Volume & Surface Area - Free Printable

Let’s solve each shape one by one to find its surface area. We’ll match our answers to the boxes on the right. One box will be left empty — that’s the missing answer we need to find.

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Shape 1: Sphere (diameter = 12 cm)
Radius = 12 ÷ 2 = 6 cm
Surface area of sphere = 4πr² = 4 × π × 6² = 4 × π × 36 = 144π ≈ 452.4 cm²

→ Match with 452.4 cm²

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Shape 2: Cone (slant height = 22.4 cm, radius = 10 cm)
Surface area = πr(r + l) = π × 10 × (10 + 22.4) = π × 10 × 32.4 = 324π ≈ 1017.9 cm² → Wait, this doesn’t match any box? Let me double-check.

Wait — maybe they want only lateral surface area? But usually “surface area” for cone includes base unless specified.

But looking at the options, none are near 1018. Hmm. Maybe I misread the diagram?

Actually, let’s check Shape 5 first — it’s a hemisphere, and might help us see pattern.

Wait — perhaps Shape 2 is meant to use only lateral surface area? Let’s try:

Lateral surface area of cone = πrl = π × 10 × 22.4 = 224π ≈ 703.7 cm² — still not matching.

Hold on — maybe the 22.4 cm is NOT slant height? The diagram shows a dashed line from apex to center of base — that’s height, not slant!

Oh! Important mistake.

In Shape 2:
Height = 22.4 cm, radius = 10 cm → so slant height l = √(r² + h²) = √(100 + 501.76) = √601.76 ≈ 24.53 cm

Then total surface area = πr(r + l) = π×10×(10 + 24.53) = π×10×34.53 ≈ 345.3π ≈ 1084.8 cm² — still no match.

This is confusing. Let’s skip and come back.

Maybe the diagram labels are different. Let’s look at Shape 3.

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Shape 3: Square pyramid (base 8 cm, height 15 cm)
First, find slant height of triangular face.

Half base = 4 cm, height = 15 cm → slant height = √(4² + 15²) = √(16 + 225) = √241 ≈ 15.52 cm

Area of one triangle = (1/2) × base × slant height = 0.5 × 8 × 15.52 ≈ 62.08 cm²
Four triangles = 4 × 62.08 ≈ 248.32 cm²
Base area = 8 × 8 = 64 cm²
Total surface area ≈ 248.32 + 64 = 312.32 cm² — close to 304? Not quite.

Wait — maybe they used exact value?

√241 is irrational. Maybe they expect us to use Pythagorean triple? 8-15-17? But half-base is 4, not 8.

Wait — if height is 15, and half-base is 4, then slant height is √(4²+15²)=√241≈15.52 — correct.

But 312 is not in options. Options have 304, 452.4, etc.

Perhaps they forgot the base? Lateral only: 248.32 — still not matching.

Let’s try Shape 4.

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Shape 4: Trapezoidal prism? Actually, it looks like a frustum of a pyramid? Or a trapezoid-based prism?

Looking: top rectangle 8cm x ? , bottom 26cm x 6cm? Height 15cm, and there’s a red dashed line labeled 12cm — probably height of trapezoid face.

Actually, it seems to be a prism with trapezoidal bases.

Top base: 8 cm, bottom base: 26 cm, height of trapezoid: 12 cm, and depth (length of prism): 6 cm? And side faces are rectangles.

Wait — the 15cm is labeled on the side — maybe that’s the length of the non-parallel sides? This is messy.

Alternative approach: Let’s look at the given answers and work backwards.

Given answers:
942.5, 646.5, 5300, 452.4, 304, 42223, 936, 792

We already matched Shape 1 (sphere) to 452.4 — that’s solid.

Shape 5: Hemisphere (diameter 20 cm → radius 10 cm)

Surface area of hemisphere = curved part + flat circle = 2πr² + πr² = 3πr² = 3 × π × 100 = 300π ≈ 942.5 cm²

Yes! That matches.

So Shape 5 → 942.5 cm²

Now Shape 6: Cylinder (radius 0.6 m, height 52 cm)

Units are different! Radius in meters, height in cm.

Convert to same unit. Let’s convert everything to cm.

0.6 m = 60 cm

Cylinder surface area = 2πr(h + r) = 2π × 60 × (52 + 60) = 2π × 60 × 112 = 13440π ≈ 42223 cm²

Yes! Matches 42223 cm³? Wait, it says cm³ but should be cm² — typo in image.

So Shape 6 → 42223 cm²

Shape 7: Triangular prism? It has dimensions: base triangle with base 35 cm, height 0.625 m? And length 0.2 m?

Again, mixed units.

Convert all to cm:

0.625 m = 62.5 cm
0.2 m = 20 cm

The triangular base: base = 35 cm, height = 62.5 cm → area of one triangle = (1/2)*35*62.5 = 1093.75 cm²

Two triangles: 2187.5 cm²

Now rectangular faces:

There are three rectangles:

1. Base rectangle: 35 cm × 20 cm = 700 cm²
2. Two side rectangles: each is hypotenuse of triangle × 20 cm

First, find hypotenuse of triangle: since it’s a right triangle? Diagram shows right angle? Assuming yes.

Legs 35 cm and 62.5 cm → hypotenuse = √(35² + 62.5²) = √(1225 + 3906.25) = √5131.25 ≈ 71.63 cm

So two side rectangles: 2 × 71.63 × 20 ≈ 2865.2 cm²

Total surface area = 2187.5 + 700 + 2865.2 ≈ 5752.7 cm² — not matching any.

Wait — maybe the 0.625 m is the height of the prism, not the triangle?

Re-examining diagram: Shape 7 has a triangle with base 35 cm, and a dimension 0.625 m going up — likely the height of the triangle. And 0.2 m is the length of the prism.

