Three-Dimensional Figures - Sixth Grade Mathematics
Three-dimensional solid figures occupy physical space, possessing length, width, and depth, bounded by flat polygonal faces or smooth curved surfaces.
Polyhedra vs. Non-Polyhedra
A solid figure bounded entirely by flat polygonal surfaces is called a polyhedron (plural: polyhedra). Solid figures with curved surfaces are non-polyhedra.
The Solid Geometry Classification:
Three-Dimensional Solids
|
+-------------------------+-------------------------+
| |
Polyhedra Non-Polyhedra
(All Flat Polygonal Faces) (Curved Surfaces)
| |
+------+------+ +--------------+--------------+
| | | | |
Prisms Pyramids Cylinders Cones Spheres
(2 Congruent (1 Base + (2 Circular (1 Circular (All points
Bases) Triangles) Bases) Base) equidistant)
Visualizing Common 3D Solids:
Rectangular Prism: Square Pyramid: Cylinder: Cone:
+-------+ /\ .---. /\
/ /| / \ / \ / \
+-------+ | / /\ \ | | / /\ \
| | + +------+ | | +------+
| |/ | | \ / (Base)
+-------+ +------+ '---'
Anatomy of a Polyhedron
Every polyhedron is defined by three interrelated geometric features:
1. Face: A flat surface of the solid shaped like a polygon.
2. Edge: A straight line segment where two adjacent faces meet.
3. Vertex (plural: vertices): A corner point where three or more edges intersect.
Anatomy Illustrated on a Cube:
Face (Top flat square)
v
+--------+ <- Vertex (Point/Corner)
/| /|
+--------+ |
| | | | <- Edge (Line segment)
| +------|-+
|/ |/
+--------+
Prisms vs. Pyramids
Prisms and pyramids are named by the polygonal shape of their base.
+-------------------+----------------------------+----------------------------+
| Characteristic | Prisms | Pyramids |
+-------------------+----------------------------+----------------------------+
| Bases | 2 congruent, parallel bases| Exactly 1 polygonal base |
+-------------------+----------------------------+----------------------------+
| Lateral Faces | Parallelograms/Rectangles | Triangles meeting at apex |
+-------------------+----------------------------+----------------------------+
| Vertex Point | No single shared peak | Opposite apex vertex |
+-------------------+----------------------------+----------------------------+
Examples:
- Triangular Prism: 2 triangular bases + 3 rectangular lateral faces = 5 faces, 9 edges, 6 vertices.
- Rectangular Prism: 2 rectangular bases + 4 rectangular lateral faces = 6 faces, 12 edges, 8 vertices.
- Triangular Pyramid (Tetrahedron): 1 triangular base + 3 triangular lateral faces = 4 faces, 6 edges, 4 vertices.
- Square Pyramid: 1 square base + 4 triangular lateral faces = 5 faces, 8 edges, 5 vertices.
Euler's Formula for Convex Polyhedra
The Swiss mathematician Leonhard Euler discovered a universal invariant relating the faces (F), vertices (V), and edges (E) of any convex polyhedron: F + V - E = 2 (or F + V = E + 2).
Verifying Euler's Formula:
Cube (Rectangular Prism): F = 6, V = 8, E = 12 --> 6 + 8 - 12 = 14 - 12 = 2.
Square Pyramid: F = 5, V = 5, E = 8 --> 5 + 5 - 8 = 10 - 8 = 2.
Triangular Prism: F = 5, V = 6, E = 9 --> 5 + 6 - 9 = 11 - 9 = 2.
Exercises
Exercise 1. Classify each solid figure as a polyhedron or non-polyhedron, and state its specific name: (a) A soda can; (b) A standard six-sided die; (c) An Egyptian monument with a square bottom and four triangular sides; (d) An ice cream sugar cone.
Exercise 2. Count the number of faces, vertices, and edges for: (a) A pentagonal prism; (b) A hexagonal pyramid; (c) A triangular prism.
Exercise 3. Verify Euler's Formula (F + V - E = 2) for: (a) A pentagonal pyramid; (b) An octagonal prism.
Exercise 4. A mystery polyhedron has 10 faces and 16 vertices. How many edges must it have?
Solutions
Solution to Exercise 1. For (a), a soda can is a non-polyhedron because its side is curved; it is a cylinder. For (b), a six-sided die is a polyhedron whose faces are squares; it is a cube (or rectangular prism). For (c), the Egyptian monument is a polyhedron with a square base and triangular sides; it is a square pyramid. For (d), an ice cream sugar cone is a non-polyhedron with a circular base and curved side meeting at a vertex; it is a cone.
Solution to Exercise 2. For (a), a pentagonal prism has 2 pentagonal bases and 5 rectangular lateral faces: Faces = 2 + 5 = 7, Vertices = 5 * 2 = 10, Edges = 5 * 3 = 15. For (b), a hexagonal pyramid has 1 hexagonal base and 6 triangular lateral faces: Faces = 1 + 6 = 7, Vertices = 6 + 1 = 7, Edges = 6 * 2 = 12. For (c), a triangular prism has 2 triangular bases and 3 rectangular lateral faces: Faces = 2 + 3 = 5, Vertices = 3 * 2 = 6, Edges = 3 * 3 = 9.
Solution to Exercise 3. For (a), a pentagonal pyramid has F = 6, V = 6, and E = 10. Euler's formula: F + V - E = 6 + 6 - 10 = 12 - 10 = 2 (Verified). For (b), an octagonal prism has F = 10, V = 16, and E = 24. Euler's formula: F + V - E = 10 + 16 - 24 = 26 - 24 = 2 (Verified).
Solution to Exercise 4. Using Euler's formula F + V - E = 2: substitute F = 10 and V = 16: 10 + 16 - E = 2. 26 - E = 2. Subtract 26: -E = -24, so E = 24 edges. The polyhedron has 24 edges.