Area 8.1 Rectangles & Parallelograms 8.2 Triangles, Trapezoids,& Kites

Rectangles & Parallelograms A= bh, where b is the length of the base and h is the height of the rectangle. Height, h = 3 units Base, b= 6 units 6 3 A = 6  3 = 18 sq. units

Rectangles & Parallelograms, A= bh, where b is the length of the base and h is the height of the parallelogram. Height, h = 3 units Base, b= 6 units 6 3 A = 6  3 = 18 sq. units

AREA OF A PARALLELOGRAM Discover the formula for area of a parallelogram. h b

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

slide it… AREA OF A PARALLELOGRAM h h b

slide it… AREA OF A PARALLELOGRAM h h b

slide it… AREA OF A PARALLELOGRAM h h b

slide it… AREA OF A PARALLELOGRAM h h b

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

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

A Closer Look at Parallelograms Height Length Side length Note that the height of the parallelogram is an altitude, perpendicular to the bases, and that it is used in determining the area and not the side lengths.

Practice Problems for cm 6 cm 4 cm 6 cm What is the area of the shaded region? A =( 12 x 6) – (4x 6) = 72 – 24 = A = 48 cm 2 5 cm 8 cm 9.4 cm What is the area of the shaded region? A = ½ ( 8 x 5) = 20 cm 2

Homework /1-13,17,20

Triangles, Trapezoids, & Kites h Base, b h b1b1 b2b2

AREA OF A TRIANGLE Discovering the formula for area of a triangle. h b

AREA OF A TRIANGLE divide the triangle so that the height is divided in two equal parts b ½ h

AREA OF A TRIANGLE Now take the top and rotate… b Remember, we divided the height into two equal parts. ½ h

AREA OF A TRIANGLE rotate… ½ h b

AREA OF A TRIANGLE b rotate… ½ h

AREA OF A TRIANGLE b rotate… ½ h

AREA OF A TRIANGLE b rotate… ½ h

AREA OF A TRIANGLE b rotate… ½ h

AREA OF A TRIANGLE b …until you have a parallelogram. How would you represent the height of this parallelogram? ½ h

AREA OF A TRIANGLE b b Remember, you divided the height in two. ½ h

AREA OF A TRIANGLE ½ h b What is the area of this parallelogram?

AREA OF A TRIANGLE The formula for the area of a triangle is… what? h b

AREA OF A TRAPEZOID Let’s derive the formula for the area for a trapezoid.

AREA OF A TRAPEZOID Remember, there are two different bases on a trapezoid. h

AREA OF A TRAPEZOID ½ h First divide the trapezoid horizontally so the height is divided in two equal parts. ½ h

AREA OF A TRAPEZOID Remember, we divided the height in two. Now, rotate… ½ h

AREA OF A TRAPEZOID rotate… ½ h

AREA OF A TRAPEZOID …until, you have a parallelogram. ½ h

AREA OF A TRAPEZOID How would you represent the height of this parallelogram? ½ h

AREA OF A TRAPEZOID Remember, we divided the height in two. ½ h

AREA OF A TRAPEZOID How would you represent the base of this parallelogram? ½ h

AREA OF A TRAPEZOID The new base is made by connecting the top and bottom bases. ½ h

AREA OF A TRAPEZOID How would you represent the area of this parallelogram? ½ h

AREA OF A TRAPEZOID The area of this trapezoid is the same as the parallelogram. What is the formula for area of a trapezoid? h

Triangles, Trapezoids,& Kites d2d2 d1d1 A 1 = ½h 1 d 1 ; A 2 = ½h 2 d 1 A = ½h 1 d 1 + ½h 2 d 1 h1h1 h2h2 A = ½ d 1 (h 1 + h 2 ) Since d 2 = (h 1 + h 2 ) A = ½ d 1 d 2

Triangles, Trapezoids, & Kites h Base, b A =½ bh h b1b1 b2b2 A = ½ h (b 1 + b 2 )

Practice Problems for 8.2 h = 4 cm b = 11 cm A= ½bh; A = ½(11)(4) A = ½(44) = 22 cm 2 h = 6 cm b 2 = 15 cm b 1 = 8 cm A = ½h(b 1 + b 2 ) A = ½(6)(8 + 15) A = (3)(23) = 69 cm 2

Homework /1-12, 20,25-28