Geometry & MensurationAptitude Questions and Answers

1. Important Formulae

1.1 Rectangle

Area:

$A = l \times b$

$A=\text{length} \times \text{breadth}$

Perimeter:

$P=2 \times (l+b)$

$P=2 \times (\text{length}+\text{breadth})$

1.2 Square

Area:

$A = l \times l=l^2$

$A = \text{length} \times \text{length}$

$A = \text{length}^2$

Perimeter:

$P=4 \times l$

$P=4 \times \text{length}$

1.3 Triangle

Consider the given figure where, $A$, $B$ and $C$ are the vertex and $\angle A, \angle B, \angle C$ are respective angles of triangles and $a , b , c$ are the respective opposite sides of the angles as shown in figure.

Height $=h$, Base $=b$

Area:

1. Formula 1:

$A = \dfrac{1}{2} \times b \times h$

$A = \dfrac{1}{2} \times \text{base} \times \text{height}$

1. Formula 2:

$A = \dfrac{1}{2} \times a \times b \angle C$

$A = \dfrac{1}{2} \times b \times c \angle A$

$A = \dfrac{1}{2} \times c \times a \angle B$

3. Formula 3 - Heron's formula:

$A = \sqrt{s(s-a)(s-b)(s-c)}$,

Where,

$s=\dfrac{a+b+c}{2}$

$s=\left(\dfrac{\text{perimeter}}{2}\right)$

$s$ is called 'semi perimeter'.

4. Area of Isosceles Triangle:

$s=\dfrac{b}{4}\sqrt{4a^2-b^2}$

Where, $a =$ length of two equal side , $b=$ length of base of isosceles triangle.

Perimeter:

$P=a+b+c$

$P=\text{Sum of sides}$

1.4 Trapezium

Consider the given figure, where $a$ and $b$ are the length of parallel sides and $h$ is the perpendicular distance between $a$ and $b$.

Area:

$A = \dfrac{1}{2} (a+b) \times h$

$A = \dfrac{1}{2} (\text{sum of parallel sides}) \times \text{perpendicular distance}$

Perimeter:

$P=\text{Sum of all sides}$

1.5 Rhombus

A Rhombus is a parallelogram with opposite equal acute angles, opposite equal obtuse angles, and four equal sides.

Consider the given figure, where $l$ is the length of sides, $d_1$ and $d_2$ are the length of diagonals.

Area:

$A = \dfrac{1}{2} (d_1 \times d_2)$

$A = \dfrac{1}{2} (\text{Product of diagonals})$

Perimeter:

$P=4 \times l$

Kite is an example of Rhombus

Area of a Kite, $A = \dfrac{1}{2} \times \text{product of it's diagonals}$

Consider the given figure, where $d$ is the length of diagonal and $o_1$, $o_2$ are the lengths of offset.

Area:

$A = \dfrac{1}{2} \times d \times (o_1 + o_2)$

$A = \dfrac{1}{2} \times \text{diagonal} \times (\text{sum of offsets})$

Perimeter:

$P=\text{Sum of all sides}$

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