Area of a Regular Polygon:

Area of Regular Polygons and Circles continued Do NOW 04-15-2014Take out your notes from yesterday and work on the two area of a polygon problems using the A = (1/2)Pa formula.Homework: Irregular Polygons WorksheetTEST FRIDAY!!!!Area of a Regular Polygon:

Lets begin with an example using a regular pentagon:What is the Perimeter of MNOPQ when QP = 12 inches? P = # of sides side length = 5 (12 in.) = 60 inchesWhat is the Area?= 247.80 in2SOHCAHTOAHow do we find the apothem(a)?360/ 5 = 72o72/ 2 = 36oTan (36o) = (6/a) a = (6/(tan(36o))) = 8.26 cm How can we use all of this information to find the area of a circle? Can we get to the area of a circle from the equation for Perimeter?

Area of a CircleCan we use the area of a regular polygon? Yes!A = (1/2) PaPC = 2rPlug in!A = (1/2)(2r)aWhats our a? Its our radius!!A = (1/2)(2r)(r) = r2If a circle has an area of A square units and a radius of r units, then: A = r2Lets Try One!Let the circle shown below have a radius of 9 centimeters, what is the perimeter and the area of the circle?

P = 2r = 2(9) = 18 cm 56.55 cm A = r2 = (9)2 = (99) cm2 = 81 cm2 254.47 cm2 Your Turn!!What if we want to know the area of the shaded region around an inscribed polygon?Let r = 12.5 inchesWhat do we need to calculate??

Area of a Circle:= r2= (12.5)2= 156. 25 in2 490.87 in2Area of the Square:= s2 s2 + s2 = (2r)2, where r = 12.5 2s2 = (25)2 = 625 s2 = 312.5 in2Square root both sides! s = 17.68 inches

Area of a Circle Area of a Square:(490.87 in2) (312.5 in2)178.37 in2Areas of Irregular Figures Objective: TLW find areas of irregular figures on and off the coordinate plane. SOL G.14b

So weve already discussed regular polygons what if we have an irregular figure? What does irregular mean?An Irregular Figure is a figure that: Is not comprised of equal side and angle measurementsCan it be classified like the other polygons weve studied?NODo irregular figures exist in our everyday lives? What are some examples?

But how do we solve for the area?

The area of an Irregular figure is the:Sum of all its distinct partsWhat do I mean by distinct parts??Non-overlapping figures that can be combined to create the irregular figureThese parts can be made up of rectangles, squares, triangles, circles, and other polygons!Lets think about our original figure, how could we break it up?How would breaking our image up help us to find the area?Lets evaluate that same figure!Do we have all of the measurements we need? What is missing? The base of our bottom How could we find this?The Pythagorean Theorem!!AE = 6 cmNow what can we do?

Lets evaluate that same figure!Area of CFB = (1/2) bh = (1/2) (8 cm) (4 cm) = 16 cm2

Area of BFDE = (sides)2 = (8 cm)2 = 64 cm2

Area of BEA = (1/2) bh = (1/2) (6 cm) (8 cm) = 24 cm2Now add it all together to get your Area!

A = 104 cm2What if we have a figure that is missing a portion of its area?

Consider the image from the previous slide Can this figure be separated into other figures?YesWhat are they?

Consider the image from the previous slide Im going to set up the problem for the area of a rectangle + a semi-circle, minus the triangle. Also, assume the triangle is an equilateral.

Area of the rectangle = lw = 19 in 6 in = 114 in2Area of the semi-circle = (1/2)r2 = (1/2) (3 in)2 = 4.5 in2Area of the = (1/2) bh = (1/2) (6) (33) = 93 in2Area of the figure= (114 in2 93 in2) + 4.5 in2= 112.5 in2YOUR TURN!!!!What if we were on the coordinate plane? What do we need to know about a figure to evaluate it?Consider the image to your left!How can we break this image into two portions that we can evaluate with our prior knowledge?

Area of Trapezoid:= (1/2) h(b1 + b2)= (1/2)(7) (8+6)= 49 u2Area of Triangle:= (1/2) bh= (1/2)(6)(3)= 9 u2Area of Irregular Figure:= 49 u2 + 9 u2= 58 u2