Composite Solids and Cross-Sections
A composite solid is made by joining or removing simpler solids, and a cross-section is the shape you get when a plane cuts through a solid. In brief: the volume of a composite solid is the sum (or difference) of the volumes of its parts. A silo made of a cylinder with radius 2 and height 6 topped by a hemisphere has volume π · 4 · 6 + (2/3)π · 8 = 24π + 16π/3 ≈ 80.9. A horizontal cut through a cone gives a circle, and a vertical cut through the apex gives a triangle.
Real objects are rarely a single standard solid. Houses, tanks, bottles and machine parts combine boxes, cylinders, cones and spheres, and engineers study cross-sections to see inside objects or to calculate volumes by slicing. This guide shows how to decompose composite solids, how to find volumes and surface areas without double counting, how to recognise the cross-sections of common solids and a table of cross-sections.
How do you handle composite solids?
- Sketch the solid and identify the simple solids it is made of (and any parts that are removed).
- Write down the dimensions of each part, noting which are shared.
- Compute the volume of each part.
- Add the volumes of joined parts and subtract the volumes of removed parts (holes, hollows).
- State the result with units.
Example 1: a house shape
A shed is a box 6 m by 4 m by 3 m with a triangular prism on top as a roof. The prism has a triangular cross-section with base 4 m and height 1.5 m, and the length is 6 m. The volume of the box is 72 m3. The prism has volume (1/2) · 4 · 1.5 · 6 = 18 m3. The total is 90 m3.
Example 2: a hollow pipe
A pipe 2 m long has the outer radius 6 cm and the inner radius 5 cm. Volume of material = π(62 − 52) · 200 = π · 11 · 200 = 2200π ≈ 6,911.5 cm3.
Example 3: an ice cream cone
A cone with radius 3 and height 10 topped by a hemisphere of radius 3. V = (1/3)π · 9 · 10 + (2/3)π · 27 = 30π + 18π = 48π ≈ 150.8 cm3.
Example 4: a cube with a cylindrical hole
A cube with edge 10 has a cylindrical hole of radius 2 drilled right through between two opposite faces. V = 1000 − π · 4 · 10 = 1000 − 40π ≈ 874.3.
How do you find the surface area of a composite solid?
Add the areas of the exposed surfaces only. Surfaces where two solids are joined are no longer on the outside, and must be left out. For the silo (cylinder plus hemisphere), the surface consists of the lateral area of the cylinder, the base circle at the bottom and the curved surface of the hemisphere. The top circle of the cylinder is hidden.
Example 5: the silo
Radius 2, cylinder height 6. Lateral area of the cylinder: 2π · 2 · 6 = 24π. Bottom base: π · 4 = 4π. Hemisphere: 2π · 4 = 8π. Total: 36π ≈ 113.1.
Example 6: two cubes
Two cubes of edge 3 are glued together face to face into a 3 by 3 by 6 box. The separate cubes have a combined surface area of 2 · 54 = 108. Gluing hides two faces of 9 each: 108 − 18 = 90. Check with the box formula: 2(9 + 18 + 18) = 90 ✓. The volume is just 54, the sum of the parts.
What are cross-sections?
A cross-section is the intersection of a solid with a plane. The shape depends on the solid and on the position of the plane.
- Cube or cuboid: a plane parallel to a face gives a rectangle (a square for a cube). A plane through opposite edges gives a rectangle. Other planes give triangles, pentagons or hexagons; the largest regular cut of a cube is a regular hexagon through the midpoints of six edges.
- Cylinder: parallel to the bases, a circle. Parallel to the axis, a rectangle. Slanted, an ellipse.
- Cone: parallel to the base, a circle. Through the apex, a triangle. Slanted, an ellipse; parallel to a side, a parabola; steeper, a hyperbola (these are the conic sections).
- Sphere: always a circle. A plane through the center gives a great circle with radius r. A plane at distance d from the center gives a circle of radius √(r2 − d2).
- Pyramid: parallel to the base, a polygon similar to the base. Through the apex, a triangle.
For a plane parallel to the base of a pyramid or cone at distance x from the apex (height h), the cross-section is similar to the base with the scale factor x/h, and its area is B(x/h)2.
Cross-sections of common solids
| Solid | Cut parallel to the base | Cut perpendicular to the base through the axis | Other cuts |
|---|---|---|---|
| Cube | square | rectangle or square | triangle, pentagon, hexagon |
| Cylinder | circle | rectangle | ellipse |
| Cone | circle | isosceles triangle | ellipse, parabola, hyperbola |
| Sphere | circle | circle (great circle) | circle |
| Square pyramid | square | isosceles triangle | trapezoid, triangle, pentagon |
| Triangular prism | triangle | rectangle | parallelogram, trapezoid |
Worked examples
Example 7: a slice of a sphere
A sphere of radius 13 is cut by a plane at distance 5 from its center. The circle has radius √(169 − 25) = 12, and its area is 144π ≈ 452.4.
Example 8: a slice of a cone
A cone has height 12 and base radius 6. A plane parallel to the base, 4 units below the apex, makes a circle of radius 6 · (4/12) = 2. Its area is 4π. The part of the cone above the plane is a small cone with volume (1/3)π · 4 · 4 = 16π/3 ≈ 16.8, and the frustum has the volume (1/3)π · 36 · 12 − 16π/3 = 144π − 5.33π ≈ 435.
Example 9: cross-section area by Cavalieri
A hemisphere of radius r and a cylinder of radius r and height r from which a cone is removed have equal cross-section areas at every height y: πr2 − πy2 for both (hemisphere: π(r2 − y2); cylinder minus cone: πr2 − πy2). By Cavalieri’s principle, the volumes are equal: (2/3)πr3 = πr3 − (1/3)πr3 ✓.
Where do these ideas come from?
Apollonius of Perga studied the conic sections, the cross-sections of a cone, in the third century BCE. Archimedes computed volumes by comparing slices, and his method anticipated both Cavalieri’s principle and integration. Cavalieri’s work in the seventeenth century systematised the idea, and Newton and Leibniz turned slicing into the integral calculus.
Where are they used today?
Medical scanners (CT and MRI) produce cross-sections of the body, which are stacked to form a three-dimensional image. Engineers draw sections of machines to show the interior. Civil engineers compute the volume of earth to move by averaging the cross-sectional areas along a road. A concrete column in the shape of a cylinder with a hole, or a bottle made of a cylinder and a cone, is analysed as a composite solid.
Which mistakes are common?
Counting hidden faces in the surface area
When solids are joined, the contact surfaces are not part of the outside.
Adding the surface areas of joined solids
Volumes add, but surface areas do not add simply. Subtract the joined faces.
Forgetting to subtract a hollow
For a pipe or a bottle with an interior hollow, subtract the volume of the empty space.
Assuming that every cross-section of a cone is a circle
Only cuts perpendicular to the axis give circles. Slanted cuts give ellipses, parabolas or hyperbolas.