overlap (because both the covered part of bottom and the bottom of top are internal now)

SA of bottom prism: 2(lw + lh + wh) = 2(12*6 + 12*4 + 6*4) = 2(72 + 48 + 24) = 2(144) = 288 cm²

SA of top prism: 2(6*6 + 6*4 + 6*4) = 2(36 + 24 + 24) = 2(84) = 168 cm²

Overlap area: 6*6 = 36 cm², and we subtract twice that (once from each prism's surface)

So total SA = 288 + 168 - 2*36 = 456 - 72 = 384 cm²

Is that correct? Let me visualize.

Bottom prism: normally 288, but top face has a 6x6 hole covered, so we lose 36 from top face. But the sides are still there.

Top prism: normally 168, but its bottom face is not exposed, so we lose 36.

So yes, total reduction is 72, so 288 + 168 - 72 = 384 cm².

We can also calculate manually:

Bottom prism:
- Bottom: 12*6 = 72
- Front/back: 2*(12*4) = 96
- Left/right: 2*(6*4) = 48
- Top: but top has a 6x6 covered, so exposed top is 12*6 - 6*6 = 72 - 36 = 36
Total for bottom: 72 + 96 + 48 + 36 = 252? Wait, that's not right because I missed something.

Standard SA formula includes all six faces. When we say SA of bottom prism is 288, that includes top face of 72. But after stacking, the top face is partially covered.

Better to think:

Exposed surfaces:

Bottom prism:
- Bottom: 12*6 = 72
- Four sides: front/back: 2*(12*4)=96, left/right: 2*(6*4)=48 → total sides 144
- Top: only the parts not covered. Since top prism covers 6x6 in the middle, and assuming it's centered, the top has two rectangles on sides: each 3cm x 6cm? Length is 12, top prism is 6 long, so overhangs 3cm on each end. Width is 6, same as top prism, so no overhang on width.

So top face of bottom prism: two strips: each 3cm (length) x 6cm (width) = 18 cm² each, total 36 cm² exposed.

So bottom prism exposed: bottom 72 + sides 144 + top exposed 36 = 252 cm²

Top prism:
- It has five faces exposed: top, front, back, left, right. Bottom is covered.
- Top: 6*6=36
- Front/back: 2*(6*4)=48
- Left/right: 2*(6*4)=48
Total: 36+48+48=132 cm²

Grand total: 252 + 132 = 384 cm². Same as before. Good.

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Problem 4: Cylinder with a hemisphere on bottom

Diagram shows a cylinder with a hemisphere attached to its bottom. Dimensions: cylinder height 9 m, diameter 7 m, so radius r = 3.5 m.

Surface area of composite figure:

- Lateral surface of cylinder: 2πrh
- Top circle of cylinder: πr² (since it's open on top? Or closed? Diagram doesn't show lid, but typically for such figures, if not specified, we assume the top is open? Wait, no — in surface area problems, unless stated, we include all external surfaces.

Looking at the figure: it's like a silo with dome bottom. Usually, the top is open? But the problem doesn't say. However, in the diagram, it might be implied that the top is open, but I think for surface area, we should include all surfaces that are exposed.

Standard interpretation: for a cylinder with hemisphere on bottom, the surface area includes:

- Lateral surface of cylinder
- Top circle of cylinder (if it's closed)
- Curved surface of hemisphere (not the flat part, since it's attached)

But if the top is open, we wouldn't include the top circle. The problem doesn't specify. Looking back at problem 1, we assumed the cube's top was covered, so we excluded it. Here, similarly, the hemisphere is attached to the bottom, so we exclude the bottom circle of cylinder and the flat part of hemisphere.

But what about the top of the cylinder? Is it open or closed? In many such problems, if it's a container, top might be open, but here no context.

To resolve, let's look at common practice. In most textbook problems for "cylinder with hemisphere on bottom", they include the top circle of the cylinder.

For example, a test tube shape.

I think we should include the top circle.

So:

Lateral cylinder: 2πrh = 2 * 3.14 * 3.5 * 9
First, 2*3.14=6.28
6.28*3.5=21.98
21.98*9=197.82

Top circle: πr² = 3.14 * (3.5)^2 = 3.14 * 12.25 = 38.465

Curved hemisphere: 2πr² = 2 * 3.14 * 12.25 = 6.28 * 12.25 = 76.93 (same as problem 1)

Now sum: 197.82 + 38.465 + 76.93 = let's add

197.82 + 38.465 = 236.285
236.285 + 76.93 = 313.215 → 313.22 m²

If the top were open, we wouldn't include the 38.465, but I think it's included.

Confirm with another way: total surface area without attachment would be cylinder SA + hemisphere SA, minus twice the area of the circle where they join (since both have a circle that is now internal).

