Important Formulas - Geometric Shapes and Solids

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Important Formulas - Geometric Shapes and Solids

Geometric Shape

Description

Formulas

Rectangle

$l=$ Length
$b=$ Breadth
$d=$ Length of diagonal

Area $=lb$
Perimeter $=2(l+b)$
$d=\sqrt{l^2+b^2}$

Square

$a=$ Length of a side
$d=$ Length of diagonal

$\text{Area =}{a}^2=\frac{1}{2}{d}^2$
Perimeter $=4a$
d = $\sqrt{2}a$

Parallelogram

$b$ and $c$ are sides
$b=$ base
$h=$ height

Area $=bh$
Perimeter $=2(b+c)$

Rhombus

$a=$ length of each side
$b=$ base
$h=$ height
$d1,d2$ are the diagonals

Area $=bh$
(Formula 1 for area)
Area $=\frac{1}{2}{d}_1{d}_2$
(Formula 2 for area)
Perimeter $=4a$

Triangle

$a,b$ and $c$ are sides
$b=$ base
$h=$ height

Area = $\frac{1}{2}\text{bh}$
(Formula 1 for area)

Area $=\sqrt{s(s-a)(s-b)(s-c)}$
where $s$ is the semiperimeter
$=\frac{a+b+c}{2}$
(Formula 2 for area - Heron's formula)

Perimeter $=a+b+c$

Radius of incircle of a triangle
of area A $=\frac{A}{s}$
where $s$ is the semiperimeter
$=\frac{a+b+c}{2}$

Equilateral Triangle

$a=$ side

Area $=\frac{\sqrt{3}}{4}{a}^2$
Perimeter $=3a$
Radius of incircle of an equilateral triangle of side $a$ $=\frac{a}{2\sqrt{3}}$
Radius of circumcircle of an equilateral triangle of side $a$ $=\frac{a}{\sqrt{3}}$

Trapezium (Trapezoid)

Base $a$ is parallel to base $b$
$h=$ height

Area $=\frac{1}{2}(a+b)h$

Circle

$r=$ radius
$d=$ diameter

$d=2r$
$\text{Area =}\pi r^2=\frac{1}{4}\pi d^2$
Circumference $=2\pi r=\pi d$
$\frac{\text{Circumference}}{d}=\pi$

Sector of Circle

$r=$ radius
$\theta$ = central angle

Area, A =
$\{\begin{array}{c}\frac{\theta }{360}\pi r^2\\ \text{(if angle is in degrees)}\\ \\ \frac{1}{2}{r}^2\theta \\ \text{(if angle is in radians)}\end{array}$

Arc Length, $s=$
$\{\begin{array}{c}\frac{\text{θ}}{180}\pi r\\ \text{(if angle is in degrees)}\\ \\ r\theta \\ \text{(if angle is in radians)}\end{array}$

In the radian system for angular measurement,
$2\pi$ radians $=360°$
⇒ $1$ radian $=\frac{180°}{\pi }$
⇒ $1°=\frac{\pi }{180}$ radians

Hence,
Angle in Degrees
= Angle in Radians × $\frac{180°}{\pi }$

Angle in Radians
= Angle in Degrees × $\frac{\pi }{180°}$

Ellipse

Major axis length $=2a$
Minor axis length $=2b$

Area $=\pi ab$
Perimeter $\approx 2\pi \sqrt{\frac{a^2+b^2}{2}}$

Rectangular Solid

$l=$ length
$w=$ width
$h=$ height

Total Surface Area
$=2lw+2wh+2hl=2(lw+wh+hl)$
Volume $=lwh$

Cube

$s=$ edge

Total Surface Area $=6s^2$
Volume $=s^3$

Right Circular Cylinder

$h=$ height
$r=$ radius of base

Lateral Surface Area
$=\left(2\pi r\right)h$
Total Surface Area
$=\left(2\pi r\right)h+2\left(\pi r^2\right)$
Volume $=\left(\pi r^2\right)h$

Pyramid

$h$ = height
$\text{B}$ = area of the base

Total Surface Area $=\text{B}$ + Sum of the areas of the triangular sides
Volume $=\frac{1}{3}\text{B}h$

Right Circular Cone

$h=$ height
$r=$ radius of base

Lateral Surface Area
$=\pi r\sqrt{r^2+h^2}=\pi rs$

where s is the slant height
$=\sqrt{r^2+h^2}$
Total Surface Area
$=\pi r\sqrt{r^2+h^2}+\pi r^2=\pi rs+\pi r^2$

Sphere

$r=$ radius
d = diameter

$d=2r$
Surface Area $=4\pi r^2=\pi d^2$
Volume $=\frac{4}{3}\pi r^3=\frac{1}{6}\pi d^3$