Here are the step-by-step solutions for each problem on the worksheet. The general rule for all these shapes is:
Volume = Area of the Base × Height.
1. Rectangular Prism
*
Formula: $Length \times Width \times Height$
*
Calculation: $17.5 \times 14 \times 6.4$
* $17.5 \times 14 = 245$
* $245 \times 6.4 = 1,568$
*
Answer: 1,568 in³
2. Triangular Prism
*
Step 1 (Base Area): Find the area of the triangle face. Formula: $\frac{1}{2} \times base \times height$.
* $\frac{1}{2} \times 10 \times 3.6 = 18 \text{ m}^2$
*
Step 2 (Volume): Multiply the base area by the length of the prism (the distance connecting the two triangles).
* $18 \times 9 = 162$
*
Answer: 162 m³
3. Cylinder
*
Step 1 (Radius): The image shows the diameter is 13 mm. Divide by 2 to get the radius ($r$).
* $r = 6.5 \text{ mm}$
*
Step 2 (Base Area): Formula: $\pi \times r^2$.
* $3.14159 \times (6.5)^2 \approx 132.73 \text{ mm}^2$
*
Step 3 (Volume): Multiply base area by height (19 mm).
* $132.73 \times 19 \approx 2,521.90$
*
Answer: 2,521.90 mm³
4. Pentagonal Prism
*
Note: This shape has extra numbers (like 25.7 and 37) that describe the sides of the pentagon, but we don't need them because the problem gives us the Area of the Base directly.
*
Step 1 (Identify Base Area): The label inside the pentagon says $A = 333$. So, Base Area = 333 cm².
*
Step 2 (Volume): Multiply Base Area by the height of the prism (the length connecting the front and back faces), which is 28 cm.
* $333 \times 28 = 9,324$
*
Answer: 9,324 cm³
5. Rectangular Prism
*
Formula: $Length \times Width \times Height$
*
Calculation: $18 \times 7 \times 25$
* $18 \times 7 = 126$
* $126 \times 25 = 3,150$
*
Answer: 3,150 ft³
6. Cylinder
*
Step 1 (Radius): The arrow points from the center to the edge, so the radius ($r$) is 3.2 km.
*
Step 2 (Base Area): Formula: $\pi \times r^2$.
* $3.14159 \times (3.2)^2 \approx 32.17 \text{ km}^2$
*
Step 3 (Volume): Multiply base area by height (8 km).
* $32.17 \times 8 \approx 257.36$
*
Answer: 257.36 km³
7. Trapezoidal Prism
*
Step 1 (Base Area): Find the area of the trapezoid face. Formula: $\frac{(base_1 + base_2)}{2} \times height$.
* The parallel sides are 11 m and 16 m. The height of the trapezoid is 9.5 m.
* Average of bases: $\frac{11 + 16}{2} = 13.5$
* Area: $13.5 \times 9.5 = 128.25 \text{ m}^2$
*
Step 2 (Volume): Multiply base area by the length of the prism (14 m).
* $128.25 \times 14 = 1,795.5$
*
Answer: 1,795.5 m³
8. Hexagonal Prism
*
Note: Like problem #4, this shape has side lengths (4 in), but we are given the Area of the Base directly ($B = 41.57$). We ignore the side length.
*
Step 1 (Identify Base Area): $B = 41.57 \text{ in}^2$.
*
Step 2 (Volume): Multiply Base Area by the height of the prism (9 in).
* $41.57 \times 9 = 374.13$
*
Answer: 374.13 in³
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Final Answer:
1. 1,568 in³
2. 162 m³
3. 2,521.90 mm³
4. 9,324 cm³
5. 3,150 ft³
6. 257.36 km³
7. 1,795.5 m³
8. 374.13 in³
Parent Tip: Review the logic above to help your child master the concept of volumes of prisms and cylinders worksheet.
