Three-Dimensional Figures | Grade 10
📐 Parts of Three-Dimensional Figures
Key Components
Vertex (Vertices): A point where three or more edges meet; the corner of a 3D shape.
Edge: A line segment where two faces meet; the boundary joining one vertex to another.
Face: A flat or curved surface of a solid object.
🔢 Euler's Formula
F + V - E = 2
Where: F = Number of Faces, V = Number of Vertices, E = Number of Edges
Note: This formula applies to all convex polyhedra.
📊 Common 3D Shapes - Parts Reference
| 3D Shape | Faces (F) | Vertices (V) | Edges (E) |
|---|---|---|---|
| Cube | 6 | 8 | 12 |
| Rectangular Prism | 6 | 8 | 12 |
| Triangular Prism | 5 | 6 | 9 |
| Pentagonal Prism | 7 | 10 | 15 |
| Hexagonal Prism | 8 | 12 | 18 |
| Triangular Pyramid | 4 | 4 | 6 |
| Square Pyramid | 5 | 5 | 8 |
| Pentagonal Pyramid | 6 | 6 | 10 |
| Hexagonal Pyramid | 7 | 7 | 12 |
| Cylinder | 3 | 0 | 2 |
| Cone | 2 | 1 | 1 |
| Sphere | 1 | 0 | 0 |
📚 Three-Dimensional Figure Vocabulary
🔹 Polyhedron
A three-dimensional shape with flat polygonal faces, straight edges, and sharp vertices.
🔹 Prism
A polyhedron with two congruent and parallel polygonal bases, and rectangular or parallelogram side faces.
For an n-sided base: Faces = n + 2, Vertices = 2n, Edges = 3n
🔹 Pyramid
A polyhedron with a polygonal base and triangular side faces that meet at a common vertex (apex).
For an n-sided base: Faces = n + 1, Vertices = n + 1, Edges = 2n
🔹 Platonic Solids
Regular polyhedra where all faces are congruent regular polygons and the same number of faces meet at each vertex.
The five Platonic solids: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron
🔹 Convex Polyhedron
A polyhedron where any line segment joining two points on its surface lies entirely inside or on the polyhedron.
🔹 Concave Polyhedron
A polyhedron where at least one line segment joining two points on its surface lies outside the polyhedron.
🔹 Apex
The highest point or vertex of a pyramid or cone, opposite to the base.
🔹 Base
The face on which a three-dimensional figure stands or rests.
🔹 Height (Altitude)
The perpendicular distance from the base to the opposite face or vertex.
🔹 Lateral Face
A face that is not a base; the side faces of a prism or pyramid.
📦 Nets and Drawings of Three-Dimensional Figures
What is a Net?
A net is a two-dimensional representation of a three-dimensional figure. It shows all the faces of a 3D shape unfolded and laid flat.
Common Nets
➤ Cube Net
Consists of 6 congruent squares arranged so they can be folded to form a cube.
➤ Rectangular Prism Net
Consists of 6 rectangles (3 pairs of congruent rectangles).
➤ Cylinder Net
Consists of 2 congruent circles (bases) and 1 rectangle (lateral surface).
➤ Pyramid Net
Consists of 1 polygonal base and triangular lateral faces meeting at the apex.
➤ Cone Net
Consists of 1 circle (base) and 1 sector of a circle (lateral surface).
💡 Key Points about Nets
✓ Multiple nets can represent the same 3D figure
✓ Nets help visualize surface area calculations
✓ When folded, nets must form a closed 3D shape with no overlaps
✂️ Cross Sections of Three-Dimensional Figures
What is a Cross Section?
A cross section is the two-dimensional shape formed when a plane intersects (cuts through) a three-dimensional figure.
Types of Cross Sections
➤ Parallel Cross Section
A plane parallel to the base creates a cross section congruent to the base.
➤ Perpendicular Cross Section
A plane perpendicular to the base creates a different shaped cross section.
Common Cross Sections by Shape
| 3D Figure | Possible Cross Sections |
|---|---|
| Cube | Square, Rectangle, Triangle, Trapezoid, Pentagon, Hexagon |
| Cylinder | Circle, Rectangle, Ellipse |
| Cone | Circle, Ellipse, Triangle, Parabola, Hyperbola |
| Sphere | Circle (all cross sections are circles) |
| Rectangular Prism | Rectangle, Square, Triangle, Trapezoid, Pentagon, Hexagon |
| Pyramid | Triangle, Rectangle (depends on base), Trapezoid |
| Triangular Prism | Triangle, Rectangle, Trapezoid, Pentagon |
🔍 Conic Sections
When a plane cuts through a cone at different angles, four types of conic sections are formed:
• Circle: Horizontal cut parallel to the base
• Ellipse: Diagonal cut that doesn't touch the base
• Parabola: Diagonal cut parallel to the slant edge that touches the base
• Hyperbola: Vertical cut through a double cone
🔄 Solids of Revolution
What is a Solid of Revolution?
A solid of revolution is a three-dimensional figure created by rotating a two-dimensional shape around a line (axis of rotation).
Common Solids of Revolution
🔸 Cylinder
2D Shape: Rectangle
Axis of Rotation: One side of the rectangle or through its center parallel to a side
🔸 Cone
2D Shape: Right Triangle
Axis of Rotation: One leg of the right triangle
🔸 Sphere
2D Shape: Semicircle or Circle
Axis of Rotation: Diameter of the semicircle or through the center of the circle
🔸 Torus (Doughnut Shape)
2D Shape: Circle
Axis of Rotation: A line external to the circle (not passing through it)
💡 Key Properties
✓ The axis of rotation can be inside, on the edge, or outside the 2D shape
✓ Different axis placements create different solids
✓ Perpendicular cross sections of solids of revolution match the original 2D shape
✓ Solids with curved surfaces (cylinders, cones, spheres) are typically created by revolution
📐 Important Formulas Summary
🔹 Euler's Formula for Polyhedra
F + V - E = 2
🔹 Prism Formulas (n-sided base)
Number of Faces: F = n + 2
Number of Vertices: V = 2n
Number of Edges: E = 3n
🔹 Pyramid Formulas (n-sided base)
Number of Faces: F = n + 1
Number of Vertices: V = n + 1
Number of Edges: E = 2n
💡 Quick Reference Tips
✅ To identify a 3D shape: Count its faces, vertices, and edges
✅ To verify a polyhedron: Use Euler's formula (F + V - E should equal 2)
✅ To find cross sections: Imagine slicing the 3D shape with a plane and observe the 2D shape formed
✅ To create solids of revolution: Rotate a 2D shape around a line (axis)
✅ To visualize nets: Imagine unfolding the 3D shape flat on a surface
📚 Master these concepts for success in Tenth Grade Geometry! 📚