Parallelograms, Triangles, and Circles

Areas of Polygons

CCS: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

Students will be able to:Find the areas of parallelograms, triangles, and circles

Find the circumference of circles

Find the area of complex figures

Objectives

AREAS OF POLYGONS

QUICK REVIEW:What is the formula to find the area of a rectangle?

A = l x w

AREA OF A PARALLELOGRAM

h

b

Let’s Discover the formula for a parallelogram!

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

Keep Sliding…..

AREA OF A PARALLELOGRAM

h

h

b

AREA OF A PARALLELOGRAM

h

h

b

Keep Sliding…..

AREA OF A PARALLELOGRAM

h

hb

Keep Sliding…..

…thus, changing it to a rectangle.What is the area of the rectangle?

AREA OF PARALLELOGRAM

h

b

lwA wlA lwA 2

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

bhA hbA bhA 2

Area of a Parallelogram

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=bhA=5(3)A=15m2

Area of a parallelogram

Video Time

AREA OF A TRIANGLE

Now we will discover the formula for area of a triangle.h

b

AREA OF A TRIANGLE

Let’s divide the triangle so that we divide the height in two.

b

?

?

AREA OF A TRIANGLE

b

?

?

Remember, we divided the height into two equal parts.

Now take the top and rotate…

AREA OF A TRIANGLE

rotate…?

?

AREA OF A TRIANGLE

?

?

b

rotate…

AREA OF A TRIANGLE

?

?

b

rotate…

AREA OF A TRIANGLE

?

?

b

rotate…

AREA OF A TRIANGLE

??

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

h2 h21h

AREA OF A TRIANGLE

??

b

b

?

?

Remember, you divided the height in two.

AREA OF A TRIANGLE

?

b

What is the area of this parallelogram?

bhA bhA 21 bhA 2

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

bhA bhA 21 bhA 2

Example

A= ½ bhA= ½ (30)(10)A= ½ (300)A= 150 km2

Finding the area of trianglesVideo Time

The circumference of a circle is

The distance around a circle

Hint: Circumference remember circle around

Now Let’s Talk Circles….

Diameter

Radius

centre

What is the formula

relating the circumferen

ce to the diameter?

People knew that the circumference is about 3 times the diameter but they wanted to find out exactly.

C = ? x d

C ≈ 3 x dThis means APPROXIMATELY EQUAL TO

Early AttemptsEgyptian Scribe Ahmes. in 1650 B.C. said

C≈3.16049 x d

Archimedes, said C ≈3.1419 x d

Fibonacci. In 1220 A.D. said C≈3.1418xdWhat is the value of the

number that multiplies the diameter to give the circumference????

The exact true value is……………

UNKNOWN!!

An approximation to ππ≈3.1415926535897932384626433832795

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CircumferenceRemember, circumference is the

distance around the circle.If you divide a circle’s circumference

by its diameter, you always get the same irrational number – pi (symbol: )

This is true of every circle.We estimate pi to be 3.14 or the

fraction 22/7.

Circumference Formulas

C = dC = 2r

Example

41 m

C = dC = (3.14)(41)C = 128.74 m

We substitute 3.14 in for pi.

The formula for the area of a circle is

We say:Area = pi times radius squared

Finding the Area of a Circle

Example

8 mm

A = r2

A = (3.14)(82)A =(3.14)(64)

A = 200.96 mm2

Example 2

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 = r2

A = (3.14)(6.52)A = (3.14)(42.25)

A = 132.665 cm2

The Area and Perimeter of a CircleA circle is defined by its diameter or radius

Diameter

radi

us 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)

diameter

ncecircumfere

π is an irrational number whose value to 15 decimal places is π = 3.14159265358979.... We usually say π≈3.14The circumference is found

using the formulaC=π d or C= 2πr (since d=2r)

The area is found using the formula

A=πr2

Finding circumferenceNaming the parts of a circleFinding the area of circles

Video Time

Complex FiguresUse the appropriate formula to find the area of each piece.

Add the areas together for the total area.

Example

| 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.

Solution

RectangleA = lwA = 24(10)A = 240 cm2

TriangleA = ½ bhA = ½ (3)(10)A = ½ (30)A = 15 cm2

Total FigureA = A1 + A2

A = 240 + 15 = 255 cm2

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