Composite Solids → Volume And Surface Area Of Combined 3D Shapes

1) A dome is a cylinder (r=10m, h=15m) with a hemisphere (r=10m) on top. Find total volume.

4712.5 m³

6803.33 m³

7854.2 m³

9425.4 m³

Cylinder = 3.14×100×15 = 4710 m³. Hemisphere = (2/3)×3.14×1000 = 2093.3 m³. Total = 4710+2093.3 = 6803.3 m³. Option A gives only the cylinder. Options C and D significantly overestimate the total.

Explanation

Cylinder = 3.14×100×15 = 4710 m³. Hemisphere = (2/3)×3.14×1000 = 2093.3 m³. Total = 4710+2093.3 = 6803.3 m³. Option A gives only the cylinder. Options C and D significantly overestimate the total.

2) A cube (side=5cm) with a cone (r=2.5cm, h=6cm) attached. Find total volume.

145.2 cm³

155.8 cm³

164.3 cm³

175.0 cm³

Cube = 5³ = 125 cm³. Cone = (1/3)×3.14×6.25×6 = 39.25 cm³. Total = 125+39.25 = 164.25 ≈ 164.3 cm³. Option A gives 145.2, option B gives 155.8, option D gives 175, none correctly apply both formulas.

Explanation

Cube = 5³ = 125 cm³. Cone = (1/3)×3.14×6.25×6 = 39.25 cm³. Total = 125+39.25 = 164.25 ≈ 164.3 cm³. Option A gives 145.2, option B gives 155.8, option D gives 175, none correctly apply both formulas.

3) A sphere of radius 5m is cut in half. What is the volume of one half?

261.8 m³

314.2 m³

523.6 m³

130.9 m³

Full sphere = (4/3)×3.14×125 = 523.3 m³. Half = 523.3÷2 = 261.65 ≈ 261.8 m³. Option C gives the full sphere volume. Option B gives π×r², not a volume formula. Option D gives one quarter of the sphere.

Explanation

Full sphere = (4/3)×3.14×125 = 523.3 m³. Half = 523.3÷2 = 261.65 ≈ 261.8 m³. Option C gives the full sphere volume. Option B gives π×r², not a volume formula. Option D gives one quarter of the sphere.

4) A cube (side=6cm) has a hemisphere (r=3cm) on top. Find total volume.

272.5 cm³

324.0 cm³

286.9 cm³

342.8 cm³

Cube = 6³ = 216 cm³. Hemisphere = (2/3)×3.14×27 = 56.5 cm³. Total = 216+56.5 = 272.5 cm³. Option B gives 324, option C gives 286.9, option D gives 342.8, none correctly apply the hemisphere formula.

Explanation

Cube = 6³ = 216 cm³. Hemisphere = (2/3)×3.14×27 = 56.5 cm³. Total = 216+56.5 = 272.5 cm³. Option B gives 324, option C gives 286.9, option D gives 342.8, none correctly apply the hemisphere formula.

5) A cube (side=8m) with a pyramid on top (base 8×8, height=6m). Find total volume.

560 m³

600 m³

620 m³

640 m³

Cube = 8³ = 512 m³. Pyramid = (1/3)×64×6 = 128 m³. Total = 512+128 = 640 m³. Option A gives 560, option B gives 600, option C gives 620, none correctly combine both volumes.

Explanation

Cube = 8³ = 512 m³. Pyramid = (1/3)×64×6 = 128 m³. Total = 512+128 = 640 m³. Option A gives 560, option B gives 600, option C gives 620, none correctly combine both volumes.

6) A cylinder (r=2cm, h=10cm) and a cone (r=2cm, h=4cm) form one solid. Find total volume.

142.4 cm³

162.3 cm³

180.0 cm³

132.9 cm³

Cylinder = 3.14×4×10 = 125.6 cm³. Cone = (1/3)×3.14×4×4 = 16.8 cm³. Total = 125.6+16.8 = 142.4 cm³. Option B gives 162.3, option C gives 180, option D gives 132.9, none correctly apply both formulas.

Explanation

Cylinder = 3.14×4×10 = 125.6 cm³. Cone = (1/3)×3.14×4×4 = 16.8 cm³. Total = 125.6+16.8 = 142.4 cm³. Option B gives 162.3, option C gives 180, option D gives 132.9, none correctly apply both formulas.

7) A rectangular prism 12×10×6 has a half-cylinder (r=5, length=10) on top. Find total volume.

985.5 cm³

1050.0 cm³

1112.5 cm³

1200.0 cm³

Rectangular prism = 12×10×6 = 720 cm³. Half-cylinder = (1/2)×3.14×25×10 = 392.5 cm³. Total = 720+392.5 = 1112.5 cm³. Option A gives 985.5, option B gives 1050, option D gives 1200, none correctly apply the half-cylinder formula.

Explanation

Rectangular prism = 12×10×6 = 720 cm³. Half-cylinder = (1/2)×3.14×25×10 = 392.5 cm³. Total = 720+392.5 = 1112.5 cm³. Option A gives 985.5, option B gives 1050, option D gives 1200, none correctly apply the half-cylinder formula.

8) The surface area of a cylinder depends only on its radius.

True

False

The answer is False. The total surface area of a cylinder includes two circular bases (2πr²) and the lateral curved surface (2πrh). The lateral area formula clearly shows dependence on height h as well as radius r. Both dimensions jointly determine the total surface area.

Explanation

The answer is False. The total surface area of a cylinder includes two circular bases (2πr²) and the lateral curved surface (2πrh). The lateral area formula clearly shows dependence on height h as well as radius r. Both dimensions jointly determine the total surface area.

9) A triangular prism (base=12cm, height=8cm, length=10cm) joined to a cube (side=10cm). Find total volume.

880 cm³

960 cm³

1480 cm³

1760 cm³

Triangular prism = (1/2)×12×8×10 = 480 cm³. Cube = 10³ = 1000 cm³. Total = 480+1000 = 1480 cm³. Option A gives 880, option B gives 960, option D gives 1760, none of which correctly apply both formulas.

Explanation

Triangular prism = (1/2)×12×8×10 = 480 cm³. Cube = 10³ = 1000 cm³. Total = 480+1000 = 1480 cm³. Option A gives 880, option B gives 960, option D gives 1760, none of which correctly apply both formulas.

10) A cylinder (r=7cm, h=14cm) with a hemisphere (r=7cm) on top. Find total volume.

2415.8 cm³

2654.3 cm³

2872.1 cm³

3102.4 cm³

Cylinder = 3.14×49×14 = 2154.04 cm³. Hemisphere = (2/3)×3.14×343 = 718.01 cm³. Total = 2154.04+718.01 = 2872.05 ≈ 2872.1 cm³. Options A, B, and D result from incorrect radius or height substitutions.

Explanation

Cylinder = 3.14×49×14 = 2154.04 cm³. Hemisphere = (2/3)×3.14×343 = 718.01 cm³. Total = 2154.04+718.01 = 2872.05 ≈ 2872.1 cm³. Options A, B, and D result from incorrect radius or height substitutions.

11) A cylinder (r=4 cm, h=10 cm) has a cone (r=4 cm, h=6 cm) on top. Find the total volume.

502.7 cm³

602.9 cm³

670.2 cm³

804.2 cm³

Cylinder = π×4²×10 = 3.14×16×10 = 502.4 cm³. Cone = (1/3)×3.14×16×6 = 100.5 cm³. Total = 502.4+100.5 = 602.9 cm³. Option A gives only the cylinder volume. Options C and D overestimate by using incorrect formulas.

Explanation

Cylinder = π×4²×10 = 3.14×16×10 = 502.4 cm³. Cone = (1/3)×3.14×16×6 = 100.5 cm³. Total = 502.4+100.5 = 602.9 cm³. Option A gives only the cylinder volume. Options C and D overestimate by using incorrect formulas.

