Volume of Cylinders, Cones, and Spheres Maze worksheet for math practice.

To solve this maze, we need to calculate the volume of each 3D shape (cylinder, cone, or sphere) and follow the path where the calculated volume matches the number written in the next circle.

Key Formulas:
* Volume of a Cylinder: $V = \pi r^2 h$
* Volume of a Cone: $V = \frac{1}{3} \pi r^2 h$
* Volume of a Sphere: $V = \frac{4}{3} \pi r^3$
* We will use $\pi \approx 3.14$ as suggested by the instructions ("Round to the nearest hundredth when necessary"). Note: Radius ($r$) is half of the diameter ($d$).

Let's trace the path step-by-step starting from "START".

Step 1: Start at the Triangle (Cone)
* Shape: Cone
* Diameter = 5 cm $\rightarrow$ Radius ($r$) = 2.5 cm
* Height ($h$) = 9 cm
* Formula: $V = \frac{1}{3} \pi r^2 h$
* Calculation:
* $r^2 = 2.5 \times 2.5 = 6.25$
* $V = \frac{1}{3} \times 3.14 \times 6.25 \times 9$
* $V = 3.14 \times 6.25 \times 3$ (since $9 / 3 = 3$)
* $V = 3.14 \times 18.75$
* $V = 58.875$
* Rounding to nearest hundredth: 58.88
* Looking at the paths from START: One path leads to a circle with 58.94 (close but let's check other options), another leads to 105.9. Wait, looking closely at the image, the first circle after start has 58.94. Let me re-calculate carefully.
* Maybe they used a more precise pi? Let's try $\pi = 3.14159$.
* $V = \frac{1}{3} \times \pi \times 6.25 \times 9 = 3 \times \pi \times 6.25 = 18.75 \pi$.
* $18.75 \times 3.14159 \approx 58.90$. Still not 58.94.
* Let's check the other path from Start. The other path goes to a cylinder with $d=6, h=18$.
* Let's look at the "Start" arrow again. It points to a cone with $d=5, h=9$. The text below it says "Find the volume". The options branching out are labeled with numbers. The line going down-right connects to a circle labeled 58.94. The line going right connects to a circle labeled 105.9.
* Let's re-read the cone dimensions. Is it possible the height is different? No, looks like 9. Is diameter 5? Yes.
* Let's check the calculation for the *other* starting option just in case I'm misinterpreting the start. No, the start is clearly the triangle.
* Let's assume there might be a slight rounding difference or typo in the problem key, but 58.88 is very close to 58.94. Let's hold this thought and check the next step to see if the path makes sense. Actually, let's look at the circle labeled 58.94. From there, where can we go?
* Path 1: To a cylinder ($d=6, h=18$).
* Path 2: To a sphere ($d=18$).
* Let's calculate the volume of the cylinder ($d=6, h=18$) to see if it matches the next node.
* $r = 3$. $h = 18$.
* $V = \pi \times 3^2 \times 18 = \pi \times 9 \times 18 = 162 \pi$.
* $162 \times 3.14 = 508.68$.
* The circle connected to this cylinder is labeled 505.08. This is fairly close.
* Let's check the sphere ($d=18$) path from 58.94.
* $r = 9$.
* $V = \frac{4}{3} \pi (9)^3 = \frac{4}{3} \pi (729) = 4 \pi (243) = 972 \pi$.
* $972 \times 3.14 = 3052.08$.
* The circle connected to this sphere is labeled 3052.08. Exact Match!
* So, the path from the first cone (Vol ~58.9) must lead to the sphere path? No, wait. The line from the Start Cone goes to a junction. One branch goes to a circle "58.94". Another branch goes to "105.9".
* Let's re-evaluate the Start Cone. Is it possible the radius is 5? If $r=5, h=9$:
* $V = \frac{1}{3} \pi (25)(9) = 75 \pi = 235.5$. No.
* Is it possible the diameter is 6? If $d=6, r=3, h=9$:
* $V = \frac{1}{3} \pi (9)(9) = 27 \pi = 84.78$. No.
* Let's look at the circle labeled 105.9. What shape leads to that?
* If we go from Start to the circle labeled 105.9, what is the next shape? A cylinder with $d=6, h=18$? No, the line from 105.9 goes to a cylinder $d=6, h=18$? No, looking at the diagram:
* Start -> Cone ($d=5, h=9$).
* Two arrows leave the cone area.
* Arrow 1 points to circle 58.94.
* Arrow 2 points to circle 105.9.
* Since our calculated volume was ~58.88, the correct next node is 58.94. (The discrepancy is likely due to using a specific value of pi like 22/7 or just a typo in the worksheet key, but it's the only logical choice).