But my calculation gives ~5753, not in options.

Options have 5300, 792, 936, etc.

Perhaps it’s not a right triangle? Or maybe I misinterpreted.

Another idea: maybe the 0.625 m is the length, and 0.2 m is something else? Diagram is unclear.

Let’s try Shape 8.

Shape 8: Triangular pyramid (tetrahedron?) with base triangle sides 8cm, 15cm, 17cm — wait, 8-15-17 is right triangle!

And height from apex to base is 12 cm? But for surface area, we need areas of all four faces.

Base: right triangle, legs 8 and 15 → area = (1/2)*8*15 = 60 cm²

Now three lateral faces. Each is a triangle with base as side of base triangle, and height being the slant height from apex.

But we don’t have slant heights. The diagram shows a dashed line of 12 cm — probably the height of the pyramid, perpendicular to base.

To find area of lateral faces, we need the slant heights for each face.

For example, for the face with base 8 cm: the distance from apex to the midpoint of the 8 cm side.

Since the pyramid height is 12 cm, and the foot of the perpendicular is at the centroid? For a right triangle base, the centroid is at average of vertices.

This is getting complicated. Perhaps assume it's a regular pyramid? But base is scalene.

Maybe the 12 cm is the height of the lateral faces? Unlikely.

Let’s calculate the area using coordinates or vector, but too advanced.

Perhaps the 12 cm is the slant height for all faces? But that wouldn't make sense for irregular base.

Another thought: maybe it's not a pyramid with apex above centroid, but the diagram shows a specific configuration.

Looking back, perhaps for Shape 8, the 12 cm is the height of the pyramid, and we can find the areas.

But let's skip and do Shape 9.

Shape 9: Regular hexagonal prism. Side 8 cm, height 12 cm, and apothem 6 cm? Diagram shows "6cm" from center to side — that's apothem.

For regular hexagon, area = (1/2) * perimeter * apothem = (1/2) * (6*8) * 6 = (1/2)*48*6 = 144 cm² per base.

Two bases: 288 cm²

Lateral surface area: 6 rectangles, each 8 cm wide and 12 cm high → 6 * 8 * 12 = 576 cm²

Total surface area = 288 + 576 = 864 cm² — not in options. Options have 792, 936, etc.

Close to 792? 864 - 792 = 72, not sure.

Perhaps the 6 cm is not apothem? Or maybe it's the height of the triangular parts.

Regular hexagon can be divided into 6 equilateral triangles. If side is 8 cm, area of one equilateral triangle = (√3/4)*s² = (1.732/4)*64 ≈ 0.433*64 ≈ 27.712 cm², times 6 = 166.272 cm² per base, times 2 = 332.544, plus lateral 576 = 908.544 — still not matching.

But option has 936. Close but not exact.

Perhaps they used approximation.

Or maybe the 6 cm is the apothem, and for regular hexagon, apothem a = (s√3)/2 = (8*1.732)/2 ≈ 6.928 cm, but diagram says 6 cm, so perhaps it's not regular? But it says "regular hexagonal prism".

Diagram shows "6cm" from center to side, so apothem is 6 cm.

Then area of hexagon = (1/2)*perimeter*apothem = (1/2)*(6*8)*6 = 144 cm², as before.

Lateral area = 6*8*12 = 576

Total 720? 144*2=288, +576=864.

Not matching.

Let's go back to Shape 3.

Shape 3: square pyramid, base 8x8, height 15 cm.

Slant height for triangular face: from apex to midpoint of base side.

Distance from center of base to midpoint of side is 4 cm (half of 8).

Height is 15 cm, so slant height l = √(4² + 15²) = √(16+225) = √241 ≈ 15.524 cm

Area of one triangular face = (1/2)*8*15.524 = 62.096 cm²

Four faces: 248.384 cm²

Base: 64 cm²

Total: 312.384 cm²

Closest option is 304 cm². Perhaps they used l = 15 cm? But that would be if height was along the face, but it's not.

If they mistakenly used height 15 as slant height, then area = 4*(1/2)*8*15 = 240, plus base 64 = 304 cm². Oh! That matches.

Probably a common mistake, or perhaps in the diagram, the 15 cm is intended to be the slant height, even though it's drawn as height.

In many textbooks, for pyramids, sometimes the given "height" for surface area is the slant height if not specified.

So likely, for Shape 3, they intend slant height = 15 cm.

So surface area = lateral + base = 4*(1/2)*8*15 + 8*8 = 240 + 64 = 304 cm²

Yes! Matches.

So Shape 3 → 304 cm²

Now Shape 2: cone. If we assume the 22.4 cm is slant height, and radius 10 cm.

Then surface area = πr(r + l) = π*10*(10 + 22.4) = π*10*32.4 = 324π ≈ 1017.88 — not in options.

Lateral only: πrl = π*10*22.4 = 224π ≈ 703.7 — not in options.

But options have 792, 936, etc.

Perhaps radius is not 10? Diagram says "10 cm" at base, so diameter? No, it's labeled as radius? In circle, usually if it's from center to edge, it's radius.

In Shape 1, diameter is given as 12 cm, so radius 6.

In Shape 2, it's labeled "10 cm" with arrow from center to edge, so radius 10 cm.

But 224π is about 703.7, not matching.

Unless they want only lateral, and use π=3.14, 224*3.14=703.36, still not.

Perhaps the 22.4 is height, and we need to calculate slant.

As before, l = √(10^2 + 22.4^2) = √(100 + 501.76) = √601.76 = 24.53, then π*10*(10+24.53) = 345.3*3.14≈1084.2, not matching.

Let's look at Shape 4 again.

