Math by First Principles
Level 2 · Intermediate ← Back to lesson

Composite and Hollow Solids: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered
Question 1 of 12
  1. 1

    A right cone of radius 4 cm and perpendicular height 3 cm is attached to one end of a right circular cylinder of radius 4 cm and height 6 cm. Find the whole volume.

    True axial section of a right cone attached to a cylinder. Common radius 4 cm, cone perpendicular height 3 cm, cylinder height 6 cm. The dashed diameter is their internal join. No volume is supplied.True axial section of a right cone attached to a cylinder. Common radius 4 cm, cone perpendicular height 3 cm, cylinder height 6 cm. The dashed diameter is their internal join. No volume is supplied.4 cm36 cm
  2. 2

    A right pipe has outer radius 7 cm, inner radius 5 cm, and length 4 cm. Both ends are open. Find the area of all exposed material surfaces, including the bore wall and both annuli.

    A right circular pipe open at both ends. The true end section has outer radius 7 cm and inner radius 5 cm; the axial section shows an empty bore through the full length 4 cm. Radii are measured from the common center. No computed measurements are shown.A right circular pipe open at both ends. The true end section has outer radius 7 cm and inner radius 5 cm; the axial section shows an empty bore through the full length 4 cm. Radii are measured from the common center. No computed measurements are shown.R 7 cmr 5 cmAxial sectionLength 4 cm
  3. 3

    A 12 cm by 9 cm by 5 cm block has a radius-2 cm bore through its 5 cm thickness, perpendicular to and strictly inside the 12 cm by 9 cm faces. Find all exposed area, including the bore wall and underside.

    True section perpendicular to a cylindrical through hole in a rectangular block. The outer rectangular face is 12 cm by 9 cm. The centered circular opening has radius 2 cm. The block thickness and bore length is 5 cm, perpendicular to this face. The hole fits strictly inside the face.True section perpendicular to a cylindrical through hole in a rectangular block. The outer rectangular face is 12 cm by 9 cm. The centered circular opening has radius 2 cm. The block thickness and bore length is 5 cm, perpendicular to this face. The hole fits strictly inside the face.2 cm12 cm9Thickness 5 cm
  4. 4

    A solid cylinder of radius 5 cm has one same-radius solid hemispherical end. Its total volume is 700π3\frac{700\pi}{3} cubic centimeters. Find the cylinder length.

    True axial section of a cylinder with one hemispherical end and one flat end. The common radius is 5 cm. The cylinder length is h. Dashed diameters mark internal joins, not exposed disks.True axial section of a cylinder with one hemispherical end and one flat end. The common radius is 5 cm. The cylinder length is h. Dashed diameters mark internal joins, not exposed disks.r 5 cmhCylinder length
  5. 5

    A solid capsule has radius 3 cm and full tip-to-tip length 18 cm. It consists of one right cylinder and two hemispheres. Find its exterior area.

    True axial section of a cylinder with two hemispherical ends. The common radius is 3 cm. The full tip-to-tip length is 18 cm. Dashed diameters mark internal joins, not exposed disks.True axial section of a cylinder with two hemispherical ends. The common radius is 3 cm. The full tip-to-tip length is 18 cm. Dashed diameters mark internal joins, not exposed disks.r 3 cm18 cmTip-to-tip length (schematic)
  6. 6

    A rectangular box has a base diagonal of 15 cm and height 8 cm. What is a space diagonal length?

  7. 7

    A right circular pipe has outer radius 5 cm, inner radius 4 cm, and material volume 108π108\pi cubic centimeters. Find its length.

    A right circular pipe open at both ends. The true end section has outer radius 5 cm and inner radius 4 cm; the axial section shows an empty bore through the full length h. Radii are measured from the common center. No computed measurements are shown.A right circular pipe open at both ends. The true end section has outer radius 5 cm and inner radius 4 cm; the axial section shows an empty bore through the full length h. Radii are measured from the common center. No computed measurements are shown.R 5 cmr 4 cmAxial sectionLength h
  8. 8

    A cube of side 6 cm has a side-2 cm cube cut from one corner, with the three cuts parallel to the cube faces. Find the total area of the remaining boundary, including all new cut faces.

    A side-6 cm cube with a side-2 cm cube removed from the front upper right corner. All cut planes are parallel to the original faces. The original removed corner edges are dashed. New cut faces are shown without their areas.A side-6 cm cube with a side-2 cm cube removed from the front upper right corner. All cut planes are parallel to the original faces. The original removed corner edges are dashed. New cut faces are shown without their areas.6 cm outer edge2 cm
  9. 9

    A cylinder has total surface area 32π32\pi square centimeters. Its radius is 2 cm. It is attached by one full circular base to a flat face of a solid box whose total surface area was 70 square centimeters. Their interiors do not overlap, and the cylinder has no other contact with the box. Find the assembly's exterior area.

    A cylinder of radius 2 cm stands on a flat face of a box. It meets the box only over its full bottom disk, marked dashed as an internal join. The separate total surface areas are stated in the question; no component height or exterior area is evaluated.A cylinder of radius 2 cm stands on a flat face of a box. It meets the box only over its full bottom disk, marked dashed as an internal join. The separate total surface areas are stated in the question; no component height or exterior area is evaluated.Cylinder radius 2 cmDashed disk is the join
  10. 10

    A rectangular box has base edges 9 cm and 12 cm, and space diagonal 25 cm. Find its height.

    A rectangular box with perpendicular edge lengths 9, 12, and h cm. Bottom A B C D and upper vertices E F G H are in matching order. AC is a base diagonal; AG is the space diagonal. Hidden back edges are dashed. No diagonal length is evaluated.A rectangular box with perpendicular edge lengths 9, 12, and h cm. Bottom A B C D and upper vertices E F G H are in matching order. AC is a base diagonal; AG is the space diagonal. Hidden back edges are dashed. No diagonal length is evaluated.ABCG9 cm12 cmh
  11. 11

    A solid right circular cylinder has outer radius 4 cm and height 9 cm. A coaxial cylindrical cavity of radius 3 cm extends 7 cm down from the top and stops before reaching the bottom. Find the remaining material volume.

    Schematic axial section of a right cylinder with outer radius 4 cm and height 9 cm. A coaxial cylindrical blind cavity has radius 3 cm and depth 7 cm. The cavity floor lies above the bottom; the dotted line is the axis.Schematic axial section of a right cylinder with outer radius 4 cm and height 9 cm. A coaxial cylindrical blind cavity has radius 3 cm and depth 7 cm. The cavity floor lies above the bottom; the dotted line is the axis.r 3R 49Depth 7Schematic axial section
  12. 12

    A closed solid consists of a right square prism with base side 6 cm and height 4 cm, topped by a regular square pyramid of base side 6 cm and perpendicular height 4 cm. The matching square bases are fully joined. Find all exterior area, including the bottom.

    Central vertical section through opposite base-edge midpoints of a regular square pyramid atop a right square prism. Square base side 6 cm, prism height 4 cm, pyramid perpendicular height 4 cm. The dashed 6 cm join is internal. Slant height is not evaluated.Central vertical section through opposite base-edge midpoints of a regular square pyramid atop a right square prism. Square base side 6 cm, prism height 4 cm, pyramid perpendicular height 4 cm. The dashed 6 cm join is internal. Slant height is not evaluated.44 cmBase width 6 cm
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