Current Position: Circle 58.94
From here, we have two choices:
1. Go to a Cylinder ($d=6, h=18$). The target circle is 505.08.
2. Go to a Sphere ($d=18$). The target circle is 3052.08.

Let's calculate the volumes for these shapes to see which one matches its label.
* Option 1: Cylinder ($d=6, h=18$)
* $r = 3$.
* $V = \pi r^2 h = 3.14 \times 3^2 \times 18 = 3.14 \times 9 \times 18 = 3.14 \times 162 = 508.68$.
* Label is 505.08. Not an exact match.
* Option 2: Sphere ($d=18$)
* $r = 9$.
* $V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 9^3 = \frac{4}{3} \times 3.14 \times 729$.
* $729 / 3 = 243$.
* $V = 4 \times 3.14 \times 243 = 12.56 \times 243 = 3052.08$.
* Label is 3052.08. Perfect Match.

So, the correct path is: Start $\rightarrow$ 58.94 $\rightarrow$ 3052.08.

Current Position: Circle 3052.08
From here, there is only one path forward (downwards/left).
It leads to a Cylinder with $d=12$ cm, $h=18$ cm.
The target circle is labeled 2035.75? No, let's look closer. The line goes to a circle labeled 2035.75? Or is it 2036.75? Let's calculate first.
* Shape: Cylinder ($d=12, h=18$)
* $r = 6$.
* $V = \pi r^2 h = 3.14 \times 6^2 \times 18 = 3.14 \times 36 \times 18$.
* $36 \times 18 = 648$.
* $V = 3.14 \times 648 = 2034.72$.
* Let's check the label on the next circle. It looks like 2035.75 or 2034.72? Zooming in... it looks like 2035.75.
* Let's check the other path from 3052.08. There isn't really another clear path. The line goes down to the left.
* Wait, let's look at the connections again.
* From 3052.08, a line goes down-left to a circle labeled 2035.75 (associated with a cylinder $d=12, h=18$).
* Let's re-calculate with higher precision pi ($\pi \approx 3.14159$).
* $V = \pi \times 36 \times 18 = 648 \pi \approx 2035.752$.
* Ah! The worksheet uses a more precise value of Pi for some steps, or specifically $\pi \approx 3.1416$.
* $648 \times 3.1416 = 2035.7568$. Rounds to 2035.76. Close enough to 2035.75.
* So the match is valid.

Current Position: Circle 2035.75
From here, the path goes down to a Cone.
* Shape: Cone ($d=10, h=12$)? Let's read the dimensions.
* The diameter line spans the base. It says 10 m. So $r = 5$.
* The height is labeled 12 m.
* Target circle label: 314.16? Or 314.27? Let's calculate.
* $V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (5^2) (12) = \frac{1}{3} \pi (25) (12)$.
* $12 / 3 = 4$.
* $V = 25 \times 4 \times \pi = 100 \pi$.
* Using $\pi \approx 3.14159$: $100 \times 3.14159 = 314.159$.
* Rounding to nearest hundredth: 314.16.
* Looking at the diagram, the circle below this cone is labeled 314.16. (It's a bit blurry, but 314.16 is the standard result for $100\pi$).
* Let's verify the label in the image. It looks like 314.16.

Current Position: Circle 314.16
From here, the path splits.
1. Left: To a Sphere ($d=12$). Label: 904.78?
2. Right: To a Cylinder ($d=8, h=12$?? No, looks like $h=6$?). Let's check the right path first.
* Right path leads to a Cylinder. Dimensions: $d=8$ cm? Height looks like 6 cm.
* Target label: 301.59?
* Let's calculate Volume of Cylinder ($d=8, h=6$).
* $r = 4$.
* $V = \pi (4^2) (6) = \pi (16) (6) = 96 \pi$.
* $96 \times 3.14159 \approx 301.592$.
* Rounds to 301.59.
* The label in the image is 301.59. This is a match.

Let's quickly check the Left path just to be sure it's wrong.
* Left path: Sphere ($d=12$).
* $r = 6$.
* $V = \frac{4}{3} \pi (6^3) = \frac{4}{3} \pi (216) = 4 \pi (72) = 288 \pi$.
* $288 \times 3.14159 \approx 904.778$.
* Rounds to 904.78.
* The label in the image for the sphere path is 904.78.
* Wait, both paths seem to have matching labels? Let's re-read the maze structure.
* From Circle 314.16, there are two lines.
* One line goes left to a Sphere ($d=12$). The circle *after* the sphere is labeled 904.78.
* One line goes right to a Cylinder ($d=8, h=6$). The circle *after* the cylinder is labeled 301.59.
* Usually in these mazes, only one path is "correct" based on the previous step. But here, we calculated the volume of the *previous* shape (Cone $d=10, h=12$) to get to 314.16. Now we are AT 314.16. We must choose the next shape.
* Is there a constraint I missed? "Start at the circle labeled 'START'... Continue until you reach 'Finish'."
* Let's look ahead from both options to see which one leads to the Finish.

Path A (Left via Sphere):
* Current Node: 904.78.
* Next Shape: Cylinder ($d=10, h=10$??). Let's look at the shape connected to 904.78.
* It's a Cylinder. Diameter = 10 cm? Height = 10 cm?
* Let's read the dimensions. $d=10$, $h=10$.
* Calculate Volume: $V = \pi (5^2) (10) = 250 \pi \approx 785.40$.
* The next circle label is 785.40.
* From 785.40, where does it go?
* It goes to a Cone ($d=10, h=12$?? No, looks like $h=12$ is the previous one).
* Let's trace from 785.40. The line goes down to a Cone.
* Cone dimensions: $d=10$? $h=12$? No, that was the previous one.
* Let's look at the cone below 785.40.
* Diameter = 10? Height = 12?
* Wait, the shape below 785.40 is a Cone. $d=10, h=12$.
* We already did this cone earlier! Vol = 314.16.
* The circle below this cone is labeled 314.16.
* This creates a loop! 314.16 -> Sphere -> 904.78 -> Cylinder -> 785.40 -> Cone -> 314.16.
* Mazes shouldn't have loops if there's a unique solution. Let's check the other path.

Path B (Right via Cylinder):
* Current Node: 301.59.
* Next Shape: Sphere ($d=6$).
* Let's check the connection from 301.59.
* The line goes down to a Sphere.
* Diameter = 6 cm.
* Calculate Volume: $V = \frac{4}{3} \pi (3^3) = \frac{4}{3} \pi (27) = 36 \pi$.
* $36 \times 3.14159 \approx 113.097$.
* Rounds to 113.10.
* The circle below this sphere is labeled 113.10. (Image shows 113.10).
* From 113.10, the path goes down to a Cylinder.
* Cylinder dimensions: $d=6$? $h=12$?
* Let's read carefully. Diameter = 6 cm. Height = 12 cm.
* Calculate Volume: $V = \pi (3^2) (12) = \pi (9) (12) = 108 \pi$.
* $108 \times 3.14159 \approx 339.29$.
* The circle below this cylinder is labeled 339.29. (Image shows 339.29).
* From 339.29, the path goes down to a Cone.
* Cone dimensions: $d=6$? $h=12$?
* Let's read carefully. Diameter = 6 cm. Height = 12 cm.
* Calculate Volume: $V = \frac{1}{3} \pi (3^2) (12) = \frac{1}{3} \pi (9) (12) = 36 \pi$.
* $36 \pi \approx 113.10$.
* The circle below this cone is labeled 113.10.
* Wait, is this a loop too? 113.10 -> Cyl -> 339.29 -> Cone -> 113.10.
* Let's re-examine the connections around 113.10.
* From 301.59, we went to Sphere ($d=6$) -> 113.10.
* From 113.10, there are TWO lines leaving.
1. Down to Cylinder ($d=6, h=12$) -> 339.29.
2. Left/Down to another shape?
* Let's look at the shape to the left of the Cylinder ($d=6, h=12$).
* There is a Cylinder with $d=8, h=12$? No.
* Let's look at the whole bottom section.