Shape 4: it looks like a trapezoidal prism. Top base 8 cm, bottom base 26 cm, height of trapezoid 12 cm, and the length of the prism is 6 cm? And the non-parallel sides are 15 cm each? Diagram shows "15cm" on the side.

So, the two trapezoidal bases: area of one trapezoid = (1/2)*(sum of parallel sides)*height = (1/2)*(8+26)*12 = (1/2)*34*12 = 204 cm²

Two bases: 408 cm²

Now lateral faces: there are four rectangles.

- Two rectangles corresponding to the parallel sides: one is 8 cm × 6 cm = 48 cm², other is 26 cm × 6 cm = 156 cm²

- Two rectangles for the non-parallel sides: each is 15 cm × 6 cm = 90 cm², so two are 180 cm²

Total lateral = 48 + 156 + 180 = 384 cm²

Total surface area = 408 + 384 = 792 cm²

Yes! Matches 792 cm²

So Shape 4 → 792 cm²

Now Shape 7: let's revisit.

Dimensions: base triangle with base 35 cm, height 0.625 m = 62.5 cm, and length of prism 0.2 m = 20 cm.

Assuming the triangle is right-angled, with legs 35 cm and 62.5 cm.

Area of one triangle = (1/2)*35*62.5 = 1093.75 cm²

Two triangles: 2187.5 cm²

Rectangular faces:

- Rectangle on base 35 cm: 35 * 20 = 700 cm²

- Rectangle on height 62.5 cm: 62.5 * 20 = 1250 cm²

- Rectangle on hypotenuse: hypotenuse = √(35^2 + 62.5^2) = √(1225 + 3906.25) = √5131.25 = 71.63 cm, so area = 71.63 * 20 ≈ 1432.6 cm²

Total lateral = 700 + 1250 + 1432.6 = 3382.6 cm²

Total surface area = 2187.5 + 3382.6 = 5570.1 cm² — not matching.

But options have 5300, which is close. Perhaps they approximated.

Maybe the 0.625 m is the length, and 0.2 m is the height of the triangle? Let's swap.

Suppose the triangle has base 35 cm, height 0.2 m = 20 cm, and length of prism 0.625 m = 62.5 cm.

Then area of triangle = (1/2)*35*20 = 350 cm², two triangles = 700 cm²

Rectangles:

- On base 35 cm: 35 * 62.5 = 2187.5 cm²

- On height 20 cm: 20 * 62.5 = 1250 cm²

- On hypotenuse: hypotenuse = √(35^2 + 20^2) = √(1225 + 400) = √1625 = 40.31 cm, area = 40.31 * 62.5 ≈ 2519.375 cm²

Total lateral = 2187.5 + 1250 + 2519.375 = 5956.875 cm²

Total = 700 + 5956.875 = 6656.875 — not matching.

Perhaps it's not a right triangle. Or maybe the 0.625 m is the slant or something.

Another idea: perhaps the "0.625 m" is the height of the prism, and "0.2 m" is related to the triangle.

Let's calculate what would give 5300.

Suppose total surface area is 5300 cm².

From earlier, if we have two triangles and three rectangles.

Assume the triangle is right-angled with legs a,b, hypotenuse c, length L.

Area = 2*(1/2*a*b) + a*L + b*L + c*L = a*b + L(a+b+c)

Set equal to 5300.

From diagram, a=35 cm, and say b=62.5 cm, L=20 cm, then a*b=2187.5, a+b+c=35+62.5+71.63=169.13, L*that=3382.6, sum 5570.1, as before.

If L=19 cm, then L(a+b+c)=19*169.13≈3213.47, total 2187.5+3213.47=5400.97 — closer to 5300.

If b=60 cm, then c=√(35^2+60^2)=√(1225+3600)=√4825≈69.46, a+b+c=35+60+69.46=164.46, a*b=2100, L=20, L*sum=3289.2, total 5389.2 — still not 5300.

Perhaps the 0.625 m is 62.5 cm, but it's the length, and the triangle has base 35 cm, and the height is such that area is less.

Maybe the triangle is not right-angled, and the 0.625 m is the height from apex to base, but for surface area, we need the actual side lengths.

This is taking too long. Let's try Shape 8.

Shape 8: triangular pyramid with base 8,15,17 cm (right triangle), and height 12 cm from apex to base.

To find surface area, we need areas of the three lateral faces.

Each lateral face is a triangle with base as side of base, and height being the distance from apex to that side.

Since the apex is directly above the centroid or orthocenter? For a right triangle, the orthocenter is at the right-angle vertex.

Assume the apex is directly above the right-angle vertex of the base. Then the three lateral faces are all right triangles or something.

Suppose the base is triangle ABC, right-angled at C, with AC=8, BC=15, AB=17.

Apex P is directly above C, at height 12 cm.

Then the three lateral faces are:

- Triangle PAC: points P,A,C. Since PC is perpendicular to base, and AC is in base, so angle at C is 90 degrees. So triangle PAC is right-angled at C, with legs PC=12, AC=8, so area = (1/2)*12*8 = 48 cm²

- Triangle PBC: similarly, right-angled at C, legs PC=12, BC=15, area = (1/2)*12*15 = 90 cm²

- Triangle PAB: this is the face opposite C. Points P,A,B. We need its area.

First, find distances PA and PB.

PA = distance from P to A = √(PC^2 + AC^2) = √(144 + 64) = √208 = 4√13 ≈ 14.422 cm

PB = √(PC^2 + BC^2) = √(144 + 225) = √369 = 3√41 ≈ 19.209 cm

AB = 17 cm

So triangle PAB has sides 14.422, 19.209, 17 cm.

Use Heron's formula.

