Symmetry and Geometric Constructions
This page explains line symmetry, rotational symmetry, and basic geometric (compass) constructions with examples, for REET. Within the Geometry section, this topic often brings a straightforward factual question about the symmetry lines of familiar shapes.
- 1 question in REET 2024
- Class 6–7 level
- 7 min read
How REET asks this topic
REET 2024 asked about the number of lines of symmetry in a square — a straightforward factual question. This topic usually brings factual questions on the lines of symmetry and order of rotation of familiar shapes (square, rectangle, triangle, circle), and occasionally on construction methods too.
Line Symmetry and Rotational Symmetry
A line (or axis) of symmetry is a line that divides a shape into two equal parts that are mirror images of each other.
Rotational symmetry occurs when a shape, rotated less than 360° about its centre point, still looks exactly the same. The number of times the shape looks like its original form during this rotation is called the order of rotation. Note that it's not necessary for every shape to have both line symmetry and rotational symmetry — some shapes have only one type, while regular shapes like a square or equilateral triangle have both together.
Symmetry in Common Shapes
| Shape | Number of Lines of Symmetry | Order of Rotational Symmetry |
|---|---|---|
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Equilateral triangle | 3 | 3 |
| Isosceles triangle (non-equilateral) | 1 | 1 (no meaningful rotational symmetry) |
| General parallelogram (not a rectangle/rhombus) | 0 | 2 |
| Circle | Infinite | Infinite |
A square's 4 lines of symmetry — 2 along the diagonals, and 2 joining the midpoints of opposite sides.
Examples of Symmetry in the English Alphabet
- Only vertical line symmetry
- Letters like A, M, T, U, V, W, Y
- Only horizontal line symmetry
- Letters like B, C, D, E, K
- Both vertical and horizontal line symmetry
- Letters like H, I, O, X
Basic Geometric Constructions
The main constructions with a compass and straightedge: the perpendicular bisector of a line segment, the bisector of an angle, and the construction of a 60° angle — from which further angles like 30°, 90° and 120° can also be built.
A triangle is constructed from given information using the SSS (all three sides), SAS (two sides and the included angle), ASA (two angles and the included side), or RHS (hypotenuse and one side in a right triangle) method — these are the same four criteria used in congruence.
Starting from a 60° angle, you can also obtain angles like 30° (bisecting 60°), 90° (by combining 60° and its bisected 30°), and 120° (60° taken twice) through construction, directly with a compass and straightedge, without using a protractor.
Steps for Basic Constructions (Summary)
| Construction | Main Steps |
|---|---|
| Perpendicular bisector | From both ends of the line segment, draw arcs with a radius more than half the segment's length on both sides; the line joining the two points where the arcs intersect is the perpendicular bisector |
| Angle bisector | From the vertex of the angle, draw an arc that marks a point on each arm; from these points, draw two arcs of equal radius and join their intersection point to the vertex |
| 60° angle | Draw an arc from a point on a line; using the same radius, cut another arc on that arc; drawing a line from the vertex to this new point gives a 60° angle |
How to Teach This to Children
Classroom activity
Check line symmetry by folding paper (if the two halves match exactly on folding, it's symmetric), and by holding a mirror up to a shape and viewing its reflection. For rotational symmetry, draw a shape on tracing paper and rotate it to see how many times it looks the same. Local patterns like kolam/rangoli can also be used to find symmetry.
Teaching aids
A mirror, tracing paper, a geometry box (compass, protractor, ruler), and dot-grid paper are useful. An activity like 'make a symmetric pattern' using cut-out coloured paper shapes is also a creative way to teach symmetry.
Common misconception
Students often treat line symmetry and rotational symmetry as the same thing, or assume that any shape with equal sides will have as many lines of symmetry as it has sides — this isn't always true for general polygons.
Assessment idea
Give an unfamiliar shape or letter (say, from a local script) and ask how many lines of symmetry it has, and to mark them. Asking students to find lines of symmetry in their own rangoli/pattern is also a meaningful assessment task.
Memory trick / exam tip
Rather than memorising the number of symmetry lines in the English alphabet and common shapes, understand it by folding paper — the exam often carries a direct question on a square, rectangle, equilateral triangle, or letters (like H, T), where simply counting the symmetry lines is enough.
REET 2024 Questions (with Solutions)
Question 93
How many lines of symmetry are there for a square ?
- A2
- B8
- C4
- D6
Explanation
वर्ग की 4 सममिति रेखाएँ होती हैं — 2 विकर्णों के अनुदिश और 2 सम्मुख भुजाओं के मध्य-बिंदुओं को मिलाने वाली रेखाएँ। आयत की केवल 2 सममिति रेखाएँ होती हैं, इसलिए 2 गलत है।
Practice Questions
Question 1
How many lines of symmetry does an equilateral triangle have?
- A3
- B1
- C2
- D0
Explanation
An equilateral triangle has all three sides and angles equal, so it has 3 lines of symmetry, each running from a vertex to the midpoint of the opposite side.
Question 2
How many lines of symmetry does a circle have?
- Aअनंत
- B4
- C1
- D0
Explanation
Every line passing through a circle's centre divides it into two equal halves, so a circle has infinitely many lines of symmetry.
Question 3
What is the order of rotational symmetry of a square?
- A4
- B2
- C8
- D1
Explanation
A square looks like its original form each time it's rotated by 90°, 180°, 270° and 360°, so its order is 4.
Question 4
Which of the following shapes has no line of symmetry but has rotational symmetry of order 2?
- Aसामान्य समांतर चतुर्भुज (आयत/समचतुर्भुज नहीं)
- Bसमचतुर्भुज
- Cआयत
- Dसमबाहु त्रिभुज
Explanation
A general parallelogram (that isn't a rectangle or rhombus) has no line of symmetry, but rotating it by 180° makes it look like its original form, so its order of rotational symmetry is 2.
Question 5
Which instruments are mainly required to construct the perpendicular bisector of a line segment?
- Aपरकार और रूलर
- Bकेवल चांदा
- Cकेवल स्केल
- Dफ्रीहैंड चित्रण
Explanation
Drawing arcs of equal radius (more than half the segment's length) from both ends of the segment, then joining the points where they intersect, gives the perpendicular bisector — a compass and ruler are enough for this.
Question 6
How many lines of symmetry does a regular hexagon have?
- A6
- B3
- C12
- D4
Explanation
A regular hexagon has 6 sides and 6 vertices; it has 3 lines along the diagonals and 3 joining the midpoints of opposite sides — 6 lines of symmetry in total.
Question 7
Which uppercase English letter has exactly 2 lines of symmetry (both horizontal and vertical)?
- AH
- BT
- CF
- DL
Explanation
The letter H has both horizontal and vertical lines of symmetry. T has only a vertical line, while F and L have no line of symmetry at all.
Question 8
Constructing a 60° angle using two equal-radius arcs with a compass is based on the property of which triangle?
- Aसमबाहु त्रिभुज (सभी भुजाएँ बराबर होने पर सभी कोण 60° होते हैं)
- Bसमकोण त्रिभुज
- Cसमद्विबाहु त्रिभुज
- Dविषमबाहु त्रिभुज
Explanation
Arcs of equal radius produce an equilateral triangle, and every angle of an equilateral triangle is 60° — this is exactly the property used to construct a 60° angle.
Question 9
Which is the most effective classroom activity to clarify the concept of symmetry?
- Aकागज मोड़कर या दर्पण रखकर आकृतियों की जाँच कराना
- Bकेवल परिभाषा बोर्ड पर लिखना
- Cपाठ्यपुस्तक से जोर से पढ़ाना
- Dकोई गतिविधि आवश्यक नहीं
Explanation
Folding paper or holding up a mirror lets children directly see and check symmetry, which isn't possible just by reading a definition.
Question 10
In rotational symmetry, what does 'order' mean?
- A360° की पूर्ण परिक्रमा में आकृति कितनी बार मूल आकृति जैसी दिखती है
- Bआकृति की भुजाओं की संख्या
- Cसममिति रेखाओं की संख्या
- Dआकृति के कोणों का योग
Explanation
The order of rotational symmetry tells you exactly how many times a shape looks like its original form during one full 360° rotation.
Frequently Asked Questions
How many questions come from symmetry-construction in REET?+
Usually 1 question from this topic appears, often on the lines of symmetry of a familiar shape; REET 2024 also had 1 question.
What's the difference between line symmetry and rotational symmetry?+
In line symmetry, folding the shape along a line makes the two halves match (a mirror image), while in rotational symmetry, rotating the shape about its centre point makes it look like its original form.
How many lines of symmetry does a square have?+
A square has 4 lines of symmetry — 2 along the diagonals and 2 joining the midpoints of opposite sides.
What tools are needed for geometric construction?+
Mainly a compass and an unmarked straightedge (ruler); a protractor can be used for verification when measuring angles.
Does a circle have infinite lines of symmetry?+
Yes, every line passing through the circle's centre divides it into two equal mirror halves, so a circle has infinitely many lines of symmetry.
How is the bisector of an angle constructed?+
Draw an arc from the vertex of the angle that marks a point on each arm, then draw two arcs of equal radius from those points; the line joining their intersection point to the vertex divides the angle into two equal parts.
How many lines of symmetry do the letters H and O have?+
Both letters have 2 lines of symmetry each — one vertical and one horizontal.
Why does a rectangle have fewer lines of symmetry than a square?+
A rectangle has only 2 lines of symmetry, joining the midpoints of opposite sides, while a square also has its two diagonals as symmetry lines (4 total), because all of a square's sides are equal.
Does every shape have an order of rotational symmetry of at least 1?+
Yes, every shape looks like its original form at the full 360° rotation, so the order of rotational symmetry is always taken to be at least 1, even if it has no meaningful line symmetry.
Why are the number of lines of symmetry and the order of rotation the same for an equilateral triangle?+
An equilateral triangle is a regular shape, and in regular polygons, both the number of lines of symmetry and the order of rotational symmetry equal the number of sides — that's why both are 3 here.
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Change Log
- 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked