1. **Rectangular prism (a): length = 10 cm, width = 6 cm, height = 8 cm** - Cross-sectional area (base area): $A = \text{length} \times \text{width} = 10 \times 6 = 60 \text{ cm}^2$ - Lateral surface area (side areas without top and bottom): $LSA = 2 \times \text{height} \times (\text{length} + \text{width}) = 2 \times 8 \times (10 + 6) = 2 \times 8 \times 16 = 256 \text{ cm}^2$ - Total surface area: $TSA = 2 \times A + LSA = 2 \times 60 + 256 = 120 + 256 = 376 \text{ cm}^2$ - Volume: $V = A \times \text{height} = 60 \times 8 = 480 \text{ cm}^3$ 2. **Cube (b): side length = 5 cm** - Cross-sectional area (face area): $A = 5 \times 5 = 25 \text{ cm}^2$ - Lateral surface area: $LSA = 4 \times \text{side}^2 = 4 \times 25 = 100 \text{ cm}^2$ - Total surface area: $TSA = 6 \times \text{side}^2 = 6 \times 25 = 150 \text{ cm}^2$ - Volume: $V = 5^3 = 125 \text{ cm}^3$ 3. **L-shaped prism (center-left): composed of rectangles with given dimensions** - Break into two rectangles: section 1: $4 \text{ cm} \times 3 \text{ cm}$; section 2: $6 \text{ cm} \times 4 \text{ cm}$ (based on the dimension data) - Cross-sectional area = sum of areas: $4 \times 3 + 6 \times 4 = 12 + 24 = 36 \text{ cm}^2$ - Height of prism given as 9 cm; volume = cross-sectional area $\times$ height: $36 \times 9 = 324 \text{ cm}^3$ - Lateral surface area = perimeter of cross-section $\times$ height - Perimeter = sum of all outer sides of the L-shape: $ (4 + 3 + 6 + 4 + 3 + 3) = 23 \text{ cm}$ (approximated from values) - LSA = $23 \times 9 = 207 \text{ cm}^2$ - Total surface area = lateral surface area + 2 \times cross-sectional area = $207 + 2 \times 36 = 207 + 72 = 279 \text{ cm}^2$ 4. **Cross-shaped prism (center-right)** - Cross-sectional area calculated by decomposing into rectangles: approximate calculation from given arms of widths 3 cm or 4 cm and height 4 cm. - Assume the cross consists of one main vertical rectangle 4 cm wide by 8 cm high plus arms 3 cm wide by 4 cm high attached on sides (by dimensions given) - Calculate total cross-sectional area = main vertical part + 2 horizontal arms: - Main vertical: $4 \times 8 = 32$ - Each arm: $3 \times 4 = 12$; two arms $= 24$ - Total cross-section area = $32 + 24 = 56 \text{ cm}^2$ - Height of prism = 4 cm; volume = $56 \times 4 = 224 \text{ cm}^3$ - Perimeter estimated as $2(4+8) + 2 \times 3$ sum of outer edges approx.: $24 + 6 = 30$ - Lateral surface area = perimeter $\times$ height = $30 \times 4 = 120 \text{ cm}^2$ - Total surface area = $120 + 2 \times 56 = 120 + 112 = 232 \text{ cm}^2$ 5. **Trapezoidal prism (bottom-left): base rectangle 4 cm by 6 cm, top trapezoid with sides 3 cm, height 10 cm** - Cross-sectional area: base rectangle + trapezoid on top - Area rectangle = $4 \times 6 = 24$ - Area trapezoid: using given sides (assumed height 3 cm): - Bases are 4 cm and 6 cm, height 3 cm, trapezoid area = $\frac{(4 + 6)}{2} \times 3 = \frac{10}{2} \times 3 = 5 \times 3 = 15$ - Total cross-sectional area = $24 + 15 = 39 \text{ cm}^2$ - Volume = cross-sectional area $\times$ prism height = $39 \times 10 = 390 \text{ cm}^3$ - Perimeter approximated as rectangle perimeter plus trapezoid sides: $2(4 + 6) + 2 \times 3 = 20 + 6 = 26$ - Lateral surface area = perimeter $\times$ height = $26 \times 10 = 260 \text{ cm}^2$ - Total surface area = lateral surface area + 2 $\times$ cross-sectional area = $260 + 78 = 338 \text{ cm}^2$ 6. **Multi-step prism (bottom-right): steps 10 cm high, one top rectangle 30 cm wide and other dimensions 10 cm** - Cross-sectional area calculated by summing rectangle areas forming the step - Steps: 3 layers each 10 cm by 10 cm area = $3 \times 10 \times 10 = 300$ plus top rectangle 30 cm by 10 cm = 300 - Total cross-sectional area = $300 + 300 = 600 \text{ cm}^2$ - Volume = cross-sectional area $\times$ depth - Without specific height, assume height equals depth slab dimension (not clearly given) - repeating 10 cm as consistent - Volume = $600 \times 10 = 6000 \text{ cm}^3$ - Perimeter counted along the edges approximately $30 + 10 + 10 + \ldots$ sum to $80$ cm (approximation) - Lateral surface area = perimeter $\times$ height = $80 \times 10 = 800 \text{ cm}^2$ - Total surface area = lateral surface area + 2 $\times$ cross-sectional area = $800 + 1200 = 2000 \text{ cm}^2$ **Final answers summary:** (a) Cross-sectional area = $60 \text{ cm}^2$, LSA = $256 \text{ cm}^2$, TSA = $376 \text{ cm}^2$, Volume = $480 \text{ cm}^3$ (b) Cross-sectional area = $25 \text{ cm}^2$, LSA = $100 \text{ cm}^2$, TSA = $150 \text{ cm}^2$, Volume = $125 \text{ cm}^3$ (c) Cross-sectional area = $36 \text{ cm}^2$, LSA = $207 \text{ cm}^2$, TSA = $279 \text{ cm}^2$, Volume = $324 \text{ cm}^3$ (d) Cross-sectional area = $56 \text{ cm}^2$, LSA = $120 \text{ cm}^2$, TSA = $232 \text{ cm}^2$, Volume = $224 \text{ cm}^3$ (e) Cross-sectional area = $39 \text{ cm}^2$, LSA = $260 \text{ cm}^2$, TSA = $338 \text{ cm}^2$, Volume = $390 \text{ cm}^3$ (f) Cross-sectional area = $600 \text{ cm}^2$, LSA = $800 \text{ cm}^2$, TSA = $2000 \text{ cm}^2$, Volume = $6000 \text{ cm}^3$