Geometric Terms and Notation
This topic covers the fundamental vocabulary and concepts of 2D geometry, from points and lines to the properties of polygons. Mastering this terminology is essential for describing shapes and understanding more complex geometric problems.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- A regular polygon has all side lengths equal and all interior angles equal. An irregular polygon does not have both of these properties.
- For any regular n-sided polygon, the order of rotational symmetry is n, and it has n lines of reflection symmetry.
- Lines of symmetry in regular polygons with an even number of sides connect opposite vertices or the midpoints of opposite sides.
- Lines of symmetry in regular polygons with an odd number of sides connect a vertex to the midpoint of the opposite side.
- An angle 'subtended' by a line segment at a point is the angle formed by drawing lines from that point to the two ends of the segment.
- Parallel lines have the same gradient and never intersect. Perpendicular lines intersect at a right angle (90 degrees).
Formulae
Srotational = Sreflection = n To find the number of symmetries for a REGULAR n-sided polygon. The order of rotational symmetry and the number of lines of reflection both equal the number of sides, n.
Definitions
- Polygon
- A closed, two-dimensional shape formed by three or more straight line segments.
- Vertex (plural: Vertices)
- A point where two or more edges of a shape meet; a corner.
- Line Segment
- A finite part of a straight line with two distinct endpoints.
- Reflection Symmetry
- A property where a shape can be divided by a line (the line of symmetry) into two halves that are mirror images of each other.
- Rotational Symmetry
- A property where a shape looks the same after being rotated by less than a full 360-degree turn around a central point. The 'order' is the number of times it matches its original outline in one full rotation.
Worked example
A polygon has 8 lines of symmetry. Four of these lines pass through opposite vertices, and four pass through the midpoints of opposite sides. What is the name of this polygon?
- 1
The polygon has 8 lines of symmetry.
For a regular polygon, the number of sides (n) equals the number of lines of symmetry.
- 2
This implies the shape is a regular octagon (n=8).
- 3
The description of the symmetry lines confirms this.
Regular polygons with an even number of sides (like 8) have lines of symmetry that connect opposite vertices and lines that connect the midpoints of opposite sides.
- 4
Therefore, the polygon is a regular octagon.
Answer: Regular octagon
Common mistakes
- ×Assuming all polygons have the same symmetry properties as regular polygons. For example, a non-square rectangle has rotational symmetry of order 2, not 4.
- ×Confusing the types of symmetry lines for regular polygons with odd vs. even numbers of sides.
- ×Incorrectly identifying the order of rotational symmetry. The starting position always counts as the first 'fit', but the order is the total number of fits in a 360-degree turn.
No-calculator tips
- ✓Quickly sketch the polygon in question. This makes it much easier to visualise and count lines of symmetry and rotational fits without complex calculations.
- ✓For regular n-gons, remember the shortcut: number of sides = order of rotational symmetry = number of reflection lines. No counting is needed.
- ✓To test for rotational symmetry mentally, pick a distinctive vertex and track its position as you rotate the shape in your mind.