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## Circumference

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**Circumference**Lesson 8-1**Vocabulary Start-Up**A circle is the set of all points in a plane that are the same distance from a point, called the center. The circumference is the distance around a circle. The diameter is the distance across a circle through its center. The radius is the distance from the center to any point on the circle. Fill in the box with one of the vocabulary terms.**Real-World Link**The table shows the approximate measurements of two sizes of hula hoops. a. Describe the relationship between the diameter and radius of each hula hoop. The diameter is twice the radius b. Describe the relationship between the circumference and diameter of each hula hoop. The circumference is about three times the diameter.**Radius and Diameter**Words: The diameter d of a circle is twice its radius r. The radius r of a circle is half of its diameter d. Symbols: d = 2r r =**Example 1 & 2**a. The diameter of a circle is 14 inches. Find the radius. r = = = 7 The radius is 7 inches. b. The radius of a circle is 8 feet. Find the diameter. d = 2r = 2(8) = 16 The diameter is 16 inches.**Got it? 1 & 2**Find the radius or diameter for each circle with the given dimensions. a. d = 23 cm b. r = 3 inches c. d = 16 yards d. r = 5.2 meters 6 inches 11.5 cm 10.4 meters 8 yards**Circumference**Words: The circumference of a circle is equal to times its diameter or times twice its radius. Symbols: C = d or C = 2r Model: The value of pi () is 3.1415926… or .**Example 3**Find the circumference of a circle with a radius of 21 inches. Since 21 is a multiple of 7, use for . C = 2r C = 2()(21) C = 22(3) C = 132 The circumference is about 132 inches.**Got it? 3**Find the circumference of each circle. Use for . a. b. 5 feet 220 inches**Example 4**Big Ben is a famous clock tower in London, England. The diameter of the clock face is 23 feet. Find the circumference of the clock face. Round to the nearest tenth. C = d C 3.14(23) C 72.2 So the distance around the clock is about 72.2 feet.**Got it? 4**A circular fence is being placed to surround a tree. The diameter of the fence is 4 feet. How much fencing is used? Use 3.14 for . Round to the nearest tenth if necessary. about 12.6 feet**Area of circles**Lesson 8-2**Real-World Link**1. Adrianne wants to find the distance the dog runs when it runs one circle with the leash fully extended. Should she calculate the circumference or area? Explain. Circumference; the circumference is the distance around the circle 2. Suppose she wants to find the amount of running room the dog has with the leash fully extended. Should she calculate the circumference or area? Explain. Area; the area is the interior region of an enclosed figure**Find the Area of a Circle**Words: The area A of a circle equals the product of and the square of its radius r. Symbols: A = r2 Model: Cross out the formula that is not used in finding area of a circle. A = r2 A = r2 A = r2 A =**Example 1**Find the area of the circle. Use 3.14 for pi. A = r2 A • 22 A 3.14 •4 A 12.56 The area of the circle is approximately 12.56 square inches.**Example 2**Find the area of the circle with a radius of 14 centimeters. Use for pi. A = r2 A • 142 A •196 A 616 The area of the circle is approximately 616 square centimeters.**Got it? 1 & 2**Find the area of a circle with a radius of 3.2 centimeters. Round to the nearest tenth. 32.2 cm2**Example 3**Find the area of the face of the Virginia quarter with a diameter of 24 millimeters. Use 3.14 for pi. Round to the nearest tenth if necessary. The radius is (24) of 12 millimeters. A = r2 A •122 A 3.14 •144 A 452.16 The area is approximately 452.2 square millimeters.**Got it? 3**The bottom of a circular swimming pool with a diameter of 30 feet is painted blue. How many square feet are blue? Round to the nearest tenth. 706.5 ft2**Example 4 – Area of Semicircles**Find the area of the semicircle. Use 3.14 for pi. Round to the nearest tenth. A = r2 A •82 A 3.14 •64 A 100.5**Got it? 4**Find the approximate area of a semicircle with a radius of 6 centimeters. Round to the nearest tenth. 56.5 cm2**Example 5**On a basketball count, there is a semicircle above the free-throw line that has a radius of 6 feet. Find the area of the semicircle. Use 3.14 for pi. Round to the nearest tenth. A = r2 A •62 A 3.14 •36 A 56.5 So, the area of the semicircle is approximately 56.5 square feet.**Area of Composite figures**Lesson 8-3**Find the Area of a Composite Figure**A composite figure is made up of two or more shapes. To find the area of a composite figure, decompose the figure into shapes with areas you know. Then find the sum of these areas.