Perimeter and Area of Plane Figures

Perimeter of a Triangle:

If a, b and c units be the lengths of the sides of a triangle, perimeter = a + b + c

or 2s = a + b + c,   where 2s denotes the perimeter of the triangle.

Heron’s Formula for Area of a Triangle:

If a, b and c are the sides of a triangle and 2s = a + b + c,  the area of the triangle is given by:

A = √s(s - a)(s - b)(s - c)

Area of an Equilateral Triangle:

Let ‘a’ be the side of an equilateral triangle. Then,

Hence,  A = ((√3)/4)side²

Sine Formula:

Area of triangle = (1/2) ab sin c = (1/2) bc sin A = (1/2) ac sin B

Cosine formula:

Perimeter and Area of a Rectangle:

P = 2(l + b)

A = l × b

Properties of a Rectangle:

1. Opposite sides are parallel, i.e., AB || DC and AD || BC.

2. Opposite sides are equal, i.e., AB = DC and AD = BC.

3. Each angle is right angle, i.e., ∠A = ∠B = ∠C = ∠D = 90º.

4. Diagonals are equal, i.e., AC = BD.

5. Diagonals bisects each other, i.e., AO = OC and BO = OD.

Perimeter and Area of a Square:

P = 4a units

A = a² square units

Properties of Square:

1. All the sides are equal.

2. Each angle is of 90°.

3. The diagonals are equal.

4. The diagonals are perpendicular to each other.

5. Angle between a diagonal and a side is 45°.

Parallelogram:

A quadrilateral, which has both pair of opposite parallel sides, is called a parallelogram.

In a parallelogram, opposite sides are equal.

The opposite angles of a parallelogram are equal.

The diagonals of a parallelogram bisect each other.

Perimeter P = 2(x + y) units

Area A = Base × Altitude

Rhombus:

If all sides of a parallelogram are equal, then it is a rhombus.

P = 4 × side

Area of rhombus = 1/2 [Product of the diagonals]

Properties of a Rhombus:

1. Opposite sides are parallel, i.e., AB || DC and AD || BC.

2. All the sides are equal, i.e., AB = BC = CD = DA.

3. Opposite angles are equal, i.e., ∠A = ∠C and ∠B = ∠D.

4. Diagonals bisect each other at right angle, i.e.,
   OA = OC = (1/2)AC; OB = OD = (1/2)BD, and
   ∠AOB = ∠BOC = ∠COD = ∠AOD = 90º.

5. Diagonals bisect and the angles at the vertices, i.e.,
   ∠OAB = ∠OAD; ∠OBA = ∠OBC;
   ∠OCB = ∠OCD; ∠ODC = ∠ODA

Trapezium:

A quadrilateral, which has one pair of opposite parallel sides, is called a trapezium.

Area of the trapezium is equal to half the altitude multiple by the sum of the parallel sides. Area = (1/2)h[a+b]

Area of a Quadrilateral:

In the given figure, ABCD is a quadrilateral whose diagonal BD divides it into two triangles, i.e., ΔABD and ΔBCD. AE and CF are perpendiculars from A and C to BD.

Area of the quadrilateral ABCD

= area of ΔABD + area of ΔBCD

= (1/2)BD.AE+(1/2)BD.CF

= (1/2)BD(AE+CF)sq.unit.