IES “LA Laboral”. Bilingual SectionAuthor: Mónica García Pinillos

Lesson 10: Areas.

Objetives: To know and use surface measurements adequately.

To work out the perimeter of a circle, circumference.

To work out the length of an arc.

To know Pythagoras Theorem and how to prove it geometrically. To apply Pythagoras Theorem to:

− The identification of acute, obtuse and right triangles− The calculation of the diagonal of a rectangle− The calculation of the height of an isosceles triangle− The calculation of the apothem of a regular polygon

To know how to deduce the formula of the area of the more common polygons and of the circle.

To learn the formula of the area of parallelograms: rectangle, square, rhombus and rhomboid

To learn the formula of the area of triangles and trapeziums.

To learn the formula of the area of a regular polygon.

To learn the formula of the area of a circle, of a circle sector and of an annulus. To apply adequately the formulae to find out the area of the more common plane figures.

To know how to calculate the area or a composite figure. To know how to find out the sum of the interior angles of a polygon. To know how to calculate the interior angle and the central angle of a regular polygon. To know the definition of some angles in a circle: interior angle, inscribed angle and interior angle. To know the measurement of the angles mentioned before. To get into the habit of writing down the measurement unit in the outcome of an exercise. To solve word problems by using Pythagoras’ Theorem.

INDEX:

1. Perimeter

1.1. Perimeter of regular polygons

1.2. Perimeter of a circle: circumference

1.3. Length of an arc of a circle

2. Pythagoras’s Theorem

3. Applications of Pythagoras’s Theorem

3.1. Identifying right triangles

3.2. Calculating the diagonal of a rectangle

3.3. Calculating the height of an isosceles triangle

3.4. Calculating the apothem of a regular polygon

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IES “LA Laboral”. Bilingual SectionAuthor: Mónica García Pinillos

4. Areas of polygons

4.1. Area of parallelograms: rectangle, square, rhombus and rhomboid

4.2. Area of triangles

4.3. Area of a trapezium

4.4. Area of a regular polygon

4.5. Area of a composite 2-D shape

5. Area of circular figures

5.1. Area of a circle

5.2. Area of circular sector

5.3. Area of an annulus

6. Area of a composite plane figure

7. Angles in polygons

7.1. Sum of the angles of a polygon

7.2. Central angle of a polygon

8. Angles in a circle

1. Perimeter

The perimeter of a polygon is the distance around the outside of the polygon.

A polygon is 2-dimensional; however, perimeter is 1-dimensional and is measured in linear units

The perimeter of a polygon is the sum of its length sides.

Examples

1. Find the perimeter of a triangle with sides measuring 5 centimetres, 9 centimetres and 11 centimetres.

P = 5 cm + 9 cm + 11 cm = 25 cm

2. A rectangle has a length of 8 centimeters and a width of 3 centimeters. Find the perimeter.

P = 8 cm + 8cm + 3 cm + 3 cm = 22 cmP = 2(8 cm) + 2(3 cm) = 16 cm + 6 cm = 22 cm

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IES “LA Laboral”. Bilingual SectionAuthor: Mónica García Pinillos

We name 8 cm + 3 cm semi-perimeter, S,.

Given a rectangle with length and width we name semi perimeter to , so:

1.1. Perimeter of a regular polygon

Regular polygons have equal sized sides. Their perimeter is easy to calculate:

The perimeter of a -sided regular polygon of length side , is.

Examples

We name to length of the sides of the following regular polygons. Find their perimeter:

Equilateral triangle Square Regular hexagon

All three sides have equal lengths

P= 3·

Parallelogram with 4 equal sides and

4 right angles: P = 4·

All six sides have equal length

P = 6·

1.2. Perimeter of a circle: Circumference

From Latin: circum "around" + ferre "to carry" Circumference is the distance around the edge of a circle. This is it is the perimeter of the circle.

If we divide the perimeter of a circle by its diameter we always obtain the same decimal number. This is a very important number, named pi and designed by . This number is non exact and non periodic. It has infinite decimal figures and they don’t repeat any “period”.

