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Parallelograms, Triangles, and Circles

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6.G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems

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Students will be able to: Find the areas of parallelograms, triangles, and circles Find the circumference of circles Find the area of complex figures

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AREAS OF POLYGONS QUICK REVIEW: What is the formula to find the area of a rectangle? A = l x w

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AREA OF A PARALLELOGRAM h b Let’s Discover the formula for a parallelogram!

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AREA OF A PARALLELOGRAM To do this let’s cut the left triangle and… h b

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slide it… AREA OF A PARALLELOGRAM h h b

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Keep Sliding….. AREA OF A PARALLELOGRAM h h b

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h h b Keep Sliding…..

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AREA OF A PARALLELOGRAM h h b Keep Sliding…..

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…thus, changing it to a rectangle. What is the area of the rectangle? AREA OF PARALLELOGRAM h b

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AREA OF A PARALLELOGRAM Since the area of the rectangle and parallelogram are the same, just rearranged, what is the formula for the area of this parallelogram? h b

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Any side of a parallelogram can be considered a base. The height of a parallelogram is the perpendicular distance between opposite bases. The area formula is A=bh A=bh A=5(3) A=15m 2

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Area of a parallelogram

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AREA OF A TRIANGLE Now we will discover the formula for area of a triangle. h b

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AREA OF A TRIANGLE Let’s divide the triangle so that we divide the height in two. b ? ?

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AREA OF A TRIANGLE b ? ? Remember, we divided the height into two equal parts. Now take the top and rotate…

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AREA OF A TRIANGLE rotate… ? ?

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AREA OF A TRIANGLE ? ? b rotate…

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AREA OF A TRIANGLE ? ? b rotate…

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AREA OF A TRIANGLE ? ? b rotate…

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AREA OF A TRIANGLE ? ? b …until you have a parallelogram. How would you represent the height of this parallelogram?

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AREA OF A TRIANGLE ? ? b b ? ? Remember, you divided the height in two.

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AREA OF A TRIANGLE ? b What is the area of this parallelogram?

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AREA OF A TRIANGLE The area of this triangle would be the same as the parallelogram. Therefore, the formula for the area of a triangle is… what? h b

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A= ½ bh A= ½ (30)(10) A= ½ (300) A= 150 km 2

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Finding the area of triangles

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The circumference of a circle is The distance around a circle Hint: Circumference remember circle around

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Diameter Radius centre What is the formula relating the circumference to the diameter?

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People knew that the circumference is about 3 times the diameter but they wanted to find out exactly. C = ? x d C ≈ 3 x d This means APPROXIMATELY EQUAL TO

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Egyptian Scribe Ahmes. in 1650 B.C. said C≈3.16049 x d ArchimedesArchimedes, said C ≈3.1419 x d Fibonacci. In 1220 A.D. said C≈3.1418xd What is the value of the number that multiplies the diameter to give the circumference????

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UNKNOWN!!

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π≈3.141592653589793238462643383279502884197169 3993751058209749445923078164062862089986280 34825342117067982148086513282306647093844609 55058223172535940812848111745028410270193852110 5559644622948954930381964428810975665933446 128475648233786783165271201909145648566923460 34861045432664821339360726024914127372458700 66063155881748815209209628292540917153643678 925903600113305305488204665213841469519415116 09................forever….

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41 m C = (3.14)(41) C = 128.74 m We substitute 3.14 in for pi.

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8 mm A = (3.14)( 8 2 ) A =(3.14)(64) A = 200.96 mm 2

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13 cm If you are given a diameter, divide it in half to find the radius. 13 divided by 2 equals 6.5 cm. A = (3.14)(6.5 2 ) A = (3.14)(42.25) A = 132.665 cm 2

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A circle is defined by its diameter or radius Diameter radius The perimeter or circumference of a circle is the distance around the outside The area of a circle is the space inside it The ratio of π (pi) π is an irrational number whose value to 15 decimal places is π = 3.14159265358979.... We usually say π≈3.14 The circumference is found using the formula C=π d or C= 2πr (since d=2r) The area is found using the formula A=πr 2

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Finding circumference Naming the parts of a circle Finding the area of circles

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Use the appropriate formula to find the area of each piece. Add the areas together for the total area.

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|27 cm | 10 cm 24 cm Split the shape into a rectangle and triangle. The rectangle is 24cm long and 10 cm wide. The triangle has a base of 3 cm and a height of 10 cm.

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Rectangle A = lw A = 24(10) A = 240 cm 2 Triangle A = ½ bh A = ½ (3)(10) A = ½ (30) A = 15 cm 2 Total Figure A = A 1 + A 2 A = 240 + 15 = 255 cm 2

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Try This Area of Parallelograms Game- You have to be QUICK!!This Area of Parallelograms Game Try This Baseball Game that finds area of trianglesThis Baseball Game Homework: Reteaching/Practice 9.4 HO

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