U5L3 - Circles and area

Albus the Dog Mr. Munnerlyn has a little fluffy dog named Albus. Sometimes when he goes out, he ties Albus up to a post in the garden so that he can run around in the fresh air but not run away. When Albus is tied up, he is on a 4-meter rope so that he can run around the post in a circle. Sometimes he runs around so much that he wears an outline of a circle in the ground! a. Calculate the circumference of the circle created by Albus running around the post. b. Calculate the area of the circle that Albus makes.

Pizza Pi! A waiter in a pizza restaurant said that the medium pizza was 25 cm in diameter and the large pizza was 38 cm in diameter. He also said that the large pizza was more than twice as big as the medium pizza. Was he correct? Explain how you know.

Half circles The figure shows two half-circles. The inner half-circle has a radius of 5 cm and its edge is 2 cm away from the edge of the outer half-circle. Find the shaded area.

Circles within circles 1 Three circles are concentric with center C. On segment CF, CD = 3 cm, DE = 2 cm, and EF = 1 cm. The picture shows three enclosed, non-overlapping regions. What is the number of square centimeters in the area of the largest region?

Smiley Face In the figure shown, AB = BC = CD = DE = 14 cm. Find the perimeter of the figure, using 22/7 for pi.

Race track! The figure shows two of the lanes at a running track that are 3 m apart. Mary and Jim ran around the track three times with Mary taking the inner lane and Jim taking the outer lane. How much further did Jim go than Mary?

Circles within circles 2 Two circles, that are exactly the same size and touch at only 1 point, are surrounded by a larger circle. The diameter of the larger circle is 24 cm. How many square centimeters are in the area of the shaded region?

Centrifuge training Astronauts train for space flight in a centrifuge, which consists of a rotating arm with a cab at the outer end of the arm. The arm, which has a length of 57 feet, is revolved about the center of the centrifuge. An astronaut sits in the cab, which is then rotated 50 times per minute. How far does the astronaut travel in one minute? If 1 foot is about one-third of a meter, about how many meters is this? Does that sound like fun?

Goat A farmer ties his goat to the corner of a 10-foot by 15-foot rectangular shed in an otherwise empty field. The length of rope from the shed to the goat is 25 feet. Over how many square feet of the field can his goat roam?

ANSWERS

Albus the dog The circumference of the circle is 8π meters. The area of the circle is 16π meters2

Pizza Pi! Pizza Comparison. The waiter was correct. The 38 cm pizza has an area of π(19)2 square inches, and the 25 cm pizza has an area of π(12.5)2 square inches. The ratio of the areas is 361/156.25, which is greater than 2/1.

Half Circles The shaded area is 12 π cm2

Circles within circles 1 The area of the circle with radius CD is π•32 = 9π. The area of the circle with radius CE is π•52 = 25π. The area of the circle with radius CF is π•62 = 36π. The area of the middle ring is 25π - 9π = 16π. The area of the outer ring is 36π - 25π = 11π. So the middle ring has the greatest area at 16π square centimeters.

Smiley face! The perimeter is 160 cm

Racetrack! Jim runs 56.52 (18π) m further than Mary

Circles within circles 2 If the diameter of the larger circle is 24, then its radius is 12. The two smaller circles’ diameters = 12 and therefore each radius is 6. The area of the large circle is 144π, and the area of each of the two smaller circles is 36π. Thus, the area of the shaded area is 144π - 36π - 36π = 72π.

Centrifuge training 17,898 (5700π) feet 5966 (1900π) meters No, it does not sound like fun.

Goat! Notice that we really have portions of three different-sized circles (see next slide). There is a quarter-circle along the left of the rectangle with radius 10; there is a quarter-circle along the bottom of the rectangle with radius 15; finally there is a three-quarter circle along the right side and top of the rectangle of radius 25. The total area of these three sectors is 25π + 56.25π + 468.75π = 550π square feet.

Goat picture

UPSC