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Understanding Elementary Shapes - Types of Quadrilaterals

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Quadrilateral is a closed polygon formed by four line segments. It has four sides, four vertices, and four interior angles. Visually, any four-sided flat shape like a sheet of paper or a kite represents a quadrilateral.

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A Parallelogram is a quadrilateral where both pairs of opposite sides are parallel and equal in length. Visually, it looks like a 'slanted' rectangle where opposite angles are also equal (e.g., ∠A=∠C\angle A = \angle C and ∠B=∠D\angle B = \angle D).

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A Rectangle is a special type of parallelogram where every interior angle is exactly a right angle (90∘90^{\circ}). Visually, it has two pairs of equal opposite sides and equal diagonals that bisect each other.

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A Square is a quadrilateral where all four sides are equal and all four angles are right angles (90∘90^{\circ}). Visually, it is the most symmetric quadrilateral, appearing as a perfectly regular box where diagonals are equal and bisect each other at 90∘90^{\circ}.

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A Rhombus is a parallelogram that has all four sides of equal length. Visually, it looks like a 'diamond' or a tilted square. While all sides are equal, the angles are only 90∘90^{\circ} if it is also a square; however, its diagonals always cross at a 90∘90^{\circ} angle.

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A Trapezium is a quadrilateral that has at least one pair of parallel sides. Visually, it often looks like a triangle with the top point cut off by a line parallel to the base. The non-parallel sides can be equal (Isosceles Trapezium) or unequal.

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The Angle Sum Property of a quadrilateral states that the sum of the four interior angles is always 360∘360^{\circ}. Visually, this is because any quadrilateral can be divided into two triangles by drawing a single diagonal, and since each triangle's angles sum to 180∘180^{\circ}, the total is 2×180∘=360∘2 \times 180^{\circ} = 360^{\circ}.

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A Kite is a quadrilateral with two distinct pairs of equal-length sides that are adjacent to each other. Visually, it looks like a traditional flying kite where the diagonals intersect at 90∘90^{\circ}, and one diagonal bisects the other.

📐Formulae

Sum of interior angles: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^{\circ}

Perimeter of a general Quadrilateral: P=side1+side2+side3+side4P = side_1 + side_2 + side_3 + side_4

Perimeter of a Rectangle: P=2×(l+b)P = 2 \times (l + b) where ll is length and bb is breadth

Perimeter of a Square: P=4×sP = 4 \times s where ss is the length of a side

Perimeter of a Rhombus: P=4×sP = 4 \times s where ss is the length of a side

💡Examples

Problem 1:

In a quadrilateral ABCDABCD, three angles are measured as 75∘75^{\circ}, 105∘105^{\circ}, and 100∘100^{\circ}. Find the measure of the fourth angle.

Solution:

  1. Let the fourth angle be xx.
  2. According to the angle sum property of a quadrilateral: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^{\circ}.
  3. Substitute the known values: 75∘+105∘+100∘+x=360∘75^{\circ} + 105^{\circ} + 100^{\circ} + x = 360^{\circ}.
  4. Add the known angles: 280∘+x=360∘280^{\circ} + x = 360^{\circ}.
  5. Subtract 280∘280^{\circ} from both sides: x=360∘−280∘=80∘x = 360^{\circ} - 280^{\circ} = 80^{\circ}.
  6. Therefore, the fourth angle is 80∘80^{\circ}.

Explanation:

We use the fundamental property that all interior angles of any quadrilateral must add up to exactly 360∘360^{\circ} to find the unknown value.

Problem 2:

Identify the quadrilateral which has all sides equal in length but does not necessarily have right angles at its vertices. If one side is 55 cm, what is its perimeter?

Solution:

  1. A quadrilateral with all sides equal is either a square or a rhombus.
  2. Since the problem states it does not necessarily have right angles, the most accurate name is a Rhombus.
  3. All sides of a rhombus are equal, so s=5s = 5 cm.
  4. Perimeter P=4×s=4×5 cm=20 cmP = 4 \times s = 4 \times 5\text{ cm} = 20\text{ cm}.

Explanation:

This problem tests the property-based identification of shapes. Since all sides are equal, we use the perimeter formula 4×side4 \times side.