Discovering circumference and Area of a circle

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The circumference and Area of a circle : 

The circumference and Area of a circle

Slide 2: 

Diameter Radius centre circumference What is the formula relating the circumference to the diameter?

Slide 3: 

People knew that the circumference is about 3 times the diameter but they wanted to find out exactly. C = ? x d C ≈ 3 x d This means APPROXIMATELY EQUAL TO

Slide 4: 

Investigating the relationship between the circumference of a circle and its diameter? Click on the link below and read the instructions. Use the applet to create circles with different diameters. Roll the circles on the number line to measure the circumference. Your goal is to discover the mystery number in the formulae by dividing the circumference by the diameter for each circle. If you measure and divide corectly you should always get the same value for the ratio circumference/diameter

The mystery ratio : 

The mystery ratio What value did you find for the ratio? 3.1-3.2 is pretty good 3.14 is very good and close to the true value For most circumstances we say Circumference ≈ 3.14 x diameter What is the true value of this mystery ratio?????

Early Attempts : 

Early Attempts Egyptian Scribe Ahmes. in 1650 B.C. said C≈3.16049 x d Archimedes, said C ≈3.1419 x d Fibonacci. In 1220 A.D. said C≈3.1418xd What is the value of the number that multiplies the diameter to give the circumference????

The exact true value is…………… : 

The exact true value is…………… UNKNOWN!!

An approximation to π : 

An approximation to π π≈3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609................forever….

What about the AREA of a circle? : 

What about the AREA of a circle? 2r 2r r First consider a square The area of this square in terms of r is A= 2r x2r = 4r2

What about the AREA of a circle? : 

What about the AREA of a circle? 2r 2r Now consider a circle inside the square The area of the circle must be less than the are of the square A < 4r2 r Area = ? xr2

Slide 11: 

Finding a formulae for the area of a circle

Slide 12: 

C= πd or C=2πr Semi-circle=πr πr r

Slide 13: 

Area of Rectangle= Base x Height Area = πr x r Area =πr2

The Area and Perimeter of a Circle : 

The Area and Perimeter of a Circle A circle is defined by its diameter or radius Diameter radius The perimeter or circumference of a circle is the distance around the outside The area of a circle is the space inside it The ratio of π (pi) The circumference is found using the formula C=π d or C= 2πr (since d=2r) The area is found using the formula A=πr2

The Area and Perimeter of a Circle : 

The Area and Perimeter of a Circle A circle is defined by its diameter or radius Diameter radius The perimeter or circumference of a circle is the distance around the outside The area of a circle is the space inside it The ratio of π (pi) The circumference is found using the formula C=π d or C= 2πr (since d=2r) The area is found using the formula C=πr2