12) In composite figures, subtract the missing part's volume if a hole is removed.

True

False

The answer is True. When material is removed — such as a drilled hole, carved cavity, or hollow region — compute the full original solid's volume and subtract the volume of the removed piece. This gives the remaining physical volume of the composite object.

Explanation

The answer is True. When material is removed — such as a drilled hole, carved cavity, or hollow region — compute the full original solid's volume and subtract the volume of the removed piece. This gives the remaining physical volume of the composite object.

13) A cone (r=3cm, h=9cm) is cut from a cylinder (r=3cm, h=9cm). What is the remaining volume?

169.6 cm³

226.2 cm³

254.5 cm³

84.8 cm³

Cylinder = 3.14×9×9 = 254.3 cm³. Cone = (1/3)×3.14×9×9 = 84.8 cm³. Remaining = 254.3-84.8 = 169.5 ≈ 169.6 cm³. Option C gives the full cylinder. Option D gives only the cone. Option B is incorrect.

Explanation

Cylinder = 3.14×9×9 = 254.3 cm³. Cone = (1/3)×3.14×9×9 = 84.8 cm³. Remaining = 254.3-84.8 = 169.5 ≈ 169.6 cm³. Option C gives the full cylinder. Option D gives only the cone. Option B is incorrect.

14) A prism 5m×5m×10m has a pyramid (base 5×5, height 6) on top. What is the total volume?

275 m³

300 m³

325 m³

350 m³

Prism = 5×5×10 = 250 m³. Pyramid = (1/3)×25×6 = 50 m³. Total = 250+50 = 300 m³. Option A gives 275, option C gives 325, option D gives 350, none of which correctly combine both volumes.

Explanation

Prism = 5×5×10 = 250 m³. Pyramid = (1/3)×25×6 = 50 m³. Total = 250+50 = 300 m³. Option A gives 275, option C gives 325, option D gives 350, none of which correctly combine both volumes.

15) A cone (r=4cm, h=9cm) sits on a cylinder (r=4cm, h=12cm). Find total volume.

603.2 cm³

753.6 cm³

870.0 cm³

452.4 cm³

Cylinder = 3.14×16×12 = 603.2 cm³. Cone = (1/3)×3.14×16×9 = 150.7 cm³. Total = 603.2+150.7 = 753.9 ≈ 753.6 cm³. Option A gives only the cylinder. Option D gives only the cone approximately. Option C overestimates.

Explanation

Cylinder = 3.14×16×12 = 603.2 cm³. Cone = (1/3)×3.14×16×9 = 150.7 cm³. Total = 603.2+150.7 = 753.9 ≈ 753.6 cm³. Option A gives only the cylinder. Option D gives only the cone approximately. Option C overestimates.

16) Volume of a hemisphere equals (1/2)(4/3)πr³.

True

False

The answer is True. A hemisphere is exactly half a sphere. The full sphere volume is (4/3)πr³, and half of that is (1/2)(4/3)πr³ = (2/3)πr³. Both expressions are equivalent and correctly represent the hemisphere's volume.

Explanation

The answer is True. A hemisphere is exactly half a sphere. The full sphere volume is (4/3)πr³, and half of that is (1/2)(4/3)πr³ = (2/3)πr³. Both expressions are equivalent and correctly represent the hemisphere's volume.

17) A silo (r=6m, h=12m) has a hemisphere (r=6m) on top. Find the total volume.

1696.5 m³

1808.64 m³

2261.9 m³

2714.3 m³

Cylinder = 3.14×36×12 = 1356.48 m³. Hemisphere = (2/3)×3.14×216 = 452.16 m³. Total = 1356.48+452.16 = 1808.64 m³. Options A, C, and D result from incorrect formulas or wrong substitution of r and h values.

Explanation

Cylinder = 3.14×36×12 = 1356.48 m³. Hemisphere = (2/3)×3.14×216 = 452.16 m³. Total = 1356.48+452.16 = 1808.64 m³. Options A, C, and D result from incorrect formulas or wrong substitution of r and h values.

18) The circular face where a cone meets a cylinder is counted in total surface area twice.

True

False

The answer is False. When two solids are joined, shared faces become internal and are no longer exposed. These faces must be excluded entirely from the surface area calculation since surface area measures only the outermost visible surfaces of the composite figure, not hidden interior connections.

Explanation

The answer is False. When two solids are joined, shared faces become internal and are no longer exposed. These faces must be excluded entirely from the surface area calculation since surface area measures only the outermost visible surfaces of the composite figure, not hidden interior connections.

19) A rectangular prism 8m×6m×4m has a triangular prism roof (base 8m×4m, height 3m). Find total volume.

192 m³

224 m³

240 m³

262 m³

Rectangular prism = 8×6×4 = 192 m³. Triangular prism = (1/2)×8×3×4 = 48 m³. Total = 192+48 = 240 m³. Option A gives only the rectangular prism volume. Options B and D result from incorrect height or base substitutions.

Explanation

Rectangular prism = 8×6×4 = 192 m³. Triangular prism = (1/2)×8×3×4 = 48 m³. Total = 192+48 = 240 m³. Option A gives only the rectangular prism volume. Options B and D result from incorrect height or base substitutions.

20) The volume of a composite solid is found by adding the volumes of its parts.

True

False

The answer is True. For joined solids, the total volume equals the sum of each individual part's volume. Subtraction is used only when material is removed or hollowed out. Adding volumes reflects how the total space occupied grows when separate solids are combined into one figure.

Explanation

The answer is True. For joined solids, the total volume equals the sum of each individual part's volume. Subtraction is used only when material is removed or hollowed out. Adding volumes reflects how the total space occupied grows when separate solids are combined into one figure.

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

1) A dome is a cylinder (r=10m, h=15m) with a hemisphere (r=10m) on top. Find total volume.

2) A cube (side=5cm) with a cone (r=2.5cm, h=6cm) attached. Find total volume.

3) A sphere of radius 5m is cut in half. What is the volume of one half?

4) A cube (side=6cm) has a hemisphere (r=3cm) on top. Find total volume.

5) A cube (side=8m) with a pyramid on top (base 8×8, height=6m). Find total volume.

6) A cylinder (r=2cm, h=10cm) and a cone (r=2cm, h=4cm) form one solid. Find total volume.

7) A rectangular prism 12×10×6 has a half-cylinder (r=5, length=10) on top. Find total volume.

8) The surface area of a cylinder depends only on its radius.

9) A triangular prism (base=12cm, height=8cm, length=10cm) joined to a cube (side=10cm). Find total volume.

10) A cylinder (r=7cm, h=14cm) with a hemisphere (r=7cm) on top. Find total volume.

11) A cylinder (r=4 cm, h=10 cm) has a cone (r=4 cm, h=6 cm) on top. Find the total volume.

12) In composite figures, subtract the missing part's volume if a hole is removed.

13) A cone (r=3cm, h=9cm) is cut from a cylinder (r=3cm, h=9cm). What is the remaining volume?

14) A prism 5m×5m×10m has a pyramid (base 5×5, height 6) on top. What is the total volume?

15) A cone (r=4cm, h=9cm) sits on a cylinder (r=4cm, h=12cm). Find total volume.

16) Volume of a hemisphere equals (1/2)(4/3)πr³.

17) A silo (r=6m, h=12m) has a hemisphere (r=6m) on top. Find the total volume.

18) The circular face where a cone meets a cylinder is counted in total surface area twice.

19) A rectangular prism 8m×6m×4m has a triangular prism roof (base 8m×4m, height 3m). Find total volume.

20) The volume of a composite solid is found by adding the volumes of its parts.