Cylinder SA (closed): 2πr(h + r) = 2πr h + 2πr²
Hemisphere SA (curved only): 2πr²
But when joined, we remove two circles: one from cylinder bottom, one from hemisphere flat part.

So total SA = [2πr h + 2πr²] + [2πr²] - 2*(πr²) = 2πr h + 2πr²

Which is lateral cylinder plus top circle plus curved hemisphere? 2πr h is lateral, 2πr² is top circle plus... wait.

2πr h + 2πr² = lateral + 2*top? No.

From above: after subtraction, we have 2πr h + 2πr²

But 2πr² could be interpreted as top circle and something else? Actually, in this case, it's lateral cylinder plus the top circle plus the curved hemisphere, since the 2πr² comes from the original cylinder's two ends minus one end (bottom removed) plus the hemisphere's curved part which is 2πr², but we subtracted the flat part.

In our calculation, we have lateral cylinder (2πrh), top circle (πr²), and curved hemisphere (2πr²), sum is 2πrh + 3πr²? No:

I think I confused myself.

From direct method: we have three parts: lateral cylinder, top circle, curved hemisphere. Sum is 2πrh + πr² + 2πr² = 2πrh + 3πr²

But earlier I calculated 197.82 + 38.465 + 76.93 = 313.215, and 3πr² = 3*3.14*12.25 = 3*38.465=115.395, plus 197.82=313.215, yes.

But is this correct? For a cylinder with hemisphere on bottom, the total surface area should be lateral cylinder + top circle + curved hemisphere. Yes, because the bottom of cylinder is covered by hemisphere, and the flat part of hemisphere is covered, so only those three surfaces are exposed.

Yes.

Numerically: r=3.5, h=9

2πrh = 2*3.14*3.5*9 = as above 197.82

πr² = 3.14*12.25=38.465

2πr²=76.93

Sum 197.82+38.465=236.285; +76.93=313.215 → 313.22 m²

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Problem 5: Hemisphere on top of a cone

Diagram shows a hemisphere on top of a cone. Both have the same radius. Given: radius = 3.8 cm, and the cone's height is not given directly, but the total height from tip to top of hemisphere is 10.2 cm? Let's see.

It says: from the center of the hemisphere to the tip of the cone is 10.2 cm? Or total height?

Looking: it shows a line from the top of hemisphere down to the tip of cone, labeled 10.2 cm, and the radius is 3.8 cm.

Since the hemisphere has radius 3.8 cm, the distance from the center of the sphere to the top of hemisphere is 3.8 cm. But the line is from the top of hemisphere to the tip of cone, passing through the center.

So, the length from top of hemisphere to center of sphere is 3.8 cm (radius). Then from center to tip of cone is the rest.

Total length given is 10.2 cm, which is from top of hemisphere to tip of cone.

So, distance from center of sphere to tip of cone = 10.2 - 3.8 = 6.4 cm.

This distance is the height of the cone, because the cone's apex is at the tip, and the base is at the center plane of the hemisphere.

So, cone height h = 6.4 cm, radius r = 3.8 cm.

Now, surface area of composite figure:

- Curved surface of hemisphere: 2πr²
- Lateral surface of cone: πrl, where l is slant height

We do not include the flat base of the hemisphere or the base of the cone, because they are attached together.

So only the outer curved surfaces.

First, slant height of cone: l = √(r² + h²) = √(3.8² + 6.4²) = √(14.44 + 40.96) = √55.4

√55.4: 7.44^2 = 55.3536, 7.45^2=55.5025, so approx 7.443

Calculate: 7.44^2 = (7.4+0.04)^2=54.76 + 2*7.4*0.04 + 0.0016=54.76+0.592+0.0016=55.3536

55.4 - 55.3536=0.0464, so increment by 0.0464/(2*7.44)≈0.0464/14.88≈0.00312, so l≈7.44312 cm

Now, curved hemisphere: 2πr² = 2 * 3.14 * (3.8)^2 = 2 * 3.14 * 14.44

First, 3.14 * 14.44 = 3.14*14 = 43.96, 3.14*0.44=1.3816, total 45.3416

Then *2 = 90.6832

Lateral cone: πrl = 3.14 * 3.8 * 7.44312

First, 3.14 * 3.8 = 11.932

Then 11.932 * 7.44312 ≈ ?

11.932 * 7 = 83.524

11.932 * 0.4 = 4.7728

11.932 * 0.043 = approximately 0.513076

11.932 * 0.00012 ≈ negligible

Better: 11.932 * 7.44312

First, 11.932 * 7.4 = 11.932*7 = 83.524, 11.932*0.4=4.7728, total 88.2968

Then 11.932 * 0.04312 ≈ 11.932*0.04=0.47728, 11.932*0.00312≈0.03722784, total ≈0.5145

So total ≈ 88.2968 + 0.5145 = 88.8113

More accurately: use calculator in mind.

