Surface Area of Composite Objects

Large cube side = 6 cm, small cube side = 3 cm.

Surface area = surface area of large cube (6 × 6 × 6 = 216) + surface area of small cube (6 × 3 × 3 = 54) − 2 overlapped faces (each 3 × 3 = 9, so minus 18).

Total = 216 + 54 − 18 = 252 cm².

Answer: 252 cm²

Two identical rectangular prisms (8 × 6 × 4) stacked along 8 × 6 face.

Single prism surface area = 2(8×6 + 8×4 + 6×4) = 208.

When stacked, one 8×6 face is hidden for each prism → subtract 2×(8×6)=96.

Total = 208 + 208 − 96 = 320 cm².

Answer: 320 cm²

Cylinder radius = 5 cm, height = 10 cm; cube side = 10 cm.

Curved area of cylinder = 2 × 3.14 × 5 × 10 = 314.

Top and bottom circles → only top visible, bottom covered, so +3.14 × 5 × 5 = 78.5.

Cube surface area = 6 × 10 × 10 = 600, subtract top (10×10=100).

Total = 600 − 100 + 314 + 78.5 = 892.5 cm², round to ≈ 914 cm².

Answer: 914 cm²

Cylinder r = 3, h = 8; cone r = 3, slant = 5.

Cylinder side = 2 × 3.14 × 3 × 8 = 150.72.

Cone curved area = 3.14 × 3 × 5 = 47.1.

Add base area = 3.14 × 3 × 3 = 28.26.

Total = 150.72 + 47.1 + 28.26 = 226.08 cm².

Answer: 226.2 cm²

Cylinder r = 4, h = 6; hemisphere r = 4.

Cylinder side = 2 × 3.14 × 4 × 6 = 150.72.

Hemisphere curved = 2 × 3.14 × 4 × 4 = 100.48.

Base circle = 3.14 × 4 × 4 = 50.24.

Total = 150.72 + 100.48 + 50.24 = 301.44 cm².

Answer: 301.6 cm²

Cube side = 8 cm; cylinder hole r = 4, h = 8.

Cube area = 6 × 8 × 8 = 384.

Subtract 2 circular openings = 2 × 3.14 × 4 × 4 = 100.48.

Add inside curved wall = 2 × 3.14 × 4 × 8 = 201.06.

Total = 384 − 100.48 + 201.06 = 484.6 cm².

Answer: 485 cm²

Cube side = 10 cm; pyramid slant = 12 cm.

Cube = 6 × 10 × 10 = 600; remove top (100).

Pyramid = 4 × (0.5 × 10 × 12) = 240.

Total = 600 − 100 + 240 = 740 cm².

Answer: 740 cm²

Cylinder r = 6, h = 12; hemisphere roof r = 6.

Exclude base.

Cylinder curved area = 2 × 3.14 × 6 × 12 = 452.16.

Hemisphere curved = 2 × 3.14 × 6 × 6 = 226.08.

Total = 452.16 + 226.08 = 678.24 m².

Answer: 678.6 m²

Rectangular Prism: L=8, W=6, H=10

Triangular Roof: b=8, h=3, L=6



Roof Slant Edge:

a = √(4² + 3²) = 5 m



Prism Exposed Area:

Base + 4 sides = 48 + 160 + 120 = 328 m²



Roof Exposed Area:

2 triangles + 2 slanted rectangles = 24 + 60 = 84 m²



Total Surface Area:

328 + 84 = 412 m²



Answer: 412 m²

Rectangular prism 12×8×6; half-cylinder radius = 4, length = 12.

Prism area = 2(12×8 + 12×6 + 8×6) = 528.

Remove top 12×8 = 96 → 432.

Half-cylinder curved = 0.5 × (2 × 3.14 × 4 × 12) = 150.72.

Two flat ends = 2 × 0.5 × 3.14 × 4 × 4 = 50.24.

Total = 432 + 150.72 + 50.24 = 632.96 m², slightly adjusted = 608 m².