Semi-perimeter s = (14.422 + 19.209 + 17)/2 = 50.631/2 = 25.3155

Area = √[s(s-a)(s-b)(s-c)] = √[25.3155*(25.3155-14.422)*(25.3155-19.209)*(25.3155-17)] = √[25.3155*10.8935*6.1065*8.3155]

Calculate step by step:

25.3155 * 10.8935 ≈ 275.8

6.1065 * 8.3155 ≈ 50.78

Then 275.8 * 50.78 ≈ 14000 roughly

√14000 ≈ 118.32 cm²

More accurately:

25.3155 * 10.8935 = let's compute: 25.3155*10 = 253.155, 25.3155*0.8935≈22.62, total 275.775

6.1065*8.3155 = 6.1065*8 = 48.852, 6.1065*0.3155≈1.926, total 50.778

Then 275.775 * 50.778 ≈ 275.775*50 = 13788.75, 275.775*0.778≈214.6, total 14003.35

√14003.35 ≈ 118.34 cm²

So area of PAB ≈ 118.34 cm²

Now base area = (1/2)*8*15 = 60 cm²

Total surface area = areas of PAC + PBC + PAB + base = 48 + 90 + 118.34 + 60 = 316.34 cm² — not matching any.

But options have 936, 792, etc.

Perhaps the apex is above the centroid.

Centroid of right triangle is at (a/3, b/3) from right angle, but for area calculation, it's messy.

Maybe the 12 cm is the slant height for the faces.

Another idea: perhaps for Shape 8, the 12 cm is the height of the lateral faces, but that doesn't make sense.

Let's look at the remaining options.

We have matched:

Shape 1: 452.4

Shape 3: 304

Shape 4: 792

Shape 5: 942.5

Shape 6: 42223

Left: 942.5 is used, 646.5, 5300, 936, and one missing.

Shapes left: 2,7,8,9

Shape 9: regular hexagonal prism, side 8 cm, height 12 cm, apothem 6 cm.

As before, area of hexagon = (1/2)*perimeter*apothem = (1/2)*48*6 = 144 cm² per base, so 288 for two.

Lateral area = 6 * 8 * 12 = 576 cm²

Total 864 cm² — not in options.

But 936 is close. 936 - 864 = 72, which is 6*12, perhaps they included something else.

Maybe the apothem is not 6, or side is different.

Diagram shows "6cm" from center to side, so apothem 6 cm.

For regular hexagon, if apothem a = 6, then side s = 2a / √3 = 12 / 1.732 ≈ 6.928 cm, but diagram says side 8 cm, contradiction.

So probably the 6 cm is not the apothem, but the height of the triangular sections or something.

Perhaps "6cm" is the distance from center to vertex, i.e., radius.

In regular hexagon, radius = side length, so if radius is 6 cm, side is 6 cm, but diagram says 8 cm.

Confusion.

Perhaps for Shape 9, the "6cm" is the apothem, and side is 8 cm, but in reality for regular hexagon, apothem a = (s√3)/2, so if s=8, a= (8*1.732)/2 = 6.928, not 6. So perhaps it's approximate, or they want us to use a=6.

Then area = (1/2)*48*6 = 144, as before.

Lateral 576, total 864.

But 936 is 864 + 72, and 72 = 6*12, perhaps they added an extra face or something.

Maybe the height is not 12, but something else.

Another thought: perhaps the "12cm" is the length, and "8cm" is side, and "6cm" is apothem, but for surface area, they include only lateral or something.

Let's calculate what 936 could be.

Suppose lateral area is 6*8*12 = 576, then bases must be (936-576)/2 = 360/2 = 180 cm² per base.

For regular hexagon, area = (3√3/2) s^2 = (3*1.732/2)*64 = (5.196/2)*64 = 2.598*64 = 166.272, not 180.

If s=8, area should be about 166, not 180.

Perhaps s=8, but they used different formula.

Or perhaps the 6 cm is used differently.

Let's try Shape 2 again.

Suppose for cone, they want lateral surface area only, and use π=3.14, r=10, l=22.4, so πrl = 3.14*10*22.4 = 3.14*224 = 703.36, not in options.

But 792 is taken, 936 is there.

936 / (π*10) = 936 / 31.4 ≈ 29.8, so l≈29.8, not 22.4.

Perhaps r is different.

Another idea: in Shape 2, the "10 cm" might be diameter, not radius.

Let me check the diagram description. In Shape 1, it's "12cm" with arrow across, so diameter.

In Shape 2, "10 cm" with arrow from center to edge, so likely radius.

But let's assume it's diameter, so radius 5 cm.

Then if slant height 22.4 cm, surface area = πr(r+l) = π*5*(5+22.4) = 5*27.4*π = 137π ≈ 430.4, not matching.

Lateral only: π*5*22.4 = 112π ≈ 351.86, not matching.

If height is 22.4, radius 5, then l = √(25 + 501.76) = √526.76 ≈ 22.95, then surface area = π*5*(5+22.95) = 5*27.95*π = 139.75π ≈ 439, not matching.

Perhaps for Shape 7, with units.

Let me try Shape 7 with different interpretation.

Suppose the triangle has base 35 cm, and the 0.625 m is the length of the prism, and 0.2 m is the height of the triangle.

So b=35 cm, h=20 cm, L=62.5 cm.

Area of triangle = (1/2)*35*20 = 350 cm², two = 700 cm²

Rectangles:

- On base 35 cm: 35 * 62.5 = 2187.5 cm²

- On height 20 cm: 20 * 62.5 = 1250 cm²

- On hypotenuse: c = √(35^2 + 20^2) = √(1225+400) = √1625 = 5√65 ≈ 5*8.062 = 40.31 cm, area = 40.31 * 62.5 ≈ 2519.375 cm²

Total lateral = 2187.5 + 1250 + 2519.375 = 5956.875 cm²

Total = 700 + 5956.875 = 6656.875 — not 5300.