Its value has been calculated with a precision of hundreds of thousands of decimal figures but we can not know the exact value. An approximated value is:

We will use:

The perimeter of a circle or circumference, , can be calculated through two formulae:

where is the diameter and the radius

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IES “LA Laboral”. Bilingual SectionAuthor: Mónica García Pinillos

Examples

3. Find the circumference of a circle with a radius 12 inches.

r = 12 C = 2 r C = 2 (12) C = 24 inches.

4. Find the circumference of a bike wheel whose diameter is 16 inches.

C = d C = (16) C = 16 inches.We can also apply the other formula

If the diameter is 16, the radius is 8 inches. (The radius is ½ of the diameter).

Now use the formula C = 2 r. C = 2 (8)

C = 16 inches.

1.3. Length of an arc of circle

If we have a circle of radius the length of the , which measures degrees is:

How did we obtain this formula?

If you observe the length of an arc of a circle and the angle intercepted by arc are in direct proportion. Remember that the arc which subtends 360º has the length of the circumference. Now think of the ”regla de tres directa simple”:

So:

Example

5. Find the length of a 24º arc of a circle with a 5 cm radius.

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IES “LA Laboral”. Bilingual SectionAuthor: Mónica García Pinillos

If we substitute in the formula:

5

2. Pythagoras’ theorem

A right-angled triangle or right triangle is a triangle having a right angle. One of the angles of the triangle measures 90 degrees. The side opposite the right angle is called the hypotenuse. The two sides that form the right angle are called cathetus (plural catheti), “cateto” in Spanish, but in English they use to say simply sides or legs.

Examples

red lines in red are the hypotenuses

A right triangle has the special property that is known as the Pythagorean theorem or Pythagoras’ theorem.

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.

To prove this theorem, we must first remember how to find the area of a square which is the product of the length of its side times it.

Let's call the length of our square's side "c" which makes its area c squared.

The same goes for a square of length "a" and a square of length "b".

Proving Pythagoras' theorem.

The following diagram proves Pythagoras' theorem is true.

In the diagram there are two identical squares whose sides measure b+c. In each square we have positioned four identical right-angled triangles in different places whose hypotenuse is a and other two sides are b and c.

In the square on the right there are two spaces that are squares whose sides are b and c. Therefore, their areas are b2 and c2 respectively.

As the original squares are exactly the same, the spaces in each square also have the same area. In the square on the left,  a2 and in the square on the right, b2+c2.

Therefore  a2 = b2+c2

In the square on the left, once the four triangles have been positioned we get a space which is a square in the middle whose sides are a, equal to the hypotenuse of the triangle. Therefore, the area of this square is a2.

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:

Example

6. Find the length of the missing side:

           

The hypotenuse (side opposite the right angle) is the missing side. If this side is , then by Pythagoras’ theorem

7. Given the hypotenuse, 26 cm length & one side 22 cm length, find the other side

Pythagorean triples

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 triangleso 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.

o 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

Example

8. Check if 7, 24, 25 is a Pythagorean triple. Is 14, 48, 50 a Pythagorean triple? What do you observe?

We have to check if they satisfy Pythagoras’ Theorem . must be the greatest number:

Similarly:

NOTE: The triples above such as 3,4,5 represent the ratios of side lengths that satisfy the Pythagorean theorem.

Therefore, you can create other triples by multiplying any of these triples by a number. (3,4,5) ×2 = (8,6,10) and 8,6,10 is also a Pythagorean triple (5,12,13) ×2 = (10,24,26) and 10,24,26 is also a Pythagorean triple

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

3. Applications of Pythagoras’ theorem

3.1. Identifying right triangles

Given any triangle ΔABC, if is the biggest side:

If The triangle is right-angled.

If The triangle is acute-angled

If The triangle is obtuse-angled

Recall only right-angled triangles verify Pythagoras’ Theorem.

Example

8. Determine if a triangle of side lengths 7, 24, 25 is a right triangle.

We have to check if they satisfy Pythagoras’ Theorem . must be the greatest number:

3.2. Calculating the diagonal of a rectangle

The triangle ΔABC is right-angled. Its hypotenuse is the diagonal of the rectangle, , and its sides are the sides of the rectangle.

