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Lesson 9: Areas (II)

May 9, 2011

4.Perimeter of a circumference and length of a an arc

The length of a circumference of radius and diameter d = 2r is L =2πr  or L=πd.

The length of an arc of aº  is Larc = 2πr · aº/360º.

You don’t need to learn this formula because it is easy to find out by applying a rule of three direct:

2πr ———- 360º

Larc ———– aº               Larc = 2πr · aº/360º

The following table revises the formulae we studied in this lesson.  

5.Area of Plane Shapes

Triangle AreaA = ½b · h
b = base
h = vertical height
     
Rectangle Area
Area = w · h
w = width
h = height
     
Trapezoid (US) Area
Trapezium (UK) Area
A = ½(a+b) × h
h = vertical height
     
Regular polygon Area
A = (p · a)/2     p = perimeter   a = apothem
     

 

Square AreaA = a2
a = length of side
     
  Parallelogram Area
A = b · h
b = base
h = vertical height 
     
  Circle AreaA = πr2
Circumference=2πr
r = radius
     
  Sector circle AreaA =  (πr2 · θ) /360º
r = radius
θº = angle in degrees
     

6.Area of a Circular Ring or Annulus

 A circular ring (annulus or crown) is plane figure bounded by the circumference of two concentric circles of two different radii. The area of a circular ring is found by subtraction the area of small circle from that of the large circle.

Ar = π(R2 – r2)

 7.Composite Figures

Composite figures (or shapes) are  figures (or shapes) that can be divided into more than one of the basic figures. Their area can be calculated by adding (or subtracting) the areas of the figures they are divided into.

One example would be the annulus.

Here you are some links to practice with areas and revise perimeters. The level is elemental.

Perimeter report 1. Perimeter

Area report 2.Area

Area and perimeter: word problems report 3. Area and perimeter: word problems

Parts of a circle report 4.Parts of a circle

Circles: calculate area, circumference, radius, and diameter report 5. Circles: calculate area, circumference, radius, and diameter

Circles: word problems report 6. Circles: word problems

8. Sum of the interior angles of a polygon

The sum of the interior angles of a n-sided polygon is (n-2) · 180º.

In this case, we have an hexagon so the sum of the interior angles is (6-2) · 180º.

In a regular n-sided  polygon all the interior angles are equal so each interior angle measures (n-2) · 180º/n.

Here you are a video on calculating the sum of the interior angles of 17-sided polygon. 

9. Central angle of a regular n-sided polygon

The central angle of a regular n-sided polygon measures 360º/n

In this case we have a pentagon so its central angle is 360º/5 =72º

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