Lesson 22. Circumference and Area of a Circle. Circumference. Chapter 2: Perimeter, Area & Volume. Radius and Diameter. Name of Lecturer: Mr. J.

Transcription

1 Lesson 22 Chapter 2: Perimeter, Area & Volume Circumference and Area of a Circle Circumference The distance around the edge of a circle (or any curvy shape). It is a kind of perimeter. Radius and Diameter The Radius is the distance from the center to the edge. The Diameter starts at one side of the circle, goes through the center and ends on the other side. So the Diameter is twice the Radius: Diameter = 2 Radius 2 Perimeter, Area & Volume Page 1

2 Circumference The Circumference is the distance around the edge of the circle. It is exactly Pi (the symbol is π) times the Diameter, so: Circumference = π Diameter And so these are also true: Circumference = 2 π Radius Circumference/Diameter = π Area The area of a circle is π times the Radius squared, which is written: A = π r 2 Or, in terms of the Diameter: A = (π/4) D 2 It is easy to remember if you think of the area of the square that the circle would fit inside. 2 Perimeter, Area & Volume Page 2

3 Names Because people have studied circles for thousands of years special names have come about. Nobody wants to say "that line that starts at one side of the circle, goes through the center and ends on the other side" when a word like "Diameter" would do. So here are the most common special names: Lines A line that goes from one point to another on the circle's circumference is called a Chord. If that line passes through the center it is called a Diameter. If a line "just touches" the circle as it passes it is called a Tangent. And a part of the circumference is called an Arc. Slices There are two main "slices" of a circle The "pizza" slice is called a Sector. And the slice made by a chord is called a Segment. 2 Perimeter, Area & Volume Page 3

4 Common Sectors The Quadrant and Semicircle are two special types of Sector: Quarter of a circle is called a Quadrant. Half a circle is called a Semicircle. Inside and Outside A circle has an inside and an outside (of course!). But it also has an "on", because you could be right on the circle. Example: "A" is outside the circle, "B" is inside the circle and "C" is on the circle. Example 1 Find the circumference of a circle of diameter 12.6 mm. Take Answer Method 1: Using C = 2 r with = and r = ½ of 12.6 = 6.3 Gives C = = 39.6 to 3 s.f. Method 2: Using C = d with = and d = mm cm Gives C = = 39.6 to 3 s.f. So Circumference = 39.6 mm to 3 s.f. 2 Perimeter, Area & Volume Page 4

5 Example 2 Find the perimeter of the given semicircle. (The prefix semi means half.) Answer 8 m Here we have to find the length of the curved part. As you can see this is half of a circle. So Circumference of a circle = 2 r = = 25.12m 1 Circumference of a Semicircle = = 12.56m The Perimeter = curved part + straight edge = ( ) m = m = 20.6 m to 3 s.f. Exercise 1 1) In the following questions, write down the length of the diameter of the circle whose radius is given. 6 cm a) b) c) 25 mm d) 2 km e) 5.4 cm 5 m 2) Using as an approximate value for, or the button on your calculator, and giving your answer correct to 3 s.f., find the circumference of a circle of radius: a) 53 m b) 1.4 cm c) 33.6 cm d) 2 mm 2 Perimeter, Area & Volume Page 5

6 3) Using as an approximate value for, or the button on your calculator, and giving your answer correct to 3 s.f., find the circumference of a circle of: a) diameter 37 cm b) radius 15 m c) diameter 598 mm d) diameter 3m 4) Find the perimeter of each of the following shapes: 25 mm a) 67 cm b) 25 mm (This is called a quadrant: it is one quarter of a circle.) 8 cm c) 10 cm 5) A circular flower bed has a diameter of 1.5 m. A metal edging is to be placed round it. Find the length of edging needed and the cost of the edging if it is sold by the metre (i.e. you can only buy a whole number of metres) and costs 60p a metre. 6) A rectangular sheet of metal measuring 50 cm by 30 cm has a semicircle of radius 15 cm cut from each short side as shown. Find the perimeter of the shape that is left. 50 cm 30 cm 2 Perimeter, Area & Volume Page 6

7 7) How far does a bicycle wheel of radius 28 cm travel in one complete revolution? How many times will the wheel turn when the bicycle travels a distance of 352 m? 8) A bucket is lowered into a well by unwinding rope from a cylindrical drum. The drum has a radius of 20 cm and with the bucket out of the well there are 10 complete turns of the rope on the drum. When the rope is fully unwound the bucket is at the bottom of the well. How deep is the well? 9) A window is in the shape of a rectangle with a semicircular top. The window frame is made of strips of metal along the straight and curved lines shown in the diagram. Calculate the total length of metal needed to make the frame. 2.5 m 10) Calculate the number of complete revolutions made by a cycle wheel of diameter 70cm in travelling a distance of ½ km. Example m This is a sector of a circle. Find its area. Answer 45 o 3 m First we have to find the fraction of this sector from a complete circle. i.e Therefore area of sector = of area of circle of radius 3 m. 8 = 1 1 r = 3.53 m 2 to 3 s.f. 2 Perimeter, Area & Volume Page 7

8 Exercise 2 1) Find the areas of the following circles: 4 cm 5 m a) b) 2) Calculate the area of a circle with: a) radius 6m b) radius 29.5cm c) radius 4.7m d) diameter 24cm e) diameter 3.8cm 3) Find the areas of the following shapes: 20 cm a) b) 5 cm 20 cm c) d) The sides of the square are 5cm long. The isosceles triangle has a base of 7cm and a height of 4cm. 2 Perimeter, Area & Volume Page 8

9 4) The inner circle has a radius of 3cm and the other circle has a diameter of 13 cm Example 4 A circular table has a radius of 75 cm. Find the area of the table top. The top of the table is to be varnished. One tin of varnish covers 4 m 2. Will one tin be enough to give the table top three coats of varnish? Answer Area of table top is r 2 = cm 2 = cm 2 to 4 s.f. or Area of table top is r 2 = m 2 = m 2 to 4 s.f. For three coats, enough varnish is needed to cover m 2 = 5.30 m 2 to 3 s.f. So, one tin of varnish is not enough. 5) A circular lawn has a radius of 5 m. A bottle of lawn weed-killer says that the contents are sufficient to cover 50 m 2. Is one bottle enough to treat the whole lawn? 6) Circular place mats of diameter 12 cm are made by stamping as many circles as possible from a rectangular strip of card measuring 24cm by 48cm. How many mats can be made from the strip of card and what area of card is wasted? Note, it would be better if you draw a diagram first. 7) Calculate the area of a semi-circular rug with diameter 90cm. 8) An ice cream wafer is in the shape of a quadrant of a circle with radius 7cm. Work out the area of the wafer. 2 Perimeter, Area & Volume Page 9

10 9) A lawn is in the shape of a rectangle with a semicircular end. Calculate the area of the lawn. 8 m 17 m 10) A circular pond has diameter 3.22m. The pond is surrounded by a path 28cm wide. Calculate: a) The area of the pond b) The area of the path. Give your answers in m 2, correct to 2 decimal places m 28 cm 2 Perimeter, Area & Volume Page 10