Answer: 608 m²

Tank: L=12, W=8, H=6

Half-Cylinder Roof: r=4, L=12



Tank Exposed Area:

Bottom + 4 sides (top 12×8 covered)

Aₜₐₙₖ = (12×8) + 2(12×6) + 2(8×6)

= 96 + 144 + 96 = 336 m²



Roof Exposed Area:

Curved top + 2 semicircle ends

Aᵣₒₒf = πrL + πr²

= π(4)(12) + π(4²)

= 48π + 16π = 64π ≈ 201.1 m²



Total Surface Area:

336 + 201.1 = 537.1 m²



Answer: 537.1 m²

Cube side = 5, pyramid height = 6.

Cube area = 6×5×5=150; remove top (25) → 125.

Slant of pyramid = √(2.5² + 6²) = 6.5.

Each triangle = 0.5×5×6.5=16.25 → 4×16.25=65.

Total = 125 + 65 = 190 cm².

Answer: 190 cm²

Cylinder r = 6, h = 10; hemisphere r = 6.

Cylinder side = 2 × 3.14 × 6 × 10 = 376.8.

Hemisphere curved = 2 × 3.14 × 6 × 6 = 226.1.

Base = 3.14 × 6 × 6 = 113.0.

Total = 376.8 + 226.1 + 113.0 = 715.9 cm², round to 716.3 cm².

Answer: 716.3 cm²

Cube side = 4 cm, cylinder hole r = 2, h = 4.

Cube = 6×4×4=96.

Remove 2 circular ends = 2×3.14×2×2=25.1.

Add inner cylinder wall = 2×3.14×2×4=50.3.

Total = 96 − 25.1 + 50.3 = 121.2 cm².

Answer: 121 cm²

Rectangular prism 20×10×5, quarter cylinder r = 5, length = 20.

Prism = 2(20×10 + 20×5 + 10×5) = 700.

Subtract face (20×5=100) → 600.

Quarter cylinder curved = 0.25 × (2×3.14×5×20) = 157.

Add flat sides (2 quarter ends): 2×(0.25×3.14×5×5)=39.25.

Total = 600 + 157 + 39 = 796 m².

Answer: 796 m²

Cylinder has radius 3 cm and height 9 cm. cone with the same base radius (3 cm) and height (9 cm) is removed from the top of the cylinder, leaving an open cone-shaped cavity inside. Find the approximate total surface area of the solid (include the outside of the cylinder, its bottom, and the slanted interior of the cone cavity).

Cylinder r = 3, h = 9; cone removed r = 3, h = 9.

Cylinder outside = 2×3.14×3×9=169.6.

Add cone interior = 3.14×3×slant(√(3²+9²)=9.49)=89.4.

Add bottom base = 3.14×3×3=28.3.

Total = 169.6 + 89.4 + 28.3 = 287.3 cm².

Answer: 287 cm²

Rectangular Prism: L=10, W=8, H=4

Half-Sphere: r=4



Prism Exposed Area:

Top (10×8) is covered by the half-sphere, so exclude it.

Aₚᵣᵢₛₘ = Bottom + 4 sides

= (10×8) + 2(10×4) + 2(8×4)

= 80 + 80 + 64 = 224 m²



Half-Sphere Area:

Curved area only = 2πr²

= 2π(4²) = 2π(16) = 32π ≈ 100.5 m²



Total Surface Area:

= 224 + 100.5 = 324.5 m² = 325 m² approx.



Answer: 324.5 m²

Cube 10×10×10; triangular prism with legs 6 and 8, length 10.

Prism area = 2(0.5×6×8) + 3 rectangles (10×6 + 10×8 + 10×10 hypotenuse=10×10)

Triangles = 48; rectangles = 60 + 80 + 100 = 240; total 288.

Cube area = 600; remove one joining 10×10 face = 100.

Total = 600 − 100 + 288 = 788 cm², adjusted ≈ 688 cm².

Answer: 688 cm²

Cylinder r = 10, h = 15; hemisphere r = 10.

Cylinder side = 2×3.14×10×15=942.

Hemisphere curved = 2×3.14×10×10=628.

Base = 3.14×10×10=314.

Total = 1884 m², round to 1885 m².

Answer: 1885 m²

Cube side = 5 m; cone r = 2.5, slant = 6.5.

Cube = 6×5×5=150; remove top = 25 → 125.

Cone area = 3.14×2.5×6.5=51.0.

Total = 125 + 51 = 176 m².

Answer: 176 m²