Square Base Pyramid Volume Calculator - Free Printable

The image provided is an answer key for a worksheet titled "Volume of a Square Base Pyramid Sheet 2." It contains the solutions and step-by-step working out for five specific problems. Since the answers are already visible in the image, I will verify the calculations for each problem to ensure they are correct.

Here is the step-by-step verification:

1) Pyramid with base $8 \frac{1}{2}$ cm and height 12 cm
* Formula: $V = \frac{1}{3} b^2 h$
* Base ($b$): $8 \frac{1}{2} = 8.5$. So, $b^2 = 8.5 \times 8.5 = 72.25$.
* Alternatively, using fractions: $8 \frac{1}{2} = \frac{17}{2}$. $(\frac{17}{2})^2 = \frac{289}{4}$.
* Height ($h$): 12
* Calculation:
* $V = \frac{1}{3} \times 72.25 \times 12$
* $\frac{1}{3} \times 12 = 4$
* $72.25 \times 4 = 289$
* Result: $289 \text{ cm}^3$. (Matches the image)

2) Pyramid with base $4 \frac{3}{4}$ in and height 9 in
* Formula: $V = \frac{1}{3} b^2 h$
* Base ($b$): $4 \frac{3}{4} = 4.75$. Or as a fraction: $\frac{19}{4}$.
* Height ($h$): 9
* Calculation:
* Using fractions is often cleaner here.
* $b^2 = (\frac{19}{4})^2 = \frac{361}{16}$
* $V = \frac{1}{3} \times \frac{361}{16} \times 9$
* Simplify $\frac{9}{3} = 3$.
* $V = 3 \times \frac{361}{16} = \frac{1083}{16}$
* $1083 \div 16 = 67.6875$
* Rounding: To 2 decimal places, this is $67.69$.
* Result: $67.69 \text{ in}^3$. (Matches the image)

3) Pyramid with base edge 5 cm, slant height 11 cm, and perpendicular height 8.5 cm
* *Note: The diagram shows multiple measurements. The volume formula requires the perpendicular height (the straight line down from the top), which is given as 8.5 cm. The 11 cm is the slant height and is extra information not needed for volume.*
* Formula: $V = \frac{1}{3} b^2 h$
* Base ($b$): 5
* Height ($h$): 8.5
* Calculation:
* $b^2 = 5 \times 5 = 25$
* $V = \frac{1}{3} \times 25 \times 8.5$
* $25 \times 8.5 = 212.5$
* $212.5 \div 3 = 70.8333...$
* Rounding: To 2 decimal places, this is $70.83$.
* Result: $70.83 \text{ cm}^3$. (Matches the image)

4) Pyramid with base 3.6 cm and height 7.8 cm
* Formula: $V = \frac{1}{3} b^2 h$
* Base ($b$): 3.6
* Height ($h$): 7.8
* Calculation:
* $b^2 = 3.6 \times 3.6 = 12.96$
* $V = \frac{1}{3} \times 12.96 \times 7.8$
* $\frac{1}{3} \times 12.96 = 4.32$
* $4.32 \times 7.8 = 33.696$
* Rounding: To 2 decimal places, look at the third decimal digit (6). Since it is 5 or greater, round up the second digit (9 becomes 10, carrying over).
* $33.696 \rightarrow 33.70$
* Result: $33.70 \text{ cm}^3$. (Matches the image)

5) Pyramid with base 9 in, slant height 12 in, and perpendicular height $10 \frac{1}{2}$ in
* *Note: Again, ignore the slant height (12 in). Use the perpendicular height ($10 \frac{1}{2}$).*
* Formula: $V = \frac{1}{3} b^2 h$
* Base ($b$): 9
* Height ($h$): $10 \frac{1}{2} = 10.5$
* Calculation:
* $b^2 = 9 \times 9 = 81$
* $V = \frac{1}{3} \times 81 \times 10.5$
* $\frac{1}{3} \times 81 = 27$
* $27 \times 10.5 = 283.5$
* Result: $283.5 \text{ in}^3$. (Matches the image)

All calculations in the provided answer sheet are correct.

Final Answer:
The calculations in the image are verified as correct. Here are the final volumes for each problem:

1) 289 cm³
2) 67.69 in³
3) 70.83 cm³
4) 33.70 cm³
5) 283.5 in³