Three-dimensional figures are the foundation for everything in this unit on surface area and volume. Before you can calculate how much space a solid occupies or how much material covers it, you need to know what kind of solid you're dealing with and how its parts fit together.
This section covers the classification of 3D figures (polyhedra vs. non-polyhedra), the specific characteristics of each type, how to sketch them from a set of properties, and how cross-sections work.
Three-Dimensional Figures
Classification of 3D figures
The first big distinction is between polyhedra and non-polyhedra. A polyhedron (plural: polyhedra) is a solid whose surfaces are all flat polygons. If any surface is curved, it's not a polyhedron.
Polyhedra:
- Prisms have two congruent, parallel bases (which are polygons) connected by lateral faces that are parallelograms. In a right prism, those lateral faces are rectangles. Prisms are named by their base shape: a triangular prism has triangular bases, a pentagonal prism has pentagonal bases, and so on.
- Pyramids have one polygonal base, and all the other faces are triangles that meet at a single point called the apex. Like prisms, pyramids are named by their base: a square pyramid has a square base, a hexagonal pyramid has a hexagonal base.
Non-polyhedra:
- Cylinders have two congruent, parallel circular bases connected by a curved lateral surface. Think of a soup can or a pipe.
- Cones have one circular base connected to a single point (the apex) by a curved lateral surface.
- Spheres are entirely curved, with every point on the surface equidistant from the center. A sphere has no bases, no edges, and no vertices.
Characteristics of geometric solids
Each type of solid has specific parts you should be able to identify and describe.
Prisms
- Bases are congruent, parallel polygons
- Lateral faces are parallelograms (rectangles in a right prism) connecting corresponding sides of the two bases
- Lateral edges are the parallel segments where two lateral faces meet
- A cube is a special rectangular prism where all 6 faces are congruent squares. A triangular prism has 2 triangular bases and 3 rectangular lateral faces.
Cylinders
- Bases are congruent, parallel circles
- The lateral surface is one continuous curved surface connecting the bases (if you "unroll" it, you get a rectangle)
- The axis is the segment connecting the centers of the two bases. In a right cylinder, the axis is perpendicular to both bases.
Pyramids
- One polygonal base (triangle, square, pentagon, etc.)
- Lateral faces are triangles connecting each edge of the base to the apex
- The number of lateral faces equals the number of sides on the base. A square pyramid has 4 triangular lateral faces; a hexagonal pyramid has 6.
Cones
- One circular base
- The lateral surface is curved, tapering from the base to the apex
- The axis connects the center of the base to the apex. In a right cone, the axis is perpendicular to the base.
Spheres
- No faces, edges, or vertices
- Every point on the surface is the same distance (the radius) from the center
Sketching 3D figures from properties
When you're given a description of a solid and asked to sketch it, work through these steps:
- Identify the figure type. Look at the number and shape of bases and the type of lateral surfaces. Two circular bases? Cylinder. One polygonal base with triangular lateral faces? Pyramid.
- Draw the base(s). Sketch the base shape with the correct dimensions. For a prism or cylinder, draw both bases so they appear parallel (use a slight slant to show depth).
- Connect the bases or apex. For prisms, draw rectangular lateral faces connecting corresponding vertices. For pyramids, draw edges from each base vertex to the apex. For cylinders and cones, draw the curved surface connecting the base(s) to the other base or apex.
- Label everything. Mark dimensions (height, base lengths, radii), right angle symbols where faces are perpendicular, and tick marks on parallel edges.
A common mistake is drawing the bases of a prism at different sizes. Remember: both bases of a prism are congruent, so they should look identical in your sketch (just offset to show depth).
Cross-sections of 3D figures
A cross-section is the 2D shape you get when a plane slices through a 3D figure. The shape of the cross-section depends on the angle of the cut relative to the solid. This is a topic that shows up frequently on tests, so pay close attention to how the cutting plane is oriented.
Prisms
- Parallel to the bases: The cross-section is congruent to the base. Slice a triangular prism parallel to its triangular bases, and you get a triangle of the same size.
- Perpendicular to the bases (and parallel to the lateral edges): The cross-section is a rectangle.
- Oblique to the bases: The cross-section is still a polygon with the same number of sides as the base, but it will be stretched compared to the base.
Cylinders
- Parallel to the bases: Circle (same size as the base)
- Perpendicular to the bases (through the axis): Rectangle
- Oblique to the bases: Ellipse
Pyramids
- Parallel to the base: A polygon similar to the base, but smaller. The closer the cut is to the apex, the smaller the cross-section. For a square pyramid, this gives you a smaller square.
- Perpendicular to the base, through the apex: Triangle (specifically, an isosceles triangle for a right pyramid cut through the apex along a line of symmetry)
- Perpendicular to the base, not through the apex: A trapezoid (not necessarily the same polygon as the base). For instance, slicing a square pyramid this way typically produces a trapezoid, not a square.
Cones
- Parallel to the base: Circle (smaller than the base)
- Perpendicular to the base, through the apex: Isosceles triangle
- Oblique to the base (not passing through the base): Ellipse
- These cross-sections are actually the conic sections you may encounter in later courses. Depending on the angle, a plane can also produce a parabola (parallel to the slant of the cone) or a hyperbola (steeper than the slant), but for this course, focus on circles, triangles, and ellipses.
Spheres
- Any plane that intersects a sphere produces a circle.
- A plane through the center creates the largest possible circle, called a great circle (its radius equals the sphere's radius).
- A plane that doesn't pass through the center still produces a circle, just a smaller one.
Quick check for cross-sections: Ask yourself two questions: (1) What solid am I cutting? (2) What is the orientation of the cut relative to the base(s)? Those two pieces of information are enough to determine the cross-section shape.