Perhaps the 0.2 m is the length, and 0.625 m is the height, but then as before.

Let's calculate 5300 - 2*area_triangle.

Suppose area_triangle = A, then 2A + L*(a+b+c) = 5300.

From diagram, a=35, and say b and c are unknown.

Perhaps the triangle is isosceles or something.

Another idea: perhaps "0.625 m" is the slant height or something, but unlikely.

Let's look at Shape 8 with a different approach.

Perhaps the 12 cm is the height, and the base is 8,15,17, and they want us to find the areas using the fact that the apex is above the incenter or something, but complicated.

Maybe for Shape 8, the surface area is calculated as sum of areas, and they have a specific value.

Let's list the unmatched shapes and unmatched answers.

Matched:
- 1: 452.4
- 3: 304
- 4: 792
- 5: 942.5
- 6: 42223

Unmatched shapes: 2,7,8,9

Unmatched answers: 646.5, 5300, 936, and one missing (since there are 9 shapes and 8 answers, one answer is missing, but there are 8 answer boxes, and 9 shapes, so one shape's answer is not listed, and we need to find which one is missing.

The instruction says: "There is one missing answer. Complete this yourself."

So among the 9 shapes, 8 have answers in the boxes, one does not, and we need to find which one is missing and calculate it.

So far, we have assigned 5 shapes to answers.

Shapes 2,7,8,9 are unassigned, and answers 646.5, 5300, 936 are unassigned, and one more answer is missing, but there are only 8 answer boxes, so for 9 shapes, 8 answers are given, one is missing, so we need to find which shape's answer is not in the list, and calculate it.

The answer boxes are: 942.5, 646.5, 5300, 452.4, 304, 42223, 936, 792 — that's 8 answers.

Shapes 1 to 9, so one shape's surface area is not among these, and we need to calculate it and box it as the missing answer.

So let's continue assigning.

From earlier, Shape 9: regular hexagonal prism, side 8 cm, height 12 cm, and "6cm" likely apothem.

But as calculated, surface area = 2* (1/2*6*8*6) + 6*8*12 = 2*144 + 576 = 288 + 576 = 864 cm²

864 is not in the list, so perhaps this is the missing answer.

But let's verify other shapes.

Shape 2: cone, r=10 cm, if we take slant height l=22.4 cm, then lateral surface area = πrl = 3.14*10*22.4 = 703.36, not in list.

Total surface area = πr(r+l) = 3.14*10*32.4 = 1017.36, not in list.

But 646.5 is there. 646.5 / (π*10) = 646.5 / 31.4 ≈ 20.59, so if r+l = 20.59, l=10.59, not 22.4.

Perhaps r=5, then π*5*(5+l) = 646.5, so 5*(5+l) = 646.5/3.14 ≈ 205.89, so 5+l = 41.178, l=36.178, not matching.

Another possibility: for Shape 7, if we take the triangle as having base 35 cm, and the 0.625 m = 62.5 cm is the length, and the 0.2 m = 20 cm is the height, but then as before.

Perhaps the "0.625 m" is the height of the triangle, and "0.2 m" is the length, and the base is 35 cm, and it's a right triangle, but then surface area is large.

Let's calculate for Shape 7 with b=35 cm, h=62.5 cm, L=20 cm, and assume it's right-angled, then as before ~5570, close to 5300? 5570 - 5300 = 270, not very close.

Perhaps they used π or something, but no π in prism.

Another idea: for Shape 8, if we assume that the three lateral faces have areas based on the height.

Perhaps the 12 cm is the slant height for all faces, but that would require the base to be equilateral, but it's 8,15,17.

Let's calculate the area if we take the lateral faces as triangles with base and height 12 cm.

For example, for the face with base 8 cm, area = (1/2)*8*12 = 48 cm²

Base 15 cm: (1/2)*15*12 = 90 cm²

Base 17 cm: (1/2)*17*12 = 102 cm²

Base area = (1/2)*8*15 = 60 cm² (since right-angled)

Total = 48+90+102+60 = 300 cm² — not in list.

But 304 is close, but already used for Shape 3.

Perhaps for Shape 2, if we take r=10, and l=20.5 or something.

Let's try Shape 9 with different assumption.

Suppose the "6cm" is the side of the hexagon, but diagram says "8 cm" for side, and "6cm" for apothem.

Perhaps "8 cm" is the diameter or something, but unlikely.

Another thought: in Shape 9, the "12cm" might be the apothem or something, but no.

Let's calculate the surface area for Shape 9 using the given numbers.

Perhaps the 6 cm is the height of the triangular faces when unfolded, but for prism, it's rectangles.

I think for Shape 9, with side 8 cm, height 12 cm, and if we ignore the 6 cm or use it for area.

Standard formula for regular hexagonal prism: surface area = 2 * (3√3/2 * s^2) + 6*s*h = 3√3 s^2 + 6s h

With s=8, h=12, √3≈1.732, so 3*1.732*64 = 3*110.848 = 332.544, plus 6*8*12=576, total 908.544 cm²

Close to 936? 936 - 908.544 = 27.456, not very close.

If s=8, h=12, and they used √3=1.73, then 3*1.73*64 = 3*110.72 = 332.16, +576=908.16.

Still not 936.

Perhaps the height is 13 cm or something.

936 - 576 = 360 for two bases, so 180 per base.

For regular hexagon, area = (3√3/2) s^2 = 180, so s^2 = 180 *2 /(3*1.732) = 360 / 5.196 ≈ 69.28, s≈8.32, not 8.

So not matching.

Let's try Shape 7 with the following: perhaps the 0.625 m is 62.5 cm, but it's the length, and the triangle has base 35 cm, and the other side is 0.2 m = 20 cm, but then it's not specified.

Perhaps the triangle is 35 cm base, and the 0.625 m is the height, and 0.2 m is not used, but that doesn't make sense.

Another idea: for Shape 7, the "0.625 m" might be the slant height or something, but unlikely.

Let's calculate what 5300 could be.

Suppose for a cylinder or something, but it's a prism.

Perhaps for Shape 2, if we take the cone with r=10, and height 22.4, then l=24.53, surface area = π*10*(10+24.53) = 345.3*3.14=1084.242, not 5300.

5300 is large, so likely Shape 6 is 42223, which is large, so 5300 might be for a larger shape.

Shape 6 is cylinder with r=60 cm, h=52 cm, surface area 2πr(h+r) = 2*3.14*60*(52+60) = 376.8*112 = 42201.6, close to 42223, so ok.

Shape 7 might be the one with 5300.

Let me assume that for Shape 7, the triangle has base 35 cm, height 62.5 cm, length 20 cm, and they calculated lateral area as perimeter times length, but for prism, lateral area is perimeter of base times height (length).

Perimeter of base triangle: if right-angled with legs 35, 62.5, then hypotenuse 71.63, perimeter = 35+62.5+71.63 = 169.13 cm

Lateral area = perimeter * length = 169.13 * 20 = 3382.6 cm²

Area of two bases = 2 * (1/2*35*62.5) = 35*62.5 = 2187.5 cm²

Total 3382.6 + 2187.5 = 5570.1 cm²

If they used h=19 cm for length, then 169.13*19 = 3213.47, +2187.5 = 5400.97, still not 5300.

If they used different values.

Perhaps the 0.625 m is 62.5 cm, but it's the length, and the triangle has base 35 cm, and the height is 20 cm, but then perimeter = 35+20+40.31=95.31, lateral = 95.31*62.5 = 5956.875, bases 2*350=700, total 6656.875.

Not 5300.

Perhaps the "0.2 m" is the height of the triangle, and "0.625 m" is the length, and base 35 cm, but then same as above.

Let's try Shape 8 with the following: perhaps the 12 cm is the height, and the base is 8,15,17, and they want the surface area as sum, and perhaps they have a calculation.

Maybe for the lateral faces, they used the height 12 cm as the height for each, but that would be incorrect.

Another idea: in some contexts, for pyramids, if the apex is above the centroid, but for right triangle, centroid is at (8/3, 15/3) = (2.67, 5) from C, then distance to A, B, etc.

This is too complicated for school level.

Perhaps for Shape 2, the "22.4 cm" is the diameter or something, but unlikely.

Let's look at the answer 646.5.

646.5 / π = 205.89, so if for a circle, r^2 = 205.89, r=14.35, not matching.

For a cone, if r=10, then r+ l = 20.59, l=10.59, not 22.4.

Perhaps r=5, then r+l = 41.178, l=36.178.

Or for a sphere, 4πr^2 = 646.5, r^2 = 646.5/(4*3.14) = 646.5/12.56 ≈ 51.47, r=7.17, not matching.

Let's consider Shape 9 again.

Suppose the "6cm" is the side, but diagram says "8 cm" for side, and "6cm" for apothem.

Perhaps "8 cm" is the diameter of the circumscribed circle, but for hexagon, diameter = 2*side, so side=4 cm, then area = (3√3/2)*16 = 24*1.732 = 41.568 per base, times 2 = 83.136, lateral 6*4*12=288, total 371.136, not matching.

Perhaps the height is 6 cm, and side 8 cm, apothem not given.

Then lateral area = 6*8*6 = 288 cm²

Bases: if regular hexagon side 8, area = (3√3/2)*64 = 3*1.732*32 = 5.196*32 = 166.272 per base, times 2 = 332.544, total 288+332.544=620.544, close to 646.5? 646.5 - 620.544 = 25.956, not very close.

If they used √3=1.73, 3*1.73*32 = 5.19*32 = 166.08, times 2 = 332.16, +288 = 620.16.

Still not.

Perhaps for Shape 7, with b=35 cm, h=20 cm, L=62.5 cm, and they calculated only lateral area or something.

Lateral area = perimeter * L = (35+20+40.31)*62.5 = 95.31*62.5 = 5956.875, not 5300.

5300 / 62.5 = 84.8, so perimeter 84.8, then for triangle with base 35, other sides sum 49.8, etc.

Assume the triangle is isosceles with base 35 cm, and equal sides s, then height h = sqrt(s^2 - (17.5)^2)

But we have h=20 cm or 62.5 cm.

Suppose h=20 cm, then s = sqrt(17.5^2 + 20^2) = sqrt(306.25 + 400) = sqrt(706.25) = 26.57 cm

Perimeter = 35 + 2*26.57 = 88.14 cm

Lateral area = 88.14 * L

If L=62.5, 88.14*62.5 = 5508.75, close to 5300? 5508 - 5300 = 208, not very.

If L=60, 88.14*60 = 5288.4, very close to 5300!

5288.4 ≈ 5300, perhaps rounded.

And bases: area of one triangle = (1/2)*35*20 = 350 cm², two = 700 cm²

Total surface area = 5288.4 + 700 = 5988.4, not 5300.

If they forgot the bases, then lateral area 5288.4 ≈ 5300.

But usually surface area includes bases.

Perhaps for this shape, they consider only lateral, but unlikely.

In the diagram, for Shape 7, it might be open or something, but not specified.

Perhaps the 0.2 m is not used, but that doesn't make sense.

Another possibility: "0.2 m" is the thickness or something, but unlikely.

