Quadrilaterals and Polygons: Properties, Angle Sum, Diagonals and Area Formulas for SSC CGL and CHSL Geometry
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PDF Notes: Quadrilaterals and Polygons: Properties, Angle Sum, Diagonals and Area Formulas for SSC CGL and CHSL Geometry
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Quadrilaterals and polygons are scoring topics in the SSC geometry and mensuration sections. This guide on quadrilaterals and polygons for SSC gives the properties of all the quadrilaterals in a table, the formulas for the angles and the diagonals of polygons and the area formulas, with solved examples, practice questions and FAQs.
Quadrilaterals: General
The sum of the four angles of a quadrilateral is 360°. The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram (Varignon’s theorem).
Properties of Special Quadrilaterals
| Quadrilateral | Sides and Angles | Diagonals | Area |
|---|---|---|---|
| Parallelogram | Opposite sides parallel and equal; opposite angles equal; adjacent angles supplementary | Bisect each other (not equal) | Base × height |
| Rhombus | All four sides equal; opposite angles equal | Bisect each other at right angles; bisect the angles | ½ d1 d2 |
| Rectangle | Opposite sides equal; all angles 90° | Equal and bisect each other | l × b |
| Square | All sides equal; all angles 90° | Equal, perpendicular and bisect each other; length a√2 | a² = d²/2 |
| Trapezium | One pair of parallel sides | Not equal in general (equal in an isosceles trapezium) | ½(a + b) h |
| Kite | Two pairs of adjacent sides equal | Perpendicular; one diagonal bisects the other | ½ d1 d2 |
- A rhombus with one right angle is a square; a parallelogram with equal diagonals is a rectangle; a parallelogram with perpendicular diagonals is a rhombus.
- A parallelogram inscribed in a circle is a rectangle.
- The parallelograms on the same base and between the same parallels are equal in area; a triangle and a parallelogram on the same base and between the same parallels: the triangle is half.
- The midpoint theorem: the line joining the midpoints of two sides of a triangle is parallel to the third side and half of it.
- Perimeter of a rhombus = 4a; the side of a rhombus = ½√(d1² + d2²).
- The sum of the squares of the diagonals of a parallelogram = the sum of the squares of its four sides.
Cyclic Quadrilateral
A quadrilateral inscribed in a circle has opposite angles that add up to 180°; its area is given by Brahmagupta’s formula: √((s − a)(s − b)(s − c)(s − d)), where s is the semi-perimeter. Ptolemy: product of the diagonals = sum of the products of the opposite sides.
Polygons: Formulas
| Item | Formula (for a polygon with n sides) |
|---|---|
| Sum of interior angles | (n − 2) × 180° |
| Each interior angle of a regular polygon | (n − 2) × 180° / n |
| Sum of the exterior angles (always) | 360° |
| Each exterior angle of a regular polygon | 360° / n |
| Number of diagonals | n(n − 3) / 2 |
| Number of triangles formed from one vertex | n − 2 |
| Interior + exterior angle at a vertex | 180° |
Regular Polygons
| Polygon | Interior Angle | Area (side a) |
|---|---|---|
| Equilateral triangle (3) | 60° | (√3/4)a² |
| Square (4) | 90° | a² |
| Regular pentagon (5) | 108° | About 1.72a² |
| Regular hexagon (6) | 120° | (3√3/2)a² = 6 equilateral triangles; the side equals the circumradius |
| Regular octagon (8) | 135° | 2(1 + √2)a² (about 4.83a²) |
| Regular decagon (10) | 144° | – |
| Regular dodecagon (12) | 150° | – |
Solved Examples
Example 1. Find the sum of the interior angles of a hexagon.
(6 − 2) × 180 = 720°.
Example 2. Find each interior angle of a regular octagon.
6 × 180/8 = 135°.
Example 3. Each exterior angle of a regular polygon is 20°. Find the number of sides.
360/20 = 18.
Example 4. Find the number of diagonals of a decagon.
10 × 7/2 = 35.
Example 5. The diagonals of a rhombus are 16 cm and 12 cm. Find its area and side.
Area = ½ × 16 × 12 = 96 sq cm; side = √(8² + 6²) = 10 cm.
Example 6. The parallel sides of a trapezium are 10 cm and 14 cm and the height is 8 cm. Find the area.
½ × 24 × 8 = 96 sq cm.
Example 7. Find the area of a regular hexagon with side 4 cm.
(3√3/2) × 16 = 24√3 = 41.57 sq cm.
Quick Tips
- Memorise the polygon formulas for the interior angle, exterior angle and diagonals.
- Remember that the exterior angles of any polygon add up to 360°.
- Use the diagonals to find the side of a rhombus.
- Remember that a regular hexagon is made of 6 equilateral triangles.
Quick Revision Points
- The sum of the interior angles = (n − 2) × 180°.
- Each exterior angle of a regular polygon = 360°/n.
- Number of diagonals = n(n − 3)/2.
- The area of a rhombus = ½ d1 d2.
- The area of a trapezium = ½(a + b)h.
Practice Questions With Answers
- What is the sum of the interior angles of a pentagon?
Answer: 540° - What is each interior angle of a regular hexagon?
Answer: 120° - How many diagonals does an octagon have?
Answer: 20 - What is the area of a rhombus with diagonals 10 cm and 24 cm?
Answer: 120 sq cm - What is the side of the rhombus above?
Answer: 13 cm - What is the number of sides of a polygon whose interior angles add up to 1,080°?
Answer: 8 - What is each exterior angle of a regular pentagon?
Answer: 72° - What is the length of the diagonal of a square of side 5 cm?
Answer: 5√2 cm - Which quadrilateral has equal and perpendicular diagonals that bisect each other?
Answer: Square - What is the area of a trapezium with parallel sides 7 and 13 and height 5?
Answer: 50 sq cm
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Mathematics notes page, read the previous topic Circles and the next topic Polynomials, Remainder and Factor Theorem.
Frequently Asked Questions
What is the sum of the angles of a quadrilateral?
360 degrees.
What is a parallelogram?
A quadrilateral with both pairs of opposite sides parallel.
What are the properties of a rhombus?
All sides are equal; diagonals bisect each other at right angles.
What is a trapezium?
A quadrilateral with one pair of parallel sides.
What is the formula for the interior angle of a regular polygon?
(n − 2) × 180° / n.
What is the sum of the exterior angles of a polygon?
360 degrees.
How many diagonals does a polygon of n sides have?
n(n − 3)/2.
What is the area of a regular hexagon?
(3√3/2) × side².
What is the midpoint theorem?
The line joining the midpoints of two sides of a triangle is parallel to the third side and half of it.
Is this topic asked in SSC?
Yes. These are asked in geometry and mensuration.
What is a kite?
A quadrilateral with two pairs of adjacent equal sides.
What is Brahmagupta’s formula?
The area of a cyclic quadrilateral, √((s − a)(s − b)(s − c)(s − d)).
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Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.