Math worksheet for calculating the volume of pyramids, cones, and spheres with provided dimensions.

Here are the step-by-step solutions for each problem on the worksheet.

Formulas needed:
* Volume of a Pyramid: $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$
* Volume of a Cone: $V = \frac{1}{3} \times \pi \times r^2 \times h$
*(Use $\pi \approx 3.14159$)*

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1) Square-based Pyramid

* Base: Square with side $4\text{ cm}$.
* Height: $9\text{ cm}$.
* Step 1: Find Base Area. $4 \times 4 = 16\text{ cm}^2$.
* Step 2: Calculate Volume. $\frac{1}{3} \times 16 \times 9$.
* Calculation: $16 \times 3 = 48$.
* Answer: $48\text{ cm}^3$

2) Rectangular-based Pyramid

* Base: Rectangle $12\text{ cm}$ by $6\text{ cm}$.
* Height: $15\text{ cm}$.
* Step 1: Find Base Area. $12 \times 6 = 72\text{ cm}^2$.
* Step 2: Calculate Volume. $\frac{1}{3} \times 72 \times 15$.
* Calculation: $\frac{1}{3}$ of $15$ is $5$. So, $72 \times 5 = 360$.
* Answer: $360\text{ cm}^3$

3) Rectangular-based Pyramid

* Base: Rectangle $5\text{ cm}$ by $7\text{ cm}$.
* Height: $20\text{ cm}$.
* Step 1: Find Base Area. $5 \times 7 = 35\text{ cm}^2$.
* Step 2: Calculate Volume. $\frac{1}{3} \times 35 \times 20$.
* Calculation: $35 \times 20 = 700$. Then $700 \div 3 = 233.333...$
* Answer: $233.33\text{ cm}^3$ (rounded to 2 decimal places)

4) Composite Shape (Pyramid on top of a Cuboid)

* Part A: The Pyramid
* Base: Square $3\text{ cm}$ by $3\text{ cm}$. Height: $7\text{ cm}$.
* Volume: $\frac{1}{3} \times (3 \times 3) \times 7 = \frac{1}{3} \times 9 \times 7 = 3 \times 7 = 21\text{ cm}^3$.
* Part B: The Cuboid (Box)
* Dimensions: $3\text{ cm} \times 3\text{ cm} \times 2\text{ cm}$.
* Volume: $3 \times 3 \times 2 = 18\text{ cm}^3$.
* Total Volume: $21 + 18 = 39$.
* Answer: $39\text{ cm}^3$

5) Cone

* Radius ($r$): $3\text{ cm}$.
* Height ($h$): $9\text{ cm}$.
* Formula: $V = \frac{1}{3} \pi r^2 h$
* Calculation:
* $r^2 = 3^2 = 9$.
* $\frac{1}{3} \times 9 = 3$.
* $3 \times 9 = 27$.
* $27 \times \pi \approx 84.823...$
* Answer: $84.82\text{ cm}^3$

6) Cone

* Radius ($r$): $1.5\text{ cm}$.
* Height ($h$): $7\text{ cm}$.
* Formula: $V = \frac{1}{3} \pi r^2 h$
* Calculation:
* $r^2 = 1.5^2 = 2.25$.
* $2.25 \times 7 = 15.75$.
* $15.75 \div 3 = 5.25$.
* $5.25 \times \pi \approx 16.493...$
* Answer: $16.49\text{ cm}^3$

7) Cone

* Diameter: $20\text{ cm}$, so Radius ($r$) is $10\text{ cm}$.
* Height ($h$): $24\text{ cm}$.
* Formula: $V = \frac{1}{3} \pi r^2 h$
* Calculation:
* $r^2 = 10^2 = 100$.
* $100 \times 24 = 2400$.
* $2400 \div 3 = 800$.
* $800 \times \pi \approx 2513.274...$
* Answer: $2513.27\text{ cm}^3$

8) Cone

* Diameter: $7.2\text{ mm}$, so Radius ($r$) is $3.6\text{ mm}$.
* Height ($h$): $7.7\text{ mm}$. *(Note: Use the vertical height, not the slant height of 8.5)*.
* Formula: $V = \frac{1}{3} \pi r^2 h$
* Calculation:
* $r^2 = 3.6^2 = 12.96$.
* $12.96 \times 7.7 = 99.792$.
* $99.792 \div 3 = 33.264$.
* $33.264 \times \pi \approx 104.499...$
* Answer: $104.50\text{ mm}^3$

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Final Answer:
1) 48 cm³
2) 360 cm³
3) 233.33 cm³
4) 39 cm³
5) 84.82 cm³
6) 16.49 cm³
7) 2513.27 cm³
8) 104.50 mm³