3.8 * 7.44312 = 3.8*7.44312

3*7.44312=22.32936, 0.8*7.44312=5.954496, total 28.283856

Then *3.14 = 3.14*28.283856

3*28.283856=84.851568, 0.14*28.283856≈3.95973984, total ≈88.81130784

So lateral cone ≈88.8113

Now sum with hemisphere: 90.6832 + 88.8113 = 179.4945 → 179.49 cm²

Round to two decimals.

Check: r=3.8, h_cone=6.4, l=√(3.8²+6.4²)=√(14.44+40.96)=√55.4≈7.443, as above.

2πr²=2*3.14*14.44=6.28*14.44

6*14.44=86.64, 0.28*14.44=4.0432, total 90.6832

πrl=3.14*3.8*7.443≈ as above 88.811

Sum 179.4942 → 179.49 cm²

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Problem 6: Rectangular prism with a pyramid on top

Prism: 10 ft long, 5 ft wide, 4 ft high
Pyramid: square base same as top of prism, so 10 ft by 5 ft? But pyramid base should match the top face. Top face is 10x5, so pyramid base is 10x5.

Height of pyramid is given as 6 ft? Diagram shows "6" from base to apex.

So, pyramid height h_pyr = 6 ft, base 10 ft by 5 ft.

Surface area of composite figure:

- Surface area of prism MINUS the top face (covered by pyramid)
- PLUS the lateral surface area of the pyramid (the four triangular faces)

First, SA of prism: 2(lw + lh + wh) = 2(10*5 + 10*4 + 5*4) = 2(50 + 40 + 20) = 2(110) = 220 ft²

Minus top face: 10*5 = 50 ft² → so 220 - 50 = 170 ft² for the prism part.

Now, lateral surface area of pyramid: it has four triangular faces. But since the base is rectangle, not square, the triangles are not all the same.

Specifically:

- Two triangles with base 10 ft and height equal to the slant height for that direction
- Two triangles with base 5 ft and height equal to the slant height for that direction

The slant height depends on the direction.

For the triangles along the length (base 10 ft): the slant height is the distance from the midpoint of a 10-ft side to the apex, along the face.

The pyramid height is 6 ft, and for the face with base 10 ft, the distance from the center of the base to the midpoint of the 10-ft side is half the width, which is 2.5 ft (since width is 5 ft).

Similarly, for the face with base 5 ft, the distance from center to midpoint of 5-ft side is half the length, 5 ft.

So, for the triangular faces with base 10 ft:

The slant height l1 = √(height² + (half-width)²) = √(6² + (2.5)²) = √(36 + 6.25) = √42.25 = 6.5 ft

For the triangular faces with base 5 ft:

Slant height l2 = √(height² + (half-length)²) = √(6² + 5²) = √(36 + 25) = √61 ≈ 7.8102 ft

Now, area of two triangles with base 10 ft: each has area (1/2)*base*slant_height = (1/2)*10*6.5 = 32.5 ft², so two of them: 65 ft²

Area of two triangles with base 5 ft: each (1/2)*5*√61 ≈ (1/2)*5*7.8102 = 2.5 * 7.8102 = 19.5255 ft², so two: 39.051 ft²

Total lateral pyramid SA = 65 + 39.051 = 104.051 ft²

Now add to prism part: 170 + 104.051 = 274.051 → 274.05 ft²

We can write exactly.

√61 is irrational, so we keep decimal.

l2 = √61 ≈ 7.81024967591

Then area for one small triangle: 0.5 * 5 * √61 = 2.5 √61

Two: 5 √61 ≈ 5 * 7.81024967591 = 39.05124837955

Large triangles: two of them, each 0.5*10*6.5=32.5, total 65

Sum lateral pyramid: 65 + 39.05124837955 = 104.05124837955

Prism without top: 220 - 50 = 170

Total: 170 + 104.05124837955 = 274.05124837955 → rounds to 274.05 ft²

Note: the slant height for the 10-ft base faces is exactly 6.5 because √(36 + 6.25)=√42.25=6.5, nice.

For the 5-ft base, √61 is fine.

So all done.

Now summarize answers:

1) 321.93 ft²
2) 96.22 ft²
3) 384.00 cm² (but since it's exact, we can write 384.00 or just 384, but problem says round to two decimal places, so 384.00)
In problem 3, all calculations were integers, so 384.00
4) 313.22 m²
5) 179.49 cm²
6) 274.05 ft²

For problem 3, since it's exactly 384, we write 384.00 to satisfy "two decimal places".

Similarly, others are already to two decimals.

Now, to present final answers.

Final Answer:
1) 321.93
2) 96.22
3) 384.00
4) 313.22
5) 179.49
6) 274.05