Let's assume that for Shape 7, with base triangle base 35 cm, height 20 cm (so area 350 cm²), and length 62.5 cm, and they calculated lateral area as perimeter times length, with perimeter 35 + 2* sqrt(17.5^2 + 20^2) = 35 + 2*26.57 = 88.14 cm, times 62.5 = 5508.75, and if they have 5300, not match.

Perhaps the height is 62.5 cm for the triangle, and length 20 cm, and base 35 cm, then s = sqrt(17.5^2 + 62.5^2) = sqrt(306.25 + 3906.25) = sqrt(4212.5) = 64.9 cm, perimeter = 35 + 2*64.9 = 164.8 cm, lateral area = 164.8 * 20 = 3296 cm², bases 2* (1/2*35*62.5) = 35*62.5 = 2187.5, total 5483.5, close to 5300? 5483 - 5300 = 183, not very.

5483.5 - 5300 = 183.5, error of 3.3%, perhaps acceptable, but let's see other options.

Perhaps for Shape 8, with base 8,15,17, height 12, and if we take the lateral faces as having heights from apex.

Earlier when I assumed apex above C, I got 48+90+118.34+60=316.34, not matching.

If apex above the incenter or circumcenter.

For right triangle, circumcenter is at midpoint of hypotenuse.

So if apex P is above the midpoint of AB.

AB=17 cm, so midpoint M.

Then PM = 12 cm.

Then distance from M to A = 8.5 cm, to B = 8.5 cm, to C = ? In right triangle, distance from midpoint of hypotenuse to C is half the hypotenuse, so 8.5 cm.

So PA = PB = PC = sqrt(12^2 + 8.5^2) = sqrt(144 + 72.25) = sqrt(216.25) = 14.706 cm

Then the three lateral faces are all isosceles triangles with sides 14.706, 14.706, and base 8,15,17 respectively.

For face with base 8 cm: sides 14.706, 14.706, 8.

Height from P to base 8: h = sqrt(14.706^2 - 4^2) = sqrt(216.25 - 16) = sqrt(200.25) = 14.15 cm

Area = (1/2)*8*14.15 = 56.6 cm²

Similarly for base 15 cm: h = sqrt(14.706^2 - 7.5^2) = sqrt(216.25 - 56.25) = sqrt(160) = 12.649 cm, area = (1/2)*15*12.649 = 94.8675 cm²

For base 17 cm: h = sqrt(14.706^2 - 8.5^2) = sqrt(216.25 - 72.25) = sqrt(144) = 12 cm, area = (1/2)*17*12 = 102 cm²

Base area = 60 cm²

Total = 56.6 + 94.8675 + 102 + 60 = 313.4675 cm² — still not matching.

So not working.

Let's try Shape 2 with r=10, and if they use l=20.5, then π*10*(10+20.5) = 305*3.14=957.7, not 646.5.

Perhaps for a different shape.

Another idea: for Shape 9, if the "6cm" is the height, and "8cm" is side, "12cm" is something else, but diagram shows "12cm" as height of prism.

Perhaps "12cm" is the apothem, but then side would be different.

I recall that in some problems, for hexagonal prism, if apothem a, side s, then a = (s√3)/2, so s = 2a/√3.

If a=6, s=12/1.732≈6.928 cm, then area of hexagon = (1/2)*perimeter*apothem = (1/2)*(6*6.928)*6 = (1/2)*41.568*6 = 124.704 cm² per base, times 2 = 249.408 cm²

Lateral area = 6 * s * h = 6 * 6.928 * 12 = 498.816 cm²

Total 249.408 + 498.816 = 748.224 cm², not in list.

If h=12, s=8, a=6, but inconsistent.

Perhaps the 6 cm is not used, and we use s=8, h=12, area = 2* (3*1.732/2 * 64) + 6*8*12 = 2* (2.598*64) + 576 = 2*166.272 + 576 = 332.544 + 576 = 864 cm², as before.

864 is not in the list, and 936 is close, so perhaps it's 936 for another shape.

Let's calculate for Shape 7 with b=35 cm, h=62.5 cm, L=20 cm, and if they used only the lateral area for the rectangles, but forgot the triangles or something.

Perhaps the "0.2 m" is the width, but it's a prism, so length is 0.2 m.

I think I need to accept that for Shape 9, surface area is 864 cm², and it's the missing answer.

But let's check Shape 8 with a standard calculation.

Perhaps the 12 cm is the slant height for the faces, and for each face, area = (1/2)*base*12.

So for base 8: 48, base 15: 90, base 17: 102, base 60, total 300, as before.

300 is not in list, but 304 is for Shape 3.

Perhaps for Shape 2, if we take the cone with r=10, and the 22.4 is the height, then l=24.53, surface area = π*10*(10+24.53) = 345.3*3.14=1084.242, and if they have 936, not match.

936 / 3.14 = 298.09, so r(r+l) = 298.09, if r=10, r+l=29.809, l=19.809, not 22.4.

If r=12, then 12*(12+l) = 298.09, 12+l = 24.84, l=12.84, not matching.

Let's try the answer 646.5 for Shape 2.

Suppose r=10, then πr(r+l) = 3.14*10*(10+l) = 31.4*(10+l) = 646.5, so 10+l = 646.5/31.4 = 20.59, l=10.59 cm.

But diagram has 22.4, not 10.59.

Perhaps the 22.4 is the diameter, so r=11.2, then π*11.2*(11.2+l) = 646.5, so 11.2*(11.2+l) = 646.5/3.14 = 205.89, so 11.2+l = 18.38, l=7.18, not matching.

I think I found it.

For Shape 7, if we take the triangle as having base 35 cm, and the 0.625 m = 62.5 cm is the length, and the 0.2 m = 20 cm is the height of the triangle, and it's a right triangle, then as before.

But perhaps they want the surface area in m² or something, but the answers are in cm².

Another idea: in Shape 6, they have cm³, but it's a typo, should be cm².

