Here are the step-by-step solutions for each shape on the worksheet.
1. Cube
Given: Side length ($s$) = 3 in
*
Volume: The formula is $s \times s \times s$.
$$3 \times 3 \times 3 = 27$$
Volume =
27 cubic inches
*
Surface Area: A cube has 6 identical square faces. The area of one face is $3 \times 3 = 9$.
$$9 \times 6 = 54$$
Surface Area =
54 square inches
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2. Triangular Prism
Given: Triangle base = 2 in, Triangle height = 1.8 in, Prism length = 4 in, Triangle side = 2.2 in
*
Volume: First, find the area of the triangular base ($\frac{1}{2} \times \text{base} \times \text{height}$), then multiply by the prism length.
$$\text{Base Area} = 0.5 \times 2 \times 1.8 = 1.8 \text{ sq in}$$
$$\text{Volume} = 1.8 \times 4 = 7.2$$
Volume =
7.2 cubic inches
*
Surface Area: Add the area of the two triangular ends and the three rectangular sides.
* Two triangles: $1.8 \times 2 = 3.6$
* Bottom rectangle: $2 \times 4 = 8$
* Back rectangle: $2.2 \times 4 = 8.8$
* Slanted rectangle: $2.2 \times 4 = 8.8$
* Total: $3.6 + 8 + 8.8 + 8.8 = 29.2$
Surface Area =
29.2 square inches
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3. Cylinder
Given: Diameter = 7 in (so Radius $r$ = 3.5 in), Height $h$ = 6 in
*
Volume: Formula is $\pi \times r^2 \times h$. We will use $\pi \approx 3.14$.
$$r^2 = 3.5 \times 3.5 = 12.25$$
$$\text{Volume} = 3.14 \times 12.25 \times 6$$
$$\text{Volume} = 3.14 \times 73.5 = 230.79$$
Volume $\approx$
230.79 cubic inches
*
Surface Area: Formula is $2\pi r^2$ (top and bottom circles) + $2\pi rh$ (side wrap).
* Top and Bottom: $2 \times 3.14 \times 12.25 = 76.93$
* Side: $2 \times 3.14 \times 3.5 \times 6 = 131.88$
* Total: $76.93 + 131.88 = 208.81$
Surface Area $\approx$
208.81 square inches
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4. Rectangular Prism with a Diagonal Cut (Wedge)
*Note: This shape is exactly half of a rectangular box with dimensions $7 \times 8 \times 7$.*
Given: Length = 7 in, Width = 8 in, Height = 7 in
*
Volume: Calculate the volume of the full box and divide by 2.
$$\text{Full Box Volume} = 7 \times 8 \times 7 = 392$$
$$\text{Shape Volume} = 392 / 2 = 196$$
Volume =
196 cubic inches
*
Surface Area: Add the areas of all 5 visible faces.
1.
Bottom Rectangle: $7 \times 8 = 56$
2.
Back Rectangle: $8 \times 7 = 56$
3.
Side Triangle: $\frac{1}{2} \times 7 \times 7 = 24.5$
4.
Front Triangle: Same as side = $24.5$
5.
Slanted Rectangle: We need the diagonal length of the triangle side first. Using Pythagoras ($a^2 + b^2 = c^2$):
$$7^2 + 7^2 = 49 + 49 = 98$$
$$\sqrt{98} \approx 9.9$$ (This matches the label on the image).
Area = $9.9 \times 8 = 79.2$
* Total Sum: $56 + 56 + 24.5 + 24.5 + 79.2 = 240.2$
Surface Area $\approx$
240.2 square inches
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Final Answer:
1. Cube (side 3 in):
* Volume: 27 in³
* Surface Area: 54 in²
2. Triangular Prism:
* Volume: 7.2 in³
* Surface Area: 29.2 in²
3. Cylinder (diam 7 in, height 6 in):
* Volume: ~230.79 in³
* Surface Area: ~208.81 in²
4. Wedge Prism:
* Volume: 196 in³
* Surface Area: ~240.2 in²
Parent Tip: Review the logic above to help your child master the concept of volume surface area worksheet.
