The areas of polygons

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Pola czworokątów

Learning objectives

You will develop the following competences: communicating in English, mathematical, IT and basic scientific and technical competence, your learning skills.
You will consolidate the formulae for calculating the area of a rectangle, square, parallelogram, trapezoid, deltoid.
You will get to know the use of the formulae for the area of quadrangles in tasks.

Learning effect

You will consolidate the formulae for calculating the area of a rectangle, square, parallelogram, trapezoid, deltoid.
You will learn to use the formulae for the area of quadrangles in tasks.

Recollect the concepts and properties connected with quadrangles.

Task 1

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nagranie abstraktu

Complete the sentences.

A trapezoidtrapezoidtrapezoid is …
A quadrangle whose one pair of sides is parallel is called…
A parallelogramparallelogramparallelogram is a quadrangle which has…
A rhombus is a parallelogramparallelogramparallelogram whose all sides are…
A parallelogramparallelogramparallelogram whose all angles are right is called…
A squaresquaresquare is a rectanglerectanglerectangle whose all sides are…
A deltoiddeltoiddeltoid is a quadrangle which has a pair of sides…
Every squaresquaresquare is a …, but not every … is a squaresquaresquare.

The aim of the class is consolidation of the formulae for the areas of quadrangles and using them in tasks.

Recollect the formulae for calculating the areas of quadrangles. Write them down.

The area of a rectanglerectanglerectangle equals the product of the lengths of its sides.
The area of a squaresquaresquare can be calculated in two ways: the length of its side squared, or half of the length of its diagonals squared.
The area of a parallelogramparallelogramparallelogram equals the product of the length of its side divided by the length of the altitude perpendicular to this side.
The area of rhombus can be calculated in two ways: the product of the length of its side divided by its altitude, or half of the product of the lengths of its diagonals.
The area of a trapezoidtrapezoidtrapezoid equals half of the product of its bases divided by the altitude of the trapezoidtrapezoidtrapezoid.

Use the information to solve the tasks.

Task 2

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nagranie abstraktu

The lengths of the sides of a rectanglerectanglerectangle are expressed with natural numbers and its area equals 20 cmIndeks górny 2. How many such rectangles are there? What are the lengths of their sides?

Task 3

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nagranie abstraktu

One of the sides of a parallelogramparallelogramparallelogram, whose area equals 28 cm2, has the length of 3,5 cm. Calculate the altitude of this parallelogramparallelogramparallelogram perpendicular to this side.

Think, are there other formulae for calculating the areas of quadrangles? Analyse the material shown in the INTERACTIVE PRESENTATION. Formulate your conclusions.

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The conclusion. The areas of quadrangles can be calculated using the trigonometric function of the acute angles between the sides of the quadrangle( in case of a parallelogramparallelogramparallelogram, a rhombus, a deltoiddeltoiddeltoid) or between the diagonals (in case of a parallelogramparallelogramparallelogram, any quadrangleany quadrangleany quadrangle).

Use the information to solve the tasks individually.

Task 4

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nagranie abstraktu

Calculate the area of the rectanglerectanglerectangle whose 12 cm long diagonals intersect at an angle of 30Indeks górny o.

Task 5

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nagranie abstraktu

Calculate the area of the rhombus whose side is 10 dm long and the sum of the diagonals equals 28 dm.

Task 6

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nagranie abstraktu

The area of a trapezoidtrapezoidtrapezoid equals 100 cmIndeks górny 2 and its altitude is 10 cm. Calculate the lengths of the trapezoid’s bases, knowing that one of them is three times longer than the other.

Task 7

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nagranie abstraktu

The diagonals of a convex quadrangle have the lengths of 16 cm and 18 cm. The area of this quadrangle equals 72 cmIndeks górny 2. Calculate the measure of the angle of the diagonal intersection.

Task 8

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nagranie abstraktu

An extra task:

Four corners were cut off a square‑shaped wooden board, whose side length is 4 dm. In this way a regular octagon was formed. Is the sum of the areas of the waste (i.e. the cut off corners) larger than 10% of the surface area of the whole wooden board?

Yes. It is $3 - 2 \sqrt{2} \approx 0, 17$ of the board’s surface.

Having solved all the tasks, the students do the consolidation tasks. Formulate the conclusion to memorize.

The areas of quadrangles can be calculated using the trigonometric function of the acute angles between the sides of the quadrangle( in case of a parallelogramparallelogramparallelogram, a rhombus, a deltoiddeltoiddeltoid) or between the diagonals (in case of a parallelogramparallelogramparallelogram, any quadrangleany quadrangleany quadrangle).

Exercises

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Exercise 1

If the diagonals of a rhombus have the lengths of 3a and 8a, the area of this rhombus equals:

$0, 5 a^{2}$
$4 a^{2}$
$12 a^{2}$
$6 a^{2}$

Source: GroMar, licencja: CC BY 3.0.

Exercise 2

In an isosceles trapezoid, one of the bases is twice as long as the other one and the diagonal of the trapezoid divides the angle at the longer base into two halves. Calculate the length of the sides of the trapezoid, knowing that its area equals $\sqrt{27}$ and the length of the leg is equal to the length of the shorter base.

Exercise 3

The diagonals of a rectangle have the length of 15 cm and they intersect at the angle of 60Indeks górny o. Describe in English how to calculate the area of this rectangle.

Exercise 4

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Indicate which pairs of expressions or words are translated correctly.

dowolny czworokąt - any quadrangle
pole czworokąta - area of the quadrangle
przekątna czworokąta - diagonal of the quadrangle
kąt między przekątnymi czworokąta - angle between the diagonals of the quadrangle
wysokość rombu - parallelogram
deltoid - rectangle

zadanie

Source: GroMar, licencja: CC BY 3.0.

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Match Polish terms with their English equivalents.

diagonal of the quadrangle
angle between the diagonals of the quadrangle
pole czworokąta
any quadrangle
altitude of the rhombus
wysokość rombu
dowolny czworokąt
przekątna czworokąta
area of the quadrangle
kąt między przekątnymi czworokąta

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

altitude of the rhombus

wysokość rombu

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wymowa w języku angielskim: altitude of the rhombus

angle between the diagonals of the quadrangle

kąt między przekątnymi czworokąta

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wymowa w języku angielskim: angle between the diagonals of the quadrangle

any quadrangle

Dowolny czworokąt

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wymowa w języku angielskