Math - Geometry Lesson Plans : Perimeter & Area of Quadrilaterals I

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Mensuration - Formulae for Perimeter and Area of Quadrilaterals I

A parallelogram is a quadrilateral whose opposite sides are parallel.
A rhombus is a parallelogram which has four equal sides.
A trapezoid (or trapezium) is a quadrilateral whose one pair of opposite sides are parallel and one pair of opposite sides are nonparallel.

Area of a quadrilateral when diagonals intersect at right angles = ½ × Product of diagonals
In the figure alongside of the quadrilateral PQRS, area = ½ × PR × QS.

Example
Find the area (in cm^{2}) of a quadrilateral whose diagonals intersect at right angles given that the lengths of its diagonals are 10 cm and 14 cm. Solution.
Area of a quadrilateral when diagonals intersect at right angles = ½ × Product of diagonals
= ½ × 10 × 14 = 70 cm^{2}.

Area of a quadrilateral when one diagonal and the lengths of the perpendiculars from its opposite vertices to this diagonal are given = ½ × Diagonal × Sum of the lengths of the perpendiculars
In the figure alongside of the quadrilateral PQRS, area = ½ × PR (QY + SX).

Example
Find the area (in m^{2}) of a quadrilateral whose diagonal is 15 m given that the lengths of the perpendiculars from the opposite vertices to this diagonal are 4 m and 8 m. Solution.
Area of a quadrilateral when one diagonal and the lengths of the perpendiculars from its opposite vertices to this diagonal are given = ½ × Diagonal × Sum of the lengths of the perpendiculars
= ½ × 15 (4 + 8) = 90 m^{2}.

Area of a parallelogram = Base × Height
In the figure alongside of the parallelogram PQRS, area = PQ × RY = QR × PX.

Example
Find the area (in cm^{2}) of a parallelogram whose base is 10 cm and height is 14 cm. Solution.
Area of a parallelogram = Base × Height
= 10 × 14 = 140 cm^{2}.

Area of a rhombus = ½ × Product of diagonals
In the figure alongside of the rhombus PQRS, area = ½ × PR × QS.

Example
Find the area (in cm^{2}) of a rhombus given that the lengths of its diagonals are 11 cm and 30 cm. Solution.
Area of a rhombus = ½ × Product of diagonals
= ½ × 11 × 30 = 165 cm^{2}.

Area of a trapezoid = ½ × Sum of parallel sides × Distance between parallel sides
In the figure alongside of the trapezoid ABCD, area = ½ (AB + CD) CM = ½ (AB + CD) DL.

Example
Find the area (in cm^{2}) of a trapezoid whose parallel sides are 10 cm and 20 cm and the distance between them is 8 cm. Solution.
Area of a trapezoid = ½ × Sum of parallel sides × Distance between parallel sides
= ½ (10 + 20) 8 = 120 cm^{2}.