**Example 1**Find the area of the composite figure. The figure can be separated into a semicircle and a triangle. The area of the figure is about 14.1 + 33 or 47.1 square meters. Area of a semicircle A = r2 A 32 A 14.1 Area of a triangle A = A = A = 33**Got it? 1**Find the area of the figure. Round to the nearest tenth if necessary. 482.5 in2**Example 2**A miniature gold hole is composed of a trapezoid and a parallelogram. How many square feet of turf does the hole cover? So, 7.5 + 15 or 22.5 square feet of turf will be needed.**Got it? 2**Pedro’s father is building a shed. How many square feet of wood are needed to build the back of the shed shown? 210 ft2**Example 3 – Find Area of Shaded Region**Find the area of the rectangle and subtract the area of the four triangles. The area of the shaded region is 60 – 2 or 58 square inches. Area of a triangle A = 4 A = 4 A = 2 Area of a rectangle A = A = A = 60**Example 4 – Find Area of Shaded Region**The blueprint for a hotel swimming area is represented by the figure shown. The shaded area represents the pool. Find the area of the pool. The area of the shaded region is 1,050 – 440 or 610 square meters. Area of the entire rectangle A = A = A = 1,050 Area not shaded A = A = A = 440**Got it? 3 & 4**A diagram for a park is shown. The shaded area represented the picnic sections. Find the area of the picnic sections. 2,250 yd2**Volume of Prisms**Lesson 8-4**Volume of a Rectangular Prism**Words: The volume V of a rectangle prism is the product of the length , the width w, and the height h. It is also the area of the base Btimes the height h. Symbols: V = wh or V = Bh Model:**Volume of a Rectangular Prism**The volume of a three-dimensional figure is the measure of space it occupies. It is measured in cubic units such as centimeters (cm3) or cubic inches (in3). It takes 2 layers of 36 cubes to fill the box. So, the volume of the box is 72 cubic centimeters.**Example 1**Find the volume of the rectangular prism. V = wh V = 5 • 4 • 3 V = 60 The volume is 60 cubic centimeters or 60 cm3.**Got it? 1**Find the volume of the rectangular prism shown below. 142.5 m3**Volume of a Triangular Prism**Words: The volume V of a triangular prism is the product of the base Btimes the height h. Symbols: V = Bh, where B is the area of the base Model:**Volume of a Triangular Prism**The diagram below shows that the volume of a triangular prism is also the product of the area of the base B and the height h of the prism.**Example 2**Find the volume of the triangular prism shown. The area of the triangle is • 6 • 8 so, replace B with • 6 • 8. V = Bh V = ( • 6 • 8)(9) V = 216**Got it? 2**Find the volume of the triangular prism shown below. 70 in3**Example 3**Which lunch box holds more food? Find the volume of each lunch box. Then compare. Since 285 in3> 281.25 in3, Lunch Box B holds more food.**Volume of Pyramids**Lesson 8-5**Real – World Link**Dion is helping his mother build a sand sculpture at the beach in the shape of a pyramid. The square pyramid has a base with length and width of 12 inches an d the height of 14 inches. 1. Label the dimensions of the sand sculpture on the square pyramid below. 2. What is the area of the base? 144 in2 3. What is the volume of a square prims with the same dimensions? 2,016 in3**Volume of a Pyramid**Words: The volume V of a pyramid is one third the area of the base Btimes the height hof the pyramid. Symbols: V = Bh, where B is the area of the base Model: VOCABULARY: In a polyhedron, any face that is not the base is called a lateral face. The lateral faces meet at a common point or vertex.**Example 1**Find the volume of the pyramid. Round to the nearest tenth. V = Bh V = (3.2 x 1.4)2.8 V 4.2 The volume is about 4.2 cubic inches.**Example 2**Find the volume of the pyramid. Round to the nearest tenth. (What is the shape of the base?) V = Bh V = (∙ 8.1 ∙ 6.4)11 V = 95.04 The volume is about 95.04 cubic meters.**Got it? 1 & 2**Find the volume of a pyramid that has a height of 9 centimeters and a rectangular base with a length of 7 centimeters and a width of 3 centimeters. 63 cm3**Example 3**The rectangular pyramid shown has a volume of 90 cubic inches. Find the height of the pyramid. V = Bh 90 = (5 ∙ 9)h 90 = 15h 6 = h The height is 6 inches.**Example 4**A triangular pyramid has a volume of 44 cubic meters. It has an 8-meter back and a 3-meter height. Find the height of the pyramid. V = Bh 44 = (∙ 8 ∙ 4)h 44 = 4h 11 = h The height is 11 meters.**Got it? 3 & 4**a. A triangular pyramid has a volume of 840 cubic inches. It has a base of 20 inches and a height of 21 inches. Find the height of the pyramid. 12 inches b. A rectangular pyramid has a volume of 525 cubic feet. It has a base of 25 feet by 18 feet. Find the height of the pyramid. 3.5 feet