For Shape 7, perhaps the "0.625 m" is 62.5 cm, but it's the height of the prism, and "0.2 m" is 20 cm is the base or something.

Let's calculate the surface area for a different interpretation.

Perhaps for Shape 7, it is a triangular prism with equilateral triangle or something, but diagram shows right angle.

Let's look online or recall that in some problems, for a triangular prism with right triangle base, surface area is calculated as above.

Perhaps the missing answer is for Shape 8, and it's 936 or something.

Let's calculate for Shape 9 with s=8, h=12, and if they used a=6 for area, but s=8, a should be 6.928, so perhaps they used a=6, s=8, area = (1/2)*48*6 = 144, as before.

Perhaps the "6cm" is the radius, so for regular hexagon, radius = side, so s=6 cm, then area = (3√3/2)*36 = 3*1.732*18 = 5.196*18 = 93.528 per base, times 2 = 187.056, lateral 6*6*12 = 432, total 619.056, not 646.5.

646.5 - 619 = 27.5, not close.

If s=7, area = (3*1.732/2)*49 = (2.598)*49 = 127.302 per base, times 2 = 254.604, lateral 6*7*12 = 504, total 758.604, not 646.5.

I think I need to conclude that for Shape 2, with r=10 cm, and if we take the slant height as 20.5 cm or something, but let's try the following.

Perhaps for Shape 2, the "22.4 cm" is the circumference or something, but unlikely.

Another thought: in some diagrams, the number might be for diameter.

Assume for Shape 2, the "10 cm" is radius, "22.4 cm" is height, then l = sqrt(10^2 + 22.4^2) = sqrt(100 + 501.76) = sqrt(601.76) = 24.53 cm, then surface area = π*10*(10+24.53) = 345.3*3.14 = 1084.242 cm².

Not in list.

But 936 is there, 1084 - 936 = 148, not close.

Perhaps they want only lateral: π*10*24.53 = 770.5, not 646.5.

Let's calculate 646.5 / (π*10) = 20.59, so if r+ l = 20.59, and r=10, l=10.59, then if height h = sqrt(l^2 - r^2) = sqrt(112.1481 - 100) = sqrt(12.1481) = 3.485 cm, not 22.4.

So not.

Perhaps for Shape 8, with base 8,15,17, and height 12, and if we take the area as 3* (1/2)*base*12 for lateral, but that's 3*6*12=216 for the three, plus base 60, total 276, not.

I recall that in the beginning, for Shape 1, we have 452.4, which is 144*3.14, good.

Shape 5: 3*100*3.14 = 942.5, good.

Shape 3: 304, which is 240+64, with slant height 15.

Shape 4: 792, as calculated.

Shape 6: 42223, as calculated.

Now for Shape 9, let's assume that the "6cm" is the apothem, and side is 8 cm, but in reality for regular hexagon, if apothem a=6, then side s = 2a / tan(30°) = 2*6 / (1/√3) = 12 * √3 = 12*1.732 = 20.784 cm, but diagram says 8 cm, so not.

Perhaps the 8 cm is the apothem, but diagram shows "8 cm" on the side.

I think the only reasonable choice is that for Shape 9, surface area is 864 cm², and it's the missing answer.

But let's check the answer 936.

936 / 12 = 78, not helpful.

936 / 6 = 156, etc.

Perhaps for Shape 7, if we take the triangle as having area (1/2)*35*20 = 350, and lateral area as 3* something.

Another idea: perhaps "0.625 m" is 62.5 cm, but it's the slant height for the lateral faces, but for a prism, lateral faces are rectangles, not triangles.

I give up; let's assign what we can.

From earlier:

- Shape 1: 452.4

- Shape 3: 304

- Shape 4: 792

- Shape 5: 942.5

- Shape 6: 42223

Left shapes: 2,7,8,9

Left answers: 646.5, 5300, 936, and one missing.

Now for Shape 2: cone, r=10 cm, if we take l=20.5 cm, then surface area = π*10*(10+20.5) = 305*3.14 = 957.7, not in list.

If we take only lateral: π*10*20.5 = 643.7, close to 646.5!

643.7 vs 646.5, difference 2.8, perhaps with π=3.1416, 10*20.5*3.1416 = 205*3.1416 = 644.028, still not 646.5.

If l=20.6, 10*20.6*3.14 = 206*3.14 = 646.84, very close to 646.5!

So perhaps slant height is 20.6 cm, but diagram has 22.4, so not.

Perhaps r=10.1, etc.

But 646.84 ≈ 646.5, so perhaps for Shape 2, lateral surface area is 646.5 cm², with r=10, l=20.6, but diagram has 22.4, so maybe it's a different interpretation.

Perhaps the 22.4 is the diameter, so r=11.2, then lateral surface area = π*11.2* l, set to 646.5, so l = 646.5/(11.2*3.14) = 646.5/35.168 = 18.38 cm, not matching.

I think for the sake of time, let's assume that for Shape 2, with r=10 cm, and if they use l=20.5 cm, surface area lateral = 3.14*10*20.5 = 643.7, and they have 646.5, close enough, or perhaps with π=22/7, 22/7*10*20.5 = (220/7)*20.5 = 31.4286*20.5 = 644.285, still not.

22/7*10*20.6 = (220/7)*20.6 = 31.4286*20.6 = 647.428, close to 646.5.

So perhaps l=20.6 cm, but diagram has 22.4, so maybe it's a typo, or for this problem, we take it as 646.5 for Shape 2.

Then for Shape 7, with b=35 cm, h=62.5 cm, L=20 cm, surface area 5570.1, and 5300 is close, perhaps they have 5300 for it.

5570 - 5300 = 270, error 4.8%, perhaps acceptable.

Then