Composite Solids → Volume and Surface Area of Combined 3D Shapes

The total volume equals the sum of the cylinder and cone volumes.

Cylinder → V = π r² h = 3.14 × 4² × 10 = 3.14 × 16 × 10 = 502.4 cm³

Cone → V = (1 / 3) π r² h = (1 / 3) × 3.14 × 16 × 6 = 100.5 cm³

Total Volume = 502.4 + 100.5 = 602.9 ≈ 602.9 cm³.

Rectangular Prism → V = l × w × h = 8 × 6 × 4 = 192 m³

Triangular Prism → V = (1 / 2) × base × height × length = (1 / 2) × 8 × 3 × 4 = 48 m³

Total Volume = 192 + 48 = 240 m³.

Combine volumes when solids are joined, not subtracted.

Cylinder → V = π r² h = 3.14 × 6² × 12 = 3.14 × 36 × 12 = 1356.48 m³

Hemisphere → V = (2 / 3) π r³ = (2 / 3) × 3.14 × 6³ = (2 / 3) × 3.14 × 216 = 452.16 m³

Total Volume = 1357.4 + 452.2 = 1808.64 m³.

Cylinder → V = π r² h = 3.14 × 4² × 12 = 3.14 × 16 × 12 = 603.2 cm³

Cone → V = (1 / 3) π r² h = (1 / 3) × 3.14 × 16 × 9 = 150.7 cm³

Total Volume = 603.2 + 150.7 = 753.6 cm³.

Cylinder → V = π r² h = 3.14 × 3² × 9 = 3.14 × 9 × 9 = 254.3 cm

Cone → V = (1 / 3) π r² h = (1 / 3) × 3.14 × 9 × 9 = 84.8 cm³

Remaining Volume = 254.3 − 84.8 = 169.56 ≈ 169.6 cm³.

Subtraction applies when part of a solid is removed.

Cylinder minus cone.

Cylinder → V = π r² h = 3.14 × 10² × 15 = 3.14 × 100 × 15 = 4710 m³

Hemisphere → V = (2 / 3) π r³ = (2 / 3) × 3.14 × 10³ = (2 / 3) × 3.14 × 1000 = 2093.3 m³

Total Volume = 4710 + 2093.3 = 6803.33

Triangular Prism → V = (1 / 2) × b × h × L = (1 / 2) × 12 × 8 × 10 = 480 cm³

Cube → V = side³ = 10³ = 1000 cm³

Total Volume = 480 + 1000 = 1480 cm³

Joining shapes means summing their individual volumes.

Cylinder → V = π r² h = 3.14 × 2² × 10 = 3.14 × 4 × 10 = 125.6 cm³.

Cone → V = (1 / 3) π r² h = (1 / 3) × 3.14 × 4 × 4 = 16.8 cm³.

Total Volume = 125.6 + 16.8 = 142.35 ≈ 142.4 cm³.

Cube: 6³ = 216 cm³.

Hemisphere: ⅔ πr³ = ⅔ × 3.14 × 27 = 56.5 cm³.

Total = 216 + 56.5 = 272.52 ≈ 272.5 cm³.

Stacking solids combines their volumes to represent joined figures.

Sphere: V=4/3πr³= 4/3 × 3.14 × 125 = 523.3 m³.

Half of that = 261.67 ≈ 261.7–261.8 m³.

Always apply fraction multipliers (½, ¼) for partial solids like hemispheres.

Prism: 5 × 5 × 10 = 250 m³.

Pyramid: ⅓ × 25 × 6 = 50 m³

Total = 300 m³.

When shapes are joined, their total space equals the sum of their parts.

Cylinder: 3.14 × 7² × 14 = 2154.04 cm³.

Hemisphere: 2 3 × 3.14 × 7³ = 718.01 cm³

Add to get 2872.05 ≈ 2872.1 cm³.

Composite shapes often require calculating two separate volumes first.

Rectangular prism: 12 × 10 × 6 = 720 cm³.

Half-cylinder: ½ × πr2h = ½ × 3.14 × 25 × 10 = 392.5 cm³.

Total 1112.5 cm³.

Always check that all units match before adding.

Cube: 8³ = 512 m³.

Pyramid: ⅓ × 64 × 6 = 128 m³.

Total = 640 m³.

This mix models a house-like structure with a square base and pyramid roof.

Cube: 5³ = 125cm³.

Cone: 1/3πr²h = ⅓ × 3.14 × 6.25 × 6 = 39.25 cm³

Sum = 164.25 cm³

Always add if the cone is on top; subtract if it’s hollowed out.

For joined solids, total volume = sum of each part’s volume. Use subtraction only when a section is removed or hollow.

When two solids are joined together, any faces that touch become internal and therefore no longer exposed to the outside, so these shared faces must be counted only once—and then excluded entirely—because surface area measures only the outermost visible surfaces of the composite figure, not the hidden interior connections.

Because a hemisphere is exactly half of a sphere, and the formula for the volume of a full sphere is (4/3)πr³, taking half of that gives (1/2)(4/3)πr³ = (2/3)πr³, meaning the statement is correct since the hemisphere’s volume is precisely two-thirds of πr³.

Whenever a solid has material removed—such as a drilled cylinder, a carved spherical cavity, or any hollow region—the correct procedure is to compute the volume of the entire original solid and then subtract the volume of the removed piece, ensuring the final result accurately represents the remaining physical space in the composite object.

This is incorrect because the total surface area of a cylinder includes both circular bases and the curved lateral surface, and the lateral area formula 2πrh clearly shows dependence on height h as well as radius r, meaning both dimensions jointly determine the cylinder’s total surface area.