Math by First Principles
Level 1 · Foundational ← Back to lesson

Composite and Hollow Solids: Practice

12 multiple-choice questions, progressively harder.

Level 1 · Foundational 0 / 12 answered
Question 1 of 12
  1. 1

    Two pieces have disjoint interiors and meet only along a face. Their volumes are 45 and 28 cubic centimeters. What is the joined volume?

  2. 2

    A solid of volume 144 cubic centimeters has a cavity of volume 35 cubic centimeters removed. How much material remains?

  3. 3

    Two solids with disjoint interiors have total surface areas 90 and 78 square centimeters. They meet over one matching face of area 12 square centimeters, with no other contact. Find the exterior area.

  4. 4

    A right circular pipe has outer radius 3 cm, inner radius 2 cm, and length 5 cm. Find its material volume.

    A right circular pipe open at both ends. The true end section has outer radius 3 cm and inner radius 2 cm; the axial section shows an empty bore through the full length 5 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 3 cm and inner radius 2 cm; the axial section shows an empty bore through the full length 5 cm. Radii are measured from the common center. No computed measurements are shown.R 3 cmr 2 cmAxial sectionLength 5 cm
  5. 5

    One end of an open pipe is an annulus with outer radius 6 cm and inner radius 4 cm. What is the area of that one end?

  6. 6

    A right circular cylinder of length 9 cm has a hemisphere of radius 4 cm attached at each end. What is the full tip-to-tip length?

    True axial section of a cylinder with two hemispherical ends. The common radius is 4 cm. The full tip-to-tip length is unknown. Dashed diameters mark internal joins, not exposed disks.True axial section of a cylinder with two hemispherical ends. The common radius is 4 cm. The full tip-to-tip length is unknown. Dashed diameters mark internal joins, not exposed disks.r 4 cm?Tip-to-tip length (schematic)Cylinder9 cm
  7. 7

    A cylinder of radius 2 cm and length 3 cm has one solid hemisphere of the same radius attached to an end. What is the total volume?

    True axial section of a cylinder with one hemispherical end and one flat end. The common radius is 2 cm. The cylinder length is 3 cm. 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 2 cm. The cylinder length is 3 cm. Dashed diameters mark internal joins, not exposed disks.r 2 cm3 cmCylinder length
  8. 8

    A radius-3 cm hemisphere is attached flush to one end of a right cylinder of radius 3 cm and height 4 cm. The other cylinder end is flat and exposed. What is the exterior area, including that flat end?

    True axial section of a cylinder with one hemispherical end and one flat end. The common radius is 3 cm. The cylinder length is 4 cm. 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 3 cm. The cylinder length is 4 cm. Dashed diameters mark internal joins, not exposed disks.r 3 cm4 cmCylinder length
  9. 9

    A rectangular box has perpendicular edges 4 cm, 4 cm, and 7 cm. Find a space diagonal.

    A rectangular box with perpendicular edge lengths 4, 4, and 7 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 4, 4, and 7 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.ABCG4 cm4 cm7
  10. 10

    A thick pipe is open at both ends and hangs freely. A coating must cover every material surface, including inside the bore. Which list is complete?

  11. 11

    Two solid parts meet only along their boundaries, with no gaps or overlapping interiors. Why can their volumes be added?

  12. 12

    A block is 5 cm by 6 cm by 7 cm. A radius-1 cm cylindrical hole passes perpendicularly through the 7 cm thickness, strictly inside the 5 cm by 6 cm faces. What is the remaining volume?

    True section perpendicular to a cylindrical through hole in a rectangular block. The outer rectangular face is 5 cm by 6 cm. The centered circular opening has radius 1 cm. The block thickness and bore length is 7 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 5 cm by 6 cm. The centered circular opening has radius 1 cm. The block thickness and bore length is 7 cm, perpendicular to this face. The hole fits strictly inside the face.1 cm5 cm6Thickness 7 cm
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