Year 8 math Name: 2h2 circles 37 Circumference and Area of Circles 1 a) Measure the diameter d of various cylindrical items (jam jars, cups, etc) and establish the circumference U of these items (for example by using a thin string.) Work out the quotient U : d in each case b) What do you notice about the result from (a) for the function U: Diameter => Circumference? For various cylindrical items, the circumference U and the diameter d of the circular cross-section are measured. Then the quotient U : d is calculated in each case. As you can see from the table on the right, this quotient is almost the same in each case. 2 A circle of radius 5cm is broken down into smaller constituent parts, which are then fitted together again differently, as illustrated on the left. U (cm) d (cm) U:d Film box 10.0 3.2 3.1 Candle 17.9 5.6 3.2 Veg. tin 26.6 8.4 3.2 Beer mat 33.4 10.7 3.1 Saucepan 52.4 16.4 3.2 The letter π is the first letter of the Greek word perifereia = "(round area") In actual fact, the quotient is identical for each circle. This value is U : d = 3.141592654…… This constant has an infinite number of decimal places and in contrast to other constants we have considered with endless decimal places, this constant does not even have a periodic repetition. It is referred to as "pi" (in German "Kreiszahl") and written using the Greek letter pi or "π", ie. π = 3.141592654… We often use the approximate value: π ≈ 3.14 or 22/7. Mathematicians have been trying for a long time to work out the value of π more exactly. In 2002, a Japanese mathematician named Yasumasa Kanada worked it out to 1241 000 000 000 decimal places! Using the relationship U : d = π you can also work out the circumference of a circle using if you know its diameter d or radius r. A circle with diameter d or radius r has the circumference (C) in German (U) (U=) C = π · d and (U=) C = 2π · r (π = 3.14) The circumference formula U = 2π · r will now be used in order to calculate the area of a circle. To do this, we can think of a cake sliced into equal smaller portions which are then joined up together in a different way, as shown below.
Transcript
Year 8 math Name: 2h2 circles
37
Circumference and Area of Circles
1 a) Measure the diameter d of various cylindrical items (jam jars, cups, etc) and establish the circumference U of these items (for example by using a thin string.) Work out the quotient U : d in each case b) What do you notice about the result from (a) for the function U: Diameter => Circumference?
For various cylindrical items, the circumference U and the diameter d of the circular cross-section are measured. Then the quotient U : d is calculated in each case. As you can see from the table on the right, this quotient is almost the same in each case.
2 A circle of radius 5cm is broken down into smaller constituent parts, which are then fitted together again differently, as illustrated on the left.
U (cm) d (cm) U:d Film box 10.0 3.2 3.1 Candle 17.9 5.6 3.2 Veg. tin 26.6 8.4 3.2 Beer mat 33.4 10.7 3.1 Saucepan 52.4 16.4 3.2
The letter π is the first letter of the Greek word perifereia = "(round area")
In actual fact, the quotient is identical for each circle. This value is U : d = 3.141592654…… This constant has an infinite number of decimal places and in contrast to other constants we have considered with endless decimal places, this constant does not even have a periodic repetition. It is referred to as "pi" (in German "Kreiszahl") and written using the Greek letter pi or "π", ie. π = 3.141592654… We often use the approximate value: π ≈ 3.14 or 22/7.
Mathematicians have been trying for a long time to work out the value of π more exactly. In 2002, a Japanese mathematician named Yasumasa Kanada worked it out to 1241 000 000 000 decimal places!
Using the relationship U : d = π you can also work out the circumference of a circle using if you know its diameter d or radius r.
A circle with diameter d or radius r has the circumference (C) in German (U)
(U=) C = π · d and (U=) C = 2π · r (π = 3.14)
The circumference formula U = 2π · r will now be used in order to calculate the area of a circle. To do this, we can think of a cake sliced into equal smaller portions which are then joined up together in a different way, as shown below.
Year 8 math Name: 2h2 circles
38
The more you increase the number of slices, the closer you get with the new shape to a rectangle with length ½ U and width r.
The circle and the rectangle have the same area: Acircle = Arectangle = ½ U · r = ½ · 2πr · r = π · r2
A circle with radius r has an area A = π · r2
Notes For any circle of radius r, the corresponding circumference and area can be represented by the functions U: r => 2π · r and A: r => π · r2. The function U: r => 2π · r is an example of a proportional function, the proportionality constant being 2π. The function A: r => π · r2 is neither proportional nor inversely proportional.
Example 1 A tree trunk at eye level has a circumference of 1.31m. The bark is 2cm thick. What is the diameter of the trunk at this height without the bark? Solution: Diameter d of the trunk with bark : d = U : r = 131cm : 3.14 = 42cm. Diameter d1 of trunk without bark: d1 = d – 4cm = 42cm – 4cm = 38cm.
Example 2 Calculate the area of the red area of the circle on the left and explain your method. Solution: To do this, we have to calculate the area of the blue semicircle (radius r1 = 2.5cm) and take this away from the area of the entire circle (radius r2 = 4cm). A = Acircle – Asemicircle = π · (rcircle)2 – ½ (π · (rsemicircle)2 ) = π · [(rcircle)2 – ½ · (rsemicircle)2] * π is taken out as a common factor. = π · [(4cm)2 – ½ · (2.5cm)2 ) ]
= 3.14 · (16cm2 – ½ · 6.25cm2) = 40.4cm2
Exercises
3 Calculate the circumference and area of a circle with radius r (or diameter d) for the following: a) r = 12cm b) d = 2.4cm c) d = 3.4cm d) r = ¾ m e) d = 2.1km f) r = 0.25cm g) What objects could have these diameters or radii? 4 Explain: The function A: r => π · r2 is not a proportional function.
5 a) Calculate the length of the line in the diagram below:
b) Draw your own similar lines using a circular shape and calculate their lengths.
Year 8 math Name: 2h2 circles
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6 Calculate the circumference and area of the coloured sections in the following diagrams:
5 The lake Pulvemaar in the Eifel area of Germany is an almost circular lake with a volcanic origin. Is it possible to walk around the lake in 1 hour?
7 Plot the graph of the functions U: r => 2π · r and A: r => π · r2 8 In order to bring the water bucket up from the old well, we have to turn the handle 13 times. In so doing, the rope is wrapped around the cylindrical axle (diameter = 20cm). Estimate the depth of the well. 9 Right or wrong: a) A pizza with twice the diameter is enough for twice as many people. b) A pizza with twice the circumference weighs twice as much. c) A pizza with double the diameter costs four times as much. 10 a) A coin with currency euro has a diameter of 6.6cm. Which coin is it? b) Expressed as a percentage, how much smaller is the surface area of a 1 cent coin than that of a 2 euro coin? (Measure their diameters) c) Expressed as a percentage, how much bigger is the surface area of a 2 euro coin than that of a 1 cent coin? d) Calculate the total surface area, including the edges, of a 1 euro coin (and a 2 euros).
12 The television satellite ASTRA stays apparently motionless in the sky, because it completes a circumnavigation of the Earth in exactly 24 hours travelling in the same direction as the rotation of the Earth. Its distance from the Earth's surface is 35 900 km. Earth's radius is 6370km. Calculate the speed of the satellite on its orbit.
13 A string is wrapped tightly around a ball of radius 11cm (for example, a football) such that a circle is created which contains the centre point of the ball. Now the string is lengthened by 1m. It should then remain equidistant from the surface of the ball all the way around. Could a mouse now walk through the gap between the ball and the string? How does the mouse react if the ball was the Earth (radius = 6370km)? Assume the Earth is a circle. G14 Calculate:
