How to Construct Quadrilaterals Using Given Sides Angles and Diagonals
A quadrilateral is a plane closed geometric figure with four sides and four angles. A quadrilateral can be constructed using a compass and a ruler. There are several real-life examples of a quadrilateral: kites, squares, rectangles, rhombus, etc. The word quadrilateral is derived from the word “quad” which means four. Hence, there are four adjoining sides of a quadrilateral at all times.
Properties
A quadrilateral always has four sides and four angles.
The length of sides may be the same or different.
The sum of all the internal angles will be 360°.
The sum of all the external angles will also be 360°.
Construction
A quadrilateral can be constructed using a ruler and a compass if any of the following conditions are provided:
Length of four sides and one diagonal are given
Measurement of three sides and two angles are given
Measurements of two sides and three angles are given.
Types
There are a lot of examples of a quadrilateral but the most common geometric figures are square, rectangle, rhombus, parallelogram and trapezium. Except for a trapezium, all the other figures have at least two parallel sides and every quadrilateral has four vertices, four sides and four angles.
Square has all four equal sides and angles. The diagonals of a square bisect each other at 90°. Both the opposite sides of the square are parallel to each other.
Rectangle has equal opposite sides and all four equal angles. The opposite sides of a rectangle are parallel to each other as well.
Rhombus has equal sides, and opposite sides are parallel to each other. Opposite angles are equal and the sum of two adjacent angles is 180°. The diagonals from opposite sides of a rhombus bisect each other and are perpendicular to each other.
A parallelogram has two opposite and equal sides which are also parallel to each other. The opposite angles are equal and the diagonals bisect with each other.
Trapezium has only one opposite parallel and the diagonal bisectors are of the same ratio.
Quadrilaterals can be further classified as convex quadrilateral, concave quadrilateral and intersecting or crossed quadrilaterals based on the diagonals former inside a quadrilateral. In convex quadrilaterals, the diagonals are completely within the boundary of the quadrilateral while in concave quadrilaterals, the diagonals are partially outside the boundary of the quadrilateral.
Formulas
There are two basic formulas of quadrilaterals, namely: area and perimeter.
Area
Area of square: side×side
Area of rectangle: length×breadth
Area of parallelogram: base×height
Area of rhombus: ½× (first diagonal length)×(second diagonal length)
Area of trapezium: ½× (length of first diagonal)×(length of second diagonal)
Perimeter
The perimeter of square: 4×length of side
The perimeter of rectangle: 2×(length of breadth + length of a side)
Perimeter of parallelogram: 2×(base+side)
The perimeter of rhombus: 4×length of side
The perimeter of trapezium: 2×(a+b), a and b are lengths of adjacent sides.
Have more questions about quadrilaterals and their construction procedures? Visit the Vedantu website or download the app and get free study materials, solved examples and questions, and a lot more.
FAQs on Construction of Quadrilaterals Complete Guide with Methods
1. What is a quadrilateral in geometry?
A quadrilateral is a polygon with four sides, four vertices, and four angles. The sum of its interior angles is always 360°. Common types of quadrilaterals include:
- Square
- Rectangle
- Parallelogram
- Rhombus
- Trapezium (Trapezoid)
- Kite
2. What is the sum of the interior angles of a quadrilateral?
The sum of the interior angles of any quadrilateral is 360°. This can be verified by dividing a quadrilateral into two triangles:
- Each triangle has angle sum = 180°
- So, 180° + 180° = 360°
3. How do you construct a quadrilateral when four sides are given?
To construct a quadrilateral when four sides are given, you need at least one diagonal or one angle for uniqueness. Basic steps are:
- Draw one side, say AB.
- Using a compass, mark arcs for adjacent sides.
- Locate the fourth vertex using the given diagonal or angle.
- Join the points to complete the figure.
4. How do you construct a quadrilateral when two diagonals and three sides are given?
A quadrilateral can be constructed uniquely if three sides and two diagonals are given. Steps:
- Draw one side, say AB.
- Using the given side lengths, construct triangle ABC.
- With given diagonal lengths, locate point D using compass arcs.
- Join AD and CD to complete the quadrilateral.
5. What are the different types of quadrilaterals?
The main types of quadrilaterals are classified based on sides and angles. They include:
- Square: All sides equal, all angles 90°
- Rectangle: Opposite sides equal, all angles 90°
- Parallelogram: Opposite sides parallel and equal
- Rhombus: All sides equal, opposite angles equal
- Trapezium: At least one pair of parallel sides
- Kite: Two pairs of adjacent equal sides
6. What conditions are required to construct a unique quadrilateral?
A quadrilateral is uniquely constructed when sufficient measurements like five independent elements (sides, angles, diagonals) are given. Valid combinations include:
- Four sides and one diagonal
- Three sides and two diagonals
- Two adjacent sides and three angles
7. How do you construct a parallelogram given two adjacent sides and the included angle?
A parallelogram can be constructed using two adjacent sides and their included angle. Steps:
- Draw side AB of given length.
- At A, construct the given angle.
- Mark side AD of given length along the angle ray.
- Through B draw a line parallel to AD.
- Through D draw a line parallel to AB to meet at C.
8. How do you construct a rectangle when length and breadth are given?
A rectangle is constructed using the given length and breadth with right angles. Steps:
- Draw AB equal to the given length.
- At A and B, construct 90° angles.
- Mark AD and BC equal to the given breadth.
- Join C and D.
9. How do you construct a rhombus when one diagonal and a side are given?
A rhombus can be constructed when one diagonal and one side are given. Steps:
- Draw the given diagonal AC.
- With A as center and radius equal to the given side, draw arcs.
- With C as center and same radius, draw arcs intersecting at B and D.
- Join A–B–C–D.
10. What are common mistakes in construction of quadrilaterals?
Common mistakes in constructing quadrilaterals include incorrect measurements and ignoring given conditions. Key errors are:
- Not using the exact compass width
- Misreading angles or diagonals
- Forgetting that angle sum must be 360°
- Drawing rough sketches instead of precise constructions
