BILINGUAL SECTION,IES P.J.MONTOYA 51

UNIT 12

area and perimeter

THE PERIMETER OF A SHAPEThe perimeter of any shape is the total distance around its edges.

For example: Find the perimeter of this shape 100m

70 m

The total distance around the shape is 100 + 70 + 70+ 100 = 340 mArea is the amount of space that a flat shape takes up.You measure it in squares. A square with 1 cm is 1 cm2. You say this as “one square centimetre”

PYTHAGORAS’ THEOREMYears ago, a man named Pythagoras found an amazing fact about triangles:If the triangle had a right angle (90°) ...... and you made a square on each of the three sides, then ...... the biggest square had the exact same area as the other two squares put together!

The longest side of the triangle is called the “hypotenuse”, so the formal definition is:

In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So the square of a (a2) plus the square of b (b2) is equal to the square of c (c2)

a2+b2=c2

ac

b

BILINGUAL SECTION,IES P.J.MONTOYA52

Let’s see if it really works using an example. A “3,4,5” triangle has a right angle in it, so the formula should work.

Let’s check if the areas are the same

32+42=52

Calculating this becomes

9+16=25

Why Is This Useful?If we know the lengths of two sides of a right angled triangle, then Pythagoras’ Theo-rem allows us to find the length of the third side. (But remember it only works on right angled triangles!)

AREA OF SHAPES

RectangleWe can calculate the area of a rectangle by multiplying the base by the height

height Area = base x height

base Area = bxh

TriangleWe can calculate the area of a triangle imagining it is half a rectangle.

Perpendicular height

Area = ( base x perpendicular height)

Area=

SquareWe can calculate the area of a square by multiplying side by side ( because its four sides are the same)

Side Area = side x side = (side )2

Area = s2

21

2bxh

BILINGUAL SECTION,IES P.J.MONTOYA 53

ParallelogramWe can calculate the area of a parallelogram by multiplying the base by the height (the same as a rectangle)

Area = base x perpendicular height Area = b x h

Perpendicular height

base RhombusWe can calculate the area of a rhombus by multiplying the big diagonal by the small diagonal and then dividing by 2

Area=(big diagonal x small diagonal)÷2 Big diagonal A=

Small diagonal

TrapeziumWe can find the area of a trapezium by breaking it into a rectangle and triangles and then calcu-lating the areas of these shapes. small base Area = ( big base + small base )x perpendicular height ÷ 2

Area = big base Perpendicular height

Area of a regular polygon

We can calculate the area of some regular polygons by multiplying the perimeter by the apo-them then dividing by 2.

Area =( perimeter x apothem)÷2

Area =

2Dxd

2)( hbB +

2Pxa

Apothem= The distance from the center of a regular polygon to the midpoint of a side

BILINGUAL SECTION,IES P.J.MONTOYA54

REMEMBER

The Pythagoras’ theorem is only used in right angled triangles.The longest side of the triangle is called the • “hypotenuse”.In a right angled triangle the square of the hypotenuse is equal to the sum of the •squares of the other two sides.

Perimeter is the distance around the edges of a shape.Area is the amount of space in a shape.You can find the area of a composite shape by adding the areas of the parts

Area a • rectangle = base x height.Area a • triangle = ( base x perpendicular height )Area a • square = side x side = (side )2

Area a • parallelogram=base x perpendicular height .Area • rhombus= (big diagonal x small diagonal)÷2Area a • trapezium =( big base + small base )x perpendicular height ÷ 2 ) Area of a • regular polygon =( perimeter x apothem)÷2 Area is measured in cm• 2 or m2

USEFUL WEBSITESwww.mathsisfun.com/pythagoraswww.mathsisfun.com/definition/apothem

RECOMMENDED READINGS

What’s your angle, Pythagoras? A math adventure by Julie Ellies; illustrated

by Phyllis Hornung. Published by Charlesbridge, 1961.

FIND OUT!Did you know that the 2 pioneers of Greek mathematics are Thales and Pythagoras?.

The Pythagoreans´ motto is “Everything is number”.

“Los griegos, que nos legaron la trigonometría, nos proporcionaron tam-bién los nombres con que designamos los lados del triángulo rectángulo: la hipotenusa (HYPOTÉINUSA, participio del verbo HYPOTÉINO, ‘sujetar fuertemente’, ‘tensar’, a su vez formado por la preposición HYPÓ y el verbo TÉINO, ‘tender’, ‘estirar’) y los catetos (KÁTHETOS, ‘perpendicular’, del verbo KATHÍEMAI ‘dejar caer’, derivado de ÍEMAI, ‘echar’, ‘lanzar’). El cambio en la acentuación es debido a que ambas palabras nos han llegado a través del latín.”

http://www.geocities.com/Athens/Delphi/3925/H.htm

AC

TIV

ITIE

S

BILINGUAL SECTION,IES P.J.MONTOYA 55

Lisa Simpson and her fantastically funny family hit the big screen in The Simpsons Movie on July 27th.Althougt only in second grade, Lisa is much smarter at maths than we are! She use the Pythagorean theorem to see an exclusive message from Lisa to maths students.

All measurements are in cm. In a right triangle where……..

A a = 8, b= 6 , h = …… 100 dOnuT 10 I

B a = 12, b= 5, h = …… 13 EvERY 14 nAP

C a= 26, b= 10 , h = …… 28 BROkE 24 STARTS

d a= 48, b= 50 , h = …… 14 SAY 28 BuRnS

E a= 16, b= 34 , h = …… 90 PLOW 30 TIMES

F a= 60, b= 45 , h = …… 75 PI 70 TELL

G a= 510, b= 64 , h = …… 514 dROOLInG 257 SAxOPHOnE

With a little help from The Simpsons Movie creators, Lisa gave us a special quote about the picture above.She said “ I keep trying to explain this to my dad, but he…………

WHAT TO dOUsing the Pythagorean Theorem, find the length of the missing side of each of the right triangles described below. Circle the word next to the correct answer and write it in the blank above the letter of a problem

C

A D

F

G B

E

AC

TIV

ITIE

S

BILINGUAL SECTION,IES P.J.MONTOYA56

FIND OUT! The Alhambra is a walled city and fortress in

Granada, Spain, built during the last Islamic sultan-

ate on the Iberian peninsula, the nasrid dynasty

(1238-1492). The palace is decorated with stone

and wood carvings and tile patterns on most of the

ceilings, walls, and floors. Islamic art does not use

representations of living beings, but heavily uses

geometric patterns, especially symmetric (repeating) pat-

terns.

You can find examples of these tiles at:

http://www.spsu.edu/math/tile/grammar/moor.htm

or http://ficus.pntic.mec.es/~jmos0028/Archivos/Mitad.PDF

IMPROVING YOUR COMPETENCIES