Applying Pythagoras’ theorem we have:

Notice the rectangle is divided into two equal right triangles.

Example

9. Find the diagonal of the rectangle whose length = 12 cm d width = 8 cm.

cm

3.3. Calculating the height of an isosceles triangle

In isosceles triangle, the height from the vertex to the unequal side divides the unequal side into equal segments.

The triangles ΔAMC and ΔMBC are right–angled and equal. Then

In equilateral triangles this happens with any height.In scalene triangle heights divide it into two right triangles (except one height in obtuse-angled), but they are not equal.

Example

10. Find the height in this isosceles triangle

This is done by Pythagoras’s Theorem

11. Find the height in this equilateral triangle

Let A, B and C be the vertices of the equilateral triangle and M the midpoint of segment BC. Since the triangle is equilateral, AMC is a right triangle. Let us find h the height of the triangle using Pythagorean theorem.

h 2 + 5 2 = 10 2

Solve the above equation for h.

3.3. Calculating the apothem of a regular polygon

An apothem of a regular polygon is a line segment from the midpoint of one of the sides of the polygon to the centre of that polygon.

This segment is always perpendicular to the side. The term "apothem" also refers to the length of this line segment.

The radius of a regular polygon is the line segment from the centre of the polygon to any vertex.

To calculate the length of the apothem, you must know the length of the side of a regular polygon, and the length of the radius of the polygon.

In the figure below – a regular pentagon – you can see that the right triangle in green has as sides

the apothem , , and hypotenuse the radius of the polygon, .

If we apply the Pythagoras’ theorem we have:

So:

We can generalise this process to a n-sided regular polygon.

The apothem of a regular polygon of length side and radius is:

A particular case is the regular hexagon. In a regular hexagon, the radius and the side have the same length.

In this case the formula is, for ,

Examples

11. Find the apothem of a regular pentagon with side 6 cm and radius 5 cm.

So: cm

12. Find the apothem of a regular hexagon of side 4 cm.

cm

4. Area of polygons

4.1. Area of parallelograms

Area of a rectangle

A rectangle is a four-sided polygon with four right angles, whose opposite sides are parallel and equal. We name the length basis , and the width height .

The area of a rectangle is the product of its width and length.this is

Area of a square

A square is a parallelogram with 4 equal sides and 4 right angles.A square is a particular case of a rectangle and a rhombus simultaneously. So, it shows both the properties of rhombus and rectangle simultaneously.

The area of a square of length side is :

Examples

13. Find the area of a square having length side five.

The area measures 25 cm2

14. Find the area of a square of diagonal 6.

We have to apply the Pythagoras’ Theorem:

Area of a rhombus

A rhombus is a four-sided polygon having all four sides of equal length and whose opposite sides are parallel.

Each rhombus has two diagonals. The biggest one is called long diagonal and the smaller one is called short diagonal. (Note that the diagonals of a rhombus are perpendicular).

The area of a rhombus is half the area of the rectangle of base and altitude or height .

The area of any rhombus is equal to one-half the product of the lengths D and d of its diagonals.

Examples

15. Find the area of the rhombus where the long diagonal is 9 units and the short diagonal is 8 units long.

The area measures 36 square units

16. Find the area of a rhombus with short diagonal 6 cm long and side 5 cm long.

The diagonals and the side are related.

The shaded triangle is right angled, so we can apply Pythagoras’ theorem

So:

cm.

Then the area will be: cm2.

Area of a rhomboid

How could we find the area of this parallelogram?

Make it into a rectangle as we do on the right side:

The rectangle is made of the same parts as the parallelogram, so their areas are the same. The area of the rectangle is , so the area of the parallelogram is also .

Warning: Notice that the height of the rhomboid is the perpendicular distance between two parallel sides of the rhomboid, not a side of the parallelogram (unless the parallelogram is also a rectangle, of course). See the picture below

.

The area of a rhomboid that has base units and height units, , is square units

Examples

17.Find the area of this rhomboid:

Area of a parallelogram =

(The 12 is a side. The height and base are always perpendicular to each other.)

