Math - Geometry Lesson Plans : Perimeter & Area of Rectangles & Squares

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Mensuration - Formulae for Perimeter and Area of Rectangles and Squares

A quadrilateral is a polygon which has four sides.
A rectangle is a quadrilateral which has four right angles.
A square is a rectangle which has four equal sides.

Perimeter of a rectangle = 2 (Length + Breadth)
Any one of the two longer sides of a rectangle is its length.
In the figure alongside of the rectangle PQRS, length = PQ = RS.
Any one of the two shorter sides of a rectangle is its breadth.
In the figure alongside of the rectangle PQRS, breadth = QR = PS.
In the figure alongside of the rectangle PQRS, perimeter = 2 (PQ + QR).

Example
Find the perimeter (in cm) of a rectangle whose length is 8 cm and breadth is 4 cm. Solution.
Perimeter of a rectangle = 2 (Length + Breadth)
= 2 (8 + 4) = 24 cm.

In the figure alongside of the rectangle PQRS, diagonal = QS = PR (shown in dashed line)

= Ö

PQ^{2} + QR^{2}

(by Pythagorean Theorem applied to ΔPQR).

Diagonal of a rectangle = Ö

Length^{2} + Breadth^{2}

Example
Find the diagonal (in cm) of a rectangle whose length is 12 cm and breadth is 5 cm. Solution.

Diagonal of a rectangle = Ö

Length^{2} + Breadth^{2}

= Ö

12^{2} + 5^{2}

= 13 cm.

Area of a rectangle = Length × Breadth
In the figure alongside of the rectangle PQRS, area = PQ × QR.

Example
Find the area (in cm^{2}) of a rectangle whose length is 10 cm and breadth is 5 cm. Solution.
Area of a rectangle = Length × Breadth
= 10 × 5 = 50 cm^{2}.

Perimeter of a square = 4 × Side
In the figure alongside of the square PQRS, side = PQ = QR = RS = PS.
In the figure alongside of the square PQRS, perimeter = 4 × PQ.

Example
Find the perimeter (in cm) of a square whose side is 5 cm. Solution.
Perimeter of a square = 4 × Side
= 4 × 5 = 20 cm.

Diagonal of a square = Ö2 Side

= Ö

PQ^{2} + QR^{2}

(by Pythagorean Theorem)

= Ö

PQ^{2} + PQ^{2}

= Ö

2 PQ^{2}

= Ö2 PQ.

In the figure alongside of the square PQRS, diagonal = QS = PR (shown in dashed line)

Example
Find the diagonal (in cm) of a square whose side is 15Ö2 cm. Solution.
Diagonal of a square = Ö2 Side
= Ö2 × 15Ö2 = 30 cm.

Area of a square = Side^{2}
In the figure alongside of the square PQRS, area = PQ^{2}.

Example
Find the area (in cm^{2}) of a square whose side is 5 cm. Solution.
Area of a square = Side^{2}
= 5^{2} = 25 cm^{2}.

Area of a square = ½ × Diagonal^{2}
In the figure alongside of the square PQRS, area = PQ^{2} = (PR / Ö2)^{2} = ½ × PR^{2}.

Example
Find the area (in cm^{2}) of a square whose diagonal is 6 cm. Solution.
Area of a square = ½ × Diagonal^{2}
= ½ × 6^{2} = 18 cm^{2}.