Here are the step-by-step solutions for each problem on the worksheet. The general rule for finding the volume of prisms and cylinders is:
Volume = Area of Base × Height.
1. Rectangular Prism
*
Formula: $V = \text{length} \times \text{width} \times \text{height}$
*
Calculation: $17.5 \times 14 \times 6.4$
* $17.5 \times 14 = 245$
* $245 \times 6.4 = 1568$
*
Answer: $1,568 \text{ in}^3$
2. Triangular Prism
*
Step 1 (Base Area): The base is a triangle. Area = $\frac{1}{2} \times \text{base} \times \text{height}$.
* $A = 0.5 \times 10 \times 3.6 = 18 \text{ m}^2$
*
Step 2 (Volume): Multiply the base area by the length of the prism ($9 \text{ m}$).
* $V = 18 \times 9 = 162$
*
Answer: $162 \text{ m}^3$
3. Cylinder
*
Step 1 (Radius): The diameter is $13 \text{ mm}$, so the radius ($r$) is half of that: $6.5 \text{ mm}$.
*
Step 2 (Base Area): Area = $\pi \times r^2$.
* $A = \pi \times 6.5^2 = 42.25\pi$
*
Step 3 (Volume): Multiply by the height ($19 \text{ mm}$).
* $V = 42.25\pi \times 19 = 802.75\pi$
* $802.75 \times 3.14159 \approx 2521.91$
*
Answer: $2,521.91 \text{ mm}^3$
4. Pentagonal Prism
*
Note: We use the "apothem" (the distance from the center to the middle of a side), which is labeled as $17 \text{ cm}$.
*
Step 1 (Base Area): Area = $\frac{1}{2} \times \text{Perimeter} \times \text{apothem}$.
* Perimeter = $5 \text{ sides} \times 20 \text{ cm/side} = 100 \text{ cm}$.
* Area = $0.5 \times 100 \times 17 = 850 \text{ cm}^2$.
*
Step 2 (Volume): Multiply by the height of the prism ($28 \text{ cm}$).
* $V = 850 \times 28 = 23,800$
*
Answer: $23,800 \text{ cm}^3$
5. Rectangular Prism
*
Formula: $V = \text{length} \times \text{width} \times \text{height}$
*
Calculation: $25 \times 18 \times 7$
* $25 \times 18 = 450$
* $450 \times 7 = 3,150$
*
Answer: $3,150 \text{ ft}^3$
6. Cylinder
*
Step 1 (Base Area): Radius is $3.2 \text{ km}$.
* $A = \pi \times 3.2^2 = 10.24\pi$
*
Step 2 (Volume): Multiply by the height ($8 \text{ km}$).
* $V = 10.24\pi \times 8 = 81.92\pi$
* $81.92 \times 3.14159 \approx 257.36$
*
Answer: $257.36 \text{ km}^3$
7. Trapezoidal Prism
*
Step 1 (Base Area): The base is a trapezoid. Area = $\frac{h(b_1 + b_2)}{2}$.
* Height of trapezoid = $11 \text{ m}$. Parallel sides are $12.7 \text{ m}$ and $16 \text{ m}$.
* Area = $\frac{11(12.7 + 16)}{2} = \frac{11(28.7)}{2} = \frac{315.7}{2} = 157.85 \text{ m}^2$.
*
Step 2 (Volume): Multiply by the length of the prism ($14 \text{ m}$).
* $V = 157.85 \times 14 = 2,209.9$
*
Answer: $2,209.9 \text{ m}^3$
8. Octagonal Prism
*
Note: The area of the base ($B$) is already given as $41.57 \text{ in}^2$.
*
Formula: $V = \text{Base Area} \times \text{Height}$
*
Calculation: $41.57 \times 9$
* $41.57 \times 9 = 374.13$
*
Answer: $374.13 \text{ in}^3$
Final Answer:
1. 1,568 in³
2. 162 m³
3. 2,521.91 mm³
4. 23,800 cm³
5. 3,150 ft³
6. 257.36 km³
7. 2,209.9 m³
8. 374.13 in³
Parent Tip: Review the logic above to help your child master the concept of surface area of prisms and cylinders worksheet answers.
