Archive for May, 2011

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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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Lesson 9: Areas of 2-D shapes (I)

May 7, 2011

Vocabulary-on-Plane Geometry 

Lesson 9: Areas (Notes)

Area of a Composite Figure and Word problems I

Area Worksheets II

I Have to Know by the End of this Lesson  

1. Pythagorean Theorem

A right-angled triangle is a triangle that has a right angle. We name hypotenuse to the side opposite the right angle (c on the picture).

 In a right-angled triangle the sum of the squares of the lengths of the sides equals the square of the length of the hypotenuse.

Notice that:

  • If we know the lengths of a side and hypotenuse, we can find the length of the other side
  • And if we know the lengths of the two sides we can find the length of the hypotenuse

When the side lengths of right triangle satisfy the Pythagorean Theorem, these three numbers are known as “Pythagorean triples or triplets”

How to identify and create Pythagorean triples?

The most common examples of Pythagorean triples are:

  •  3,4,5 triangles
  • A 3,4,5 triple simply stands for a triangle that has a side of length  3, a side of length 4 and a side of length 5.
  • If a triangle has these side length then it must be a right triangle
  • 5,12,13 right triangles
  • 7,24,25 right triangles
  •  8,15,17 right triangles

 We can conclude the following:

  • Only right triangles verify the Pythagorean Theorem
  • If a triangle has a triple as side lengths then it must be a right triangle

 Any multiple of the ratios above represent the sides of a right triangle.

 Here you are two interesting pages:

 On this page you can check that:

  • if you have that if you have an right-angled triangle:  

a2 + b2 = c2

  • if you have that if you have an acute-angled triangle:  

a2 + b2 < c2

  • if you have that if you have an obtuse-angled triangle:

a2 + b2 > c2

If you click on the picture  below you will find an animation the geometric demonstration of the Pythagorean Theorem (go down the page because the animation is at the end)

Finally this is a video on Pythagorean Teorem and its Converse (Is this triangle a right triangle?)

 2. Applications of the Pythagorean Theorem

 There are many applications of this famous theorem. One of them are the following:

  • Identifying right triangles
  • Calculating the diagonal of a rectangle
  • Calculating the height of an isosceles  triangle
  • Calculating the apothem of a regular polygon

If we remember the first I recommended, we can write the conditions that led us to classify the triangles according their angles:

  • a2 + b2 = c2 , then it is a right-angled triangle
  • a2 + b2 < c2 , then it is an acute-angled triangle
  • a2 + b2 > c2 , then it is an obtuse-angled triangle

On the following links you can practice exercises on Pythagorean Theorem.

Pythagorean theorem: find the length of the hypotenuse report 1 Pythagorean theorem: find the length of the hypotenuse

Pythagorean theorem: find the missing leg length report 2 Pythagorean theorem: find the missing leg length

Pythagorem theorem: find the perimeter report 3 Pythagorem theorem: find the perimeter

Pythagorean theorem: word problems report 4 Pythagorean theorem: word problems

Converse of the Pythagorean theorem: is it a right triangle? report 5 Converse of the Pythagorean theorem: is it a right triangle?

This is  a video where is solved a problem by using this theorem