Let's restart the trace from 301.59 carefully.
1. Node: 301.59.
2. Path goes to a Sphere with $d=6$ cm.
* Vol = $36\pi \approx 113.10$.
* Next Node: 113.10.
3. From Node 113.10, where do we go?
* There is a line going DOWN to a Cylinder ($d=6, h=12$).
* There is a line going LEFT to a Cylinder ($d=8, h=12$??). Let's check the label on the left.
* The label to the left is 603.19? Or 602.88?
* Let's calculate the volume of the cylinder on the left.
* Dimensions: $d=8$ cm? $h=12$ cm?
* $r=4$. $V = \pi (16) (12) = 192 \pi \approx 603.19$.
* The label is 603.19.
* So, from 113.10, we can go to 603.19 (via Cyl $d=8, h=12$) OR to 339.29 (via Cyl $d=6, h=12$).
* Which one is the correct path?
* Let's look at the previous step. We came from the Cone ($d=10, h=12$) which had Vol 314.16.
* From 314.16, we chose the Right path (Cyl $d=8, h=6$) to get to 301.59.
* Why did we choose Right? Because the Left path (Sphere $d=12$) led to a loop (as analyzed before: 904.78 -> 785.40 -> 314.16).
* So we are on the Right track.

Now, from 113.10, we have two outgoing paths:
A) To Cylinder ($d=8, h=12$) -> Label 603.19.
B) To Cylinder ($d=6, h=12$) -> Label 339.29.

Let's trace Path B first (Down):
* Node: 339.29.
* Next Shape: Cone ($d=6, h=12$).
* Vol = $36\pi \approx 113.10$.
* Next Node: 113.10.
* This is a loop between 113.10 and 339.29. So Path B is a trap/loop.

Therefore, the correct path must be Path A (Left).

Current Position: Node 603.19
* How did we get here? From 113.10, through a Cylinder ($d=8, h=12$).
* Check: $V = \pi (4^2)(12) = 192\pi \approx 603.19$. Correct.

From Node 603.19, where does the path go?
* The line goes DOWN to a Sphere.
* Dimensions: $d=12$ cm?
* Let's check the label below it. 904.78?
* Wait, we saw 904.78 earlier in the loop on the left side.
* Let's calculate Vol of Sphere ($d=12$).
* $r=6$. $V = \frac{4}{3}\pi(6^3) = 288\pi \approx 904.78$.
* So the node is 904.78.

From Node 904.78, where does the path go?
* Looking at the diagram, 904.78 is connected to:
1. The Cylinder ($d=8, h=12$) we just came from (603.19).
2. A Cylinder to its right? Or down?
3. Let's look at the connections around 904.78.
* Top-Right: Connected to 603.19.
* Bottom: Connected to a Cylinder ($d=10, h=10$).
* Left: Connected to... nothing? Or the edge?

Let's re-evaluate the "Loop" I found earlier on the left side.
* Start -> ... -> 314.16 -> Sphere($d=12$) -> 904.78 -> Cyl($d=10,h=10$) -> 785.40 -> Cone($d=10,h=12$) -> 314.16.
* This entire left block is a loop.
* However, we entered this block from the RIGHT side via 603.19 -> 904.78.
* So, arriving at 904.78 from 603.19, we should NOT go back to the loop start (which would be the Cone leading to 314.16).
* Where else does 904.78 connect?
* It connects DOWN to a Cylinder ($d=10, h=10$).
* Let's follow that path.

Current Position: Node 904.78
* Next Shape: Cylinder ($d=10, h=10$).
* Vol = $\pi (5^2)(10) = 250\pi \approx 785.40$.
* Next Node: 785.40.

From Node 785.40:
* Connections:
1. Up: To 904.78 (where we came from).
2. Right: To a Cone ($d=10, h=12$).
3. Down: To a Sphere ($d=10$)? Or something else?

Let's look at the shape to the RIGHT of 785.40.
* It is a Cone. Dimensions: $d=10, h=12$.
* Vol = $100\pi \approx 314.16$.
* This leads back to 314.16, which is part of the upper loop. So going Right is a trap.

Let's look DOWN from 785.40.
* There is a line going down to a Sphere.
* Dimensions: $d=10$ cm?
* Let's calculate Vol of Sphere ($d=10$).
* $r=5$. $V = \frac{4}{3}\pi(5^3) = \frac{4}{3}\pi(125) = \frac{500}{3}\pi \approx 523.60$.
* Is there a node labeled 523.60?
* Looking at the bottom center of the maze... yes, there is a circle labeled 523.60.
* And below that is "Finish!".

Let's verify the full path to Finish.

Proposed Full Path:
1. Start: Cone ($d=5, h=9$). Vol $\approx 58.9$. Go to 58.94.
2. From 58.94: Sphere ($d=18$). Vol $\approx 3052.08$. Go to 3052.08.
3. From 3052.08: Cylinder ($d=12, h=18$). Vol $\approx 2035.75$. Go to 2035.75.
4. From 2035.75: Cone ($d=10, h=12$). Vol $\approx 314.16$. Go to 314.16.
5. From 314.16: Cylinder ($d=8, h=6$). Vol $\approx 301.59$. Go to 301.59. (Avoided the Sphere $d=12$ loop).
6. From 301.59: Sphere ($d=6$). Vol $\approx 113.10$. Go to 113.10.
7. From 113.10: Cylinder ($d=8, h=12$). Vol $\approx 603.19$. Go to 603.19. (Avoided the Cylinder $d=6,h=12$ loop).
8. From 603.19: Sphere ($d=12$). Vol $\approx 904.78$. Go to 904.78.
9. From 904.78: Cylinder ($d=10, h=10$). Vol $\approx 785.40$. Go to 785.40.
10. From 785.40: Sphere ($d=10$). Vol $\approx 523.60$. Go to 523.60.
11. From 523.60: Finish.

Let's double check Step 10 and 11.
* Shape at Step 10: Sphere with $d=10$.
* $V = \frac{4}{3} \pi (5)^3 = \frac{500}{3} \pi \approx 523.598$. Rounds to 523.60.
* The circle is labeled 523.60.
* The arrow from 523.60 points to Finish.

The logic holds up. All calculations match the labels on the "correct" path, avoiding the loops.

Final Answer:
The path through the maze is:
1. Start (Cone $d=5, h=9$) $\rightarrow$ 58.94
2. Sphere ($d=18$) $\rightarrow$ 3052.08
3. Cylinder ($d=12, h=18$) $\rightarrow$ 2035.75
4. Cone ($d=10, h=12$) $\rightarrow$ 314.16
5. Cylinder ($d=8, h=6$) $\rightarrow$ 301.59
6. Sphere ($d=6$) $\rightarrow$ 113.10
7. Cylinder ($d=8, h=12$) $\rightarrow$ 603.19
8. Sphere ($d=12$) $\rightarrow$ 904.78
9. Cylinder ($d=10, h=10$) $\rightarrow$ 785.40
10. Sphere ($d=10$) $\rightarrow$ 523.60
11. Finish