Curvaceous Circles BUT IT DOES WORK! Yep we can still relate the formula for the area of a circle to the formula for the area of a rectangle

Transcription

1 Curvaceous Circles So our running theme on our worksheets has been that all the formulas for calculating the area of various shapes comes back to relating that shape to a rectangle. But how can that possibly work for a circle? I mean it s a circle it doesn t have length and width! It doesn t even have straight lines! BUT IT DOES WORK! Yep we can still relate the formula for the area of a circle to the formula for the area of a rectangle But before we start talking about circle we need to get our heads around the terminology for circles Question 1) Research the definition of each of the following; a) Radius b) Diameter c) Circumference d) Arc e) Chord f) Segment g) Sector Question 2) Draw a diagram of a circle indicating each of the properties of circles defined in question 1.

2 Look at the picture below. Each square is 1cm 2 in area. Question 3) What is the radius of the circle? Question 4) What is the area of the large square (made up of the yellow and blue) in the top right hand side of the picture? Question 5) How many of these squares does it take to make the full square?

3 If we are trying to find the area of the circle we can see that this is less than the area of the large square. So if we are looking at the blue and yellow square we can see that if we need 4 of them to make the large square then we need less than 4 to make up the area of the circle. Question 7) How many of the blue and yellow squares do you think you need to make the full circle? Question 6) Estimate the area of the circle by counting the square. Remember each square is 1cm 2. Take your time and try to match up the part squares as best as you can because we want this area as accurate as possible. Now we want to know how many yellow and blue squares it takes to make up the circle Question 7) Divide the area of the circle by the area of the blue and yellow square? This will tell us how many we need to make up the circle. Question 8) Compare your answer to question 7 with your estimate in question 6. How close were you? So let s look at where we are at. You have found the area of the blue and yellow square and then multiplied it by your answer to question 7 and that will give you the area of the circle. But we need to neaten this up a bit. Firstly to work out the area of the blue and yellow square you should have multiplied the length by the width which was 10 by 10 to get 100cm 2. But the width (or length) that we used to find the area of the small square has a special name

4 Question 9) Look back at your answers to question 1. What name do we give to the length (or width) we used to find the area of the blue and yellow square? So to find the area of the yellow square we multiply the radius by itself, so we have squared the radius. We usually represent radius with the letter r. So the area of the yellow and blue square is radius x radius or r x r or simply r 2 But the area of the circle is found by multiplying r 2 by your answer to question 7. Ideally your answer to question 7 was somewhere between 3 and 3.4. In actual fact the number we need is called π (pi). Pi is a special number that we need to multiply the radius squared by in order to find the area of a circle. So here it is the formula for the area of a circle is; A=πr 2 Pi is what we call an irrational number because we can t write it as a fraction, which means pi has an infinite number of decimal places. Question 10) Look up the value for pi and give it correct to 10 decimal places. It s impractical to actually use pi to an infinite number of decimal places but our calculators usually have a pi button (π) that we can use, otherwise we usually just use 3.14 as pi because it s close enough for our purposes. Question 11) Why is pi only a little over 3? (Think about how we got to pi)

5 Now this is where pi gets interesting. You would probably think that pi only relates to the area of a circle by it also relates to the circumference of a circle. Look at the images above. In both cases we have used the radius to estimate the circumference of the circle. In the image on the left we have used 6 radii (the plural for radius is radii, read as raid-e-i) to make a hexagon within the circle whose perimeter is smaller than the circumference of the circle. In the image on the right we have used 7 radii to form a heptagon around the circle, which means it s perimeter is bigger than the circle. Note that we have actually used a little less than 7 as the radius marked with a blue r is actually too long. Question 12) How does radius and diameter relate to each other? (look back to question 1 if needed) So the diagram on the left uses 6 radii, and since the diameter is twice the radius, than we have used 3 diameters to make the hexagon, remember this hexagon is smaller than the circle. The diagram on the right uses just under 7 radii which would be a little under three and a half diameters so let s say 3.4 diameters, which is bigger than the circumference. Question 13) What is the value of π to 2 decimal places?

6 So π is between 3 and 3.4, (the number of diameters used above) Coincidence? I think not! We actually need π lots of the diameter to find the circumference of a circle. To find the circumference of a circle we multiply the π by the diameter. Circumference = π x Diameter C=π x D C=πD Let s recap for a second. - The area of a circle is A=πr 2 - The circumference if a circle is C=πD - π (pi) is approximately The value of pi comes from needing less than 4 times the area of a quarter of a square that surrounds a circle in order to get the area of the enclosed circle. Question 14) Write a summary of circles in your work book which should include the 4 dot points above, and the definition of radius, diameter and circumference. Question 15) Find the area AND circumference on the following circles

7 Question 16) Draw 5 circles. Find the radius, diameter, area and circumference of each Question 17) a) If the circumference of a circle is 12m what is the diameter (correct to 2 decimal places) b) If the circumference of a circle is 29cm what is the radius? (correct to 2 decimal places) c) If the circumference of a circle is 23km what is its area? (correct to 2 decimal places) Question 18) a) If the area of a circle is 12.7m 2 what is the radius? (correct to 2 decimal places) b) If the area of a circle is 25.8cm 2 what is the diameter? (correct to 2 decimal places) c) If the area of a circle is 78.34mm 2 what is its circumference? (correct to 2 decimal places) Question 19*) If the circumference is n meters what is the; a) Diameter of the circle b) The radius of the circle c) The area of the circle.