# 6.3 Polygons

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## Other polygons

There are many other polygons, some of which are given in the table below.

 Sides Name 5 pentagon 6 hexagon 7 heptagon 8 octagon 10 decagon 15 pentadecagon

## Angles of regular polygons

You can calculate the size of the interior angle of a regular polygon by using:

$\stackrel{^}{A}=\frac{n-2}{n}×{180}^{\circ }$

where $n$ is the number of sides and $\stackrel{^}{A}$ is any angle.

## Areas of polygons

1. Area of triangle: $\frac{1}{2}×$ base $×$ perpendicular height
2. Area of trapezium: $\frac{1}{2}×$ (sum of $\parallel$ (parallel) sides) $×$ perpendicular height
3. Area of parallelogram and rhombus: base $×$ perpendicular height
4. Area of rectangle: length $×$ breadth
5. Area of square: length of side $×$ length of side
6. Area of circle: $\pi$ x radius ${}^{2}$

## Polygons

1. For each case below, say whether the statement is true or false. For false statements, give a counter-example to prove it:
1. All squares are rectangles
2. All rectangles are squares
3. All pentagons are similar
4. All equilateral triangles are similar
5. All pentagons are congruent
6. All equilateral triangles are congruent
2. Find the areas of each of the given figures - remember area is measured in square units (cm ${}^{2}$ , m ${}^{2}$ , mm ${}^{2}$ ).

## Summary

• Make sure you know what: quadrilaterals, vertices, sides, angles, parallel lines, perpendicular lines,diagonals, bisectors and transversals mean.
• Similarities and differences between quadrilaterals
• Properties of triangles and quadrilaterals
• Congruency of triangles
• Classification of angles into acute, right, obtuse, straight, reflex or revolution
• Theorem of Pythagoras which is used to calculate the lengths of sides of a right-angled triangle
• Angles:
• Acute angle: An angle 0 and 90
• Right angle: An angle measuring 90
• Obtuse angle: An angle 90 and 180
• Straight angle: An angle measuring 180◦
• Reflex angle: An angle 180 and 360
• Revolution: An angle measuring 360
• Angle properties and names
• Equilateral, isoceles, right-angled, scalene triangles
• Triangles angles = 180
• Congruent and similar triangles
• Pythagoras
• Trapezium, parm, rectangle, square, rhombus, kite and properties
• Areas of particular figures

## Exercises

1. Find all the pairs of parallel lines in the following figures, giving reasons in each case.
2. Find angles $a$ , $b$ , $c$ and $d$ in each case, giving reasons.
3. Which of the following claims are true? Give a counter-example for those that are incorrect.
1. All equilateral triangles are similar.
2. All regular quadrilaterals are similar.
3. In any $▵ABC$ with $\angle ABC={90}^{\circ }$ we have $A{B}^{3}+B{C}^{3}=C{A}^{3}$ .
4. All right-angled isosceles triangles with perimeter 10 cm are congruent.
5. All rectangles with the same area are similar.
4. Say which of the following pairs of triangles are congruent with reasons.
5. For each pair of figures state whether they are similar or not. Give reasons.

## Challenge problem

1. Using the figure below, show that the sum of the three angles in a triangle is 180 ${}^{\circ }$ . Line $DE$ is parallel to $BC$ .

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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