Perimeter and Area of Polygons Mostly Area

1. What is the perimeter of a square with side lengths of 15 in.? 2. What is the perimeter of a rectangle with length 11 cm and width 11 cm? P = 4s P=4(15) P=60 in. P = 2(l + w) P = 2( ) P = 2(22) P = 44 cm

Perimeter is the distance around a shape. We worked with rectangle perimeter earlier in the year using formulas. Let’s work with perimeter of graphed polygons before moving on to area.

Graph the following points: (7,2) (7,7) (-3,2) (-3,7).... Find the length and the width. What is the shape’s perimeter?

If you are designing a square park for a local contest, and the design on the coordinate grid has one point located at (3, 5), where could the other three points be located? What is the perimeter of your design on the grid?. (3,0) (8,0) (8,5) (3,10) (8,10) (8,5) Regardless of the color choice of the points, your perimeter would still be 20 units. If the scale of the drawing is 1 unit: 7 yds, what are the real dimensions of the playground?

Finding Unknown Side Lengths and the Perimeter of a Polygon Find each unknown measure. What is the perimeter of the polygon? First find the unknown side length. Find the sides opposite side b. The length of side b = Side b is 72 cm long. P = P = 254 Find the perimeter. The perimeter of the polygon is 254 cm. 72 cm

w 4w-5 Algebra in Geometry P = 2l + 2w P = 2(l + w) P = SOS P = 2(4w-5)+ 2w P = 2(4w-5 + w) P = w+w+4w – 5 + 4w - 5 Find the width of this rectangle if the perimeter is 70 units. P = 8w – w P = 2(5w-5) P = 10w - 10 P = 10w = 10w = 10w___ ____ = width Now let’s check….. 70 = 2(8) + 2(4 ∙8 – 5) 70 = (32-5) 70 = (27) 70 = = 70

The area of a figure is the amount of surface it covers. We measure area in square units.

FORMULAS will be used on ALL problems!

Area of a Rectangle Always start with a formula! Write the formula. Substitute 15 for l. Substitute 9 for w. A = lw A = 15in 9in A = 135 sq. in. The area is 135 in in. 9 in.

Rectangle ABCD is graphed with these coordinates: (-2, 4), (4, 4), (4, 9), ( ? ) Determine what the 4 th ordered pair must be and find the area of the rectangle. (-2, 9) A = lw A = 6(5) A = 30 sq. units.

You can use the formula for the area of a rectangle to write a formula for the area of a parallelogram. Imagine cutting off the triangle drawn in the parallelogram and sliding it to the right to form a rectangle. The area of a parallelogram = bh. The area of a rectangle = lw. The base of the parallelogram is the length of the rectangle. The height of the parallelogram is the width of the rectangle. A = bh

Find the area of the parallelogram. Write the formula. Multiply. A = bh A = __ 1 2 Substitute 1 for b and 3 for h. 1 2 __ 1 2 A = 3 2 __ 7 2 A = or ___ 1 4 __ The area is 5 ft __ 1 1 ft ft 2 A = bh A = 3.5(1.5) A = 5.25 sq. ft. Oh, No!!! Fractions !!

Graph the following points: (1,-7) (1,3) (-4,8) (-4,-2).... Now let’s find the dimensions of the shape. base=10 units height=5 units A = bh A = (10)5 A = 50 sq. units

What about triangles? Look! A triangle is just half of a parallelogram! So we just divide the parallelogram formula in half!

1 2 __ Area of Triangles: Since you can make a parallelogram out of two congruent triangles, the formula for a triangle’s area is half of that of a parallelogram. It can be written two ways…….. A = bh or A = bh 2

Finding the Area of a Triangle Write the formula. Substitute 12 for b. Substitute 8 for h. Multiply. The area is 48 cm 2.A = 48 A = bh 1 2 __ A = (12)(8)cm 1 2 __ A = (96) 1 2 __ 12 cm 8 cm The height is the vertical distance between the base and the top of the triangle. A = (12) A = 48 sq. cm.

What is the height of a triangle with an area of 1200 mm and a base of 80mm?. Write the formula. Substitute 80 for b. Substitute 1200 for Area. 30 mm 80 mm ____ A = bh 2 __ 1200 = 2 (80)h 1200 = 40h cm = h 2400/2 = 1200 ??

. Graph the following points: (10,-5) (-7,6) (2,-5). Find the area... We have to locate the height. Imagine the other half…… The base is 8 units. The height is 11.Can you see the mathematical value????

A B Triangle A and Triangle B both have the same area. Use the following information to find the missing dimensions: Triangle ATriangle B base = 12 cmbase = 9 cm Height = 6 cmheight = ??????

A B Triangle A and Triangle B both have the same area. Use the following information to find the missing dimensions: Triangle ATriangle B base = 12 cmbase = 9 cm Height = 6 cmheight = ??????

Working with Trapezoids

Area of trapezoids This formula is slightly different because the base lines are not the same length The b 1 and b 2 just mean that there are two bases.... that’s all. b1b1 b2b2 14 cm 18 cm 12 cm A = 6 ( ) A = 6(32) A = 192 cm 2

Graph the following points: (-4,6) (-4,-2) (3,6) (8,-2).... We have to find both bases and the height

Graph the following points: (-4,6) (-4,-2) (3,6) (8,-2).... We have to find both bases and the height

Lesson Quiz Find the area of each figure What is the area of a parallelogram with base 16 in. and height 10 in.? 4. What is the area of a triangle with base 10 in. and height 7 in.? 4.5 m 9.2 m A =41.4 m 2 A =24 ft 2 A =160 in 2 A =35 in 2 A = ½ bh A = ½ (4)(12) A = ½ 48 A = lw or A = bh A = (4.5)(9.2) A = bh A = 16in(10in) A = ½ bh A = ½ (10)(7) A = 5(7)

Find the area of this composite figure. 12 cm 9 cm 4 cm This lower shape is a square (rectangle). 4 cm A = lw + lw + ½ bh A = 4(12) cm 2 + (9)(9) cm 2 + ½ (4)(4)cm 2 A = 48cm cm 2 + 8cm 2 A = 137 cm 2 Identify the shapes. Write the formulas. A = lw This upper shape is a rectangle. A = lw This upper shape is a triangle. A = ½ bh Review and Extend

What is the height of a parallelogram that has an area of 84 sq. cm and a base of 6 cm? What is the base of a triangle with an area is 20 sq. feet and a height of 8 ft?

12 m 18 m 8 m 3 m Find the area of the shaded portion of the rectangle. A = lw – (lw) A = 12(18) – (3)8 A = 216 – (24) A = 192 m 2

What have we accomplished? We worked briefly with perimeter and worked with area formulas for polygons. Do you remember each formula? Rectangle? A = lw Parallelogram? A = bh Triangle? A = ½ bh Trapezoid? A = ½ h(b 1 + b 2 )