4.2. Area of a Triangle

We define as altitude or height of a triangle as the perpendicular distance from one side (its base) to the opposite vertex.

IMPORTANT FACTS:

In equilateral triangles all the altitudes divide the opposite side into two halves. In isosceles triangles the height over the base divides the opposite side into two halves.

How could we find the area of this triangle?

Make it into a parallelogram. This can be done by making a copy of the original triangle and putting the copy together with the original.

The area of the parallelogram is , so the area of the triangle is

Warning: Notice that the height (also often called the altitude) of the triangle is the perpendicular distance between a vertex and the opposite side of the triangle.

The area of a triangle that has base units and corresponding altitude is.

Examples

17. Find the area of an obtuse triangle with bas 8 cm and corresponding height 5 cm.

18. Find the area of an equilateral triangle with side a cm. Apply it to the case a=8 cm

We draw a diagram of an equilateral triangle.

Draw a height from the vertex to the midpoint of the base. If a is the length of a side of the equilateral triangle and h the height then the triangle below, which is half of the rectangle, is a right triangle

and hence, by Pythagoras' theorem:

Then the area will be:

19. Find the area of the given right triangle.

In a right triangle, the base and height are the two perpendicular legs.

So one side is a base and the other the corresponding height

4.3. Area of a trapezium

A trapezium is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The nonparallel sides are called legs (or sides).

The longer base (the bottom) is big B and the smaller base (the top) is little b...

The height is h

How can I find the area of a trapezium?

Take two copies of the trapezoid (one blue trapezium and one green trapezoid)... Tip one upside down and stick them together... Now, you've got a parallelogram.

The area of the parallelogram is:

But, this is double of what we need... So, multiply by 1/2!

The area of a trapezium, , that has and lengths of the two parallel bases, and h is its height, is:

Example

20. Find the area of an isosceles trapezium of bases 19 cm and 9 cm.

A special trapezium is the isosceles trapezium (like an isosceles triangle)

Properties of the sides of an isosceles trapezoid:

The bases (top and bottom) of an isosceles trapezium are parallel.

An isosceles trapezium has opposite sides of the same length.

The angles on either side of the bases have the same size/measure.

The diagonals (not show here) have the same length.

We know both bases but we have to know the height too. If we draw the heights (which have the same length) we observe two equal right triangles with leg 5, because and the two triangles are equal. They have also the same hypotenuse, 13, and the other leg is the height h.

19

So the height is: cm

The area will be:

cm2

4.4. Area of regular polygon

A regular polygon is a figure that lies on a plane with sides of equal length and angles of equal measure.

The area of any polygon is the sum of the areas of the triangles it can be divided into.

In the figure on the right, the polygon can be broken up into four triangles by drawing all the diagonals from one of the vertex.

13

5

h

.

The calculation is easier in a n-sided regular polygon as it can be divided into n identical triangles. One of the points (vertices) of the triangle is the centre of the polygon and the other two are two consecutive vertices of the polygon. Let s the side and its length.

Hexagon It is broken up into six equal triangles

The height of each of these triangles (apothem) is the perpendicular line from the centre of the polygon to one of its sides.

Therefore, we can work out the area of each triangle by multiplying the length of the side by half of the apothem. Remember the area of a triangle is

in our case

If we multiple this result by the numbers of sides we can work out the area of the polygon:

But , the perimeter of the regular polygon. So we have:

The area of a regular polygon is half the perimeter multiplied by the apothem:

Example

21. Find the perimeter and area of the regular pentagon in figure, with apothem approximately 5.5 in.

4.5. Area of a composite 2-D shape

A figure (or shape) that can be divided into more than one of the basic figures is said to be a composite figure (or shape).

For example, figure ABCD is a composite figure as it consists of two basic figures.  That is, a figure is formed by a rectangle and triangle as shown below. We can work out its area by adding up the area of the rectangle and the area of the triangle whose formulas are well known.

We can also find this area by using the formula for a trapezium area.

The area of a composite figure is calculated by dividing the composite figure into basic figures and then using the relevant area formula for each basic figure.

Example

21. Find the area of the following composite figure:

We see this figure consists of two basic figures:

The triangle

The trapezium

Area of the triangle :

Area of the trapezium :

Area of the polygon:

5. Area of circular figures

5.1. DefinitionsA circle is a plane figure, bounded by a single curve line verifying that every part of which is equally distant from a point within it, called the centre. We name to the distance from the centre to any point of the boundary radius.

We defined circumference as the distance around the edge of a circle.

In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure (known as the perimeter) or to the whole figure including its interior.

In academic use when we refer to the whole figure including its interior the term disk is used.

In Spanish we name circunferencia to a line forming a closed loop, every point on which is a fixed distance from a centre point. This distance is named radius.

There are other figures related to circle.

A circle sector is any piece of the circle between two radial lines (shaded in both dark and clear grey).

A segment of a circle is the region between a chord of a circle and its associated arc (shaded in dark grey).

An annulus is the region lying between two concentric circles. It is also named crown.

5.2. Area of a circle

We can think of a circle as a n-sided regular polygon with n a very large number. This polygon can be inscribed in a circumference. The perimeter of such a polygon approximates to the

circumference when n increases, and its apothem approximates to the radius of the circumference. You can see it below.

As the circumference is , then:

The area of the circle is:

5.3. Area of a circle sector

We want to find the area of a circle sector. First we observe that the area of a circle sector and the measure of its arc in degrees are in direct proportion. The circle sector of arc 360º is the whole circle. Now think of the ”regla de tres directa simple”:

So:

If we substitute by its value, , we obtain the formula:

The area A of any circle sector with an arc that has degree measure and with radius is equal to:

5.4. Area of an annulus

The following formula whish comes from the definition of annulus as the region lying between two concentric circles. of radii and .Then we work out this area a the difference between the area of the biggest circle and the smallest circle.

The area of an annulus of radii and is:

Example

22. Find the area of the crown with radius 4 cm and 6 cm.

To work out the area of the crown we find the area of the large circle minus the area of the small circle.

A = r = 6 = 36 = 36 cm

A = r = (4) = 16= 16 cm

A = 36 - 16 = 20 cm

We can also apply directly the formula.

Example

22. Find the shaded area

We can find this area by subtracting the circular sectors of the circles of radii cm and cm.

cm2

cm2

cm2

6. Angles in polygons

6.1. Sum of the interior angles of a polygon

The sum of the angles of a triangle is 180º.If you look at the n-sided polygons below you see that can be divided into triangles.

Four-sided polygon, quadrilateral triangles

Five-sided polygon, pentagon triangles

Six-sided polygon, hexagon triangles

The sum of the measures of all the interior angles in a triangle is 180º. We can obtain the sum of the interior angles of a n-sided polygon by adding the sum of the interior angles of each triangle, this is (n - 2) · 180°.

Moreover in a regular polygon all the angles measure the same.

The sum of the interior angles of a convex n sided polygon is

Each interior angle of a regular n-sided polygon is

Example

23. Here's a regular pentagon. Find the sum of its interior angles and the measurement of each interior angle.

From vertex A we can draw two diagonals which separate the pentagon into three triangles. We multiply 3 times 180 degrees to find the sum of all the interior angles of a pentagon, which is 540 degrees.

sum of angles = =

6.2. Central angle of an inscribed polygon:

A central angle of a regular polygon is the angle formed by two consecutive radii.All the central angles add up to 360º

If a regular polygon has n sides:

The central angle measures

The measure of an interior angles is

Observe the first result is obvious because all the central angles, which are , add up to 360º. The second result was shown in point 6.1.

Example

24. Find1. The sum of the interior angles of a hexagon2. The measure of the interior angle of a regular hexagon3. The measure of the central angle of a regular hexagon

We can reason about this questions and get the solution without using the formulae

Sum of the Interior Angles of a Hexagon:

To find the sum of the interior angles of a hexagon, we divide it up into triangles... There are four triangles. Because the sum of the angles of each triangle is 180 degrees, we get

4Applying the formula: for , we obtain the same result.

So, the sum of the interior angles of a hexagon is 720 degrees.

Regular Hexagons:

All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the interior angles, we know that the sum of all the angles is 720 degrees (from above)...  And there are six angles...

Applying the formula:

So, the measure of the interior of an hexagon is 120º

The measure of the central angles of a regular hexagon:

To find the measure of the central angle of a regular hexagon, make a circle in the middle... A circle is 360 degrees around... Divide that by six angles...

So, the measure of the central angle of a regular hexagon is 60 degrees.

7. Angles in a circle

You can see this table is the same as the one in your text book. All these angles are interesting but this year the most important are central angles, inscribed angles and interior angles.

Central angle Inscribed angle Semi inscribed angle

It has its vertex on the centre of the circle

Its measurement is the same as the arc it intersects

It has its vertex on the circle (Spanish circumferencia) and its sides are secant lines

It measures half the arc it intersects

It has its vertex on the circle (Spanish circumferencia) and one of its sides is tangent to the circle and the other is secant

It measures half the arc it intersects

Interior angle Exterior angle Circumscribed angle

It has its vertex on an interior point of the circle.

It measures half the sum of the two arcs it intersects

It has its vertex on an exterior point and its sides are secants

It measures half the difference of the two arcs it intersects

It has its vertex on an exterior point and its sides are tangent to the circle:

It measures half the sum of the two arcs it divides the circle into.

P = 8 cm + 8cm + 3 cm + 3 cm = 22 cmP = 2(8 cm) + 2(3 cm) = 16 cm + 6 cm = 22 cm

We name 8 cm + 3 cm Semi-perimeter, S,.

Given a rectangle with length and width we name semi-perimeter to , so:

1.2. Perimeter of a regular polygon

The perimeter of a -sided regular polygon of length side , is.

EXAMPLE

We name to length of the sides of the following regular polygons. Find their perimeter:

Equilateral triangle Square Regular hexagon

All three sides have equal lengths

P= 3·

Parallelogram with 4 equal sides and

4 right angles: P = 4·

All six sides have equal length

P = 6·

1.3. Perimeter of a circle: Circumference

From Latin: circum "around" + ferre "to carry" Circumference is the distance around the edge of a circle. Also 'periphery' , 'perimeter'.

If we divide the perimeter of a circle by its diameter we always obtain the same decimal number. This is a very important number, name Pi and designed by . This number is non exact anon periodic. It has infinite decimal figures and they don’t repeat any “period”.

Its value has been calculated with a precision of hundreds of thousands of decimal figures but we can not know the exact value. An approximated value is:

We will use:

The perimeter of a circle or circumference, , can be calculated through two formulae:

where is the diameter and the radius

EXAMPLES:1. Find the circumference of a circle with a radius 12 inches.

r = 12 C = 2 r C = 2 (12) C = 24 inches.

2. Find the circumference of a bike wheel whose diameter is 16 inches.

C = d C = (16) C = 16 inches.Or If the diameter is 16, the radius is 8 inches. (The radius is ½ of the diameter).

Now use the formula C = 2 r. C = 2 (8)

C = 16 inches.

1.4. Length of an arc of circle

If we have a circle of radius the length of the , which measures degree is:

How did we obtain this formula?

If you observe the length of an arc of a circle and the angle intercepted by arc are in direct proportion. Remember that the arc which subtends 360º has the length of the circumference. Now think of the ”regla de tres directa simple”:

So:

EXAMPLE:Find the length of a 24º arc of a circle with a 5 cm radius

BLOG

Circumference video http://www.mathtutordvd.com/public/124.cfm

http://www.mathopenref.com/arcsector.html circular sector or circle sector

http://www.mathopenref.com/annulus.html

http://www.mathopenref.com/rectanglediagonals.html

You can see a video about how to work out the area of a regular hexagonhttp://es.video.yahoo.com/watch/3009502/8602185

HATy mas videos para el blog que he ido quitando. Buscar en la leccion 11 del 1º

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Given an equilateral  triangle ABC with side length = s.Find hBy Pythagorean Theorem (PT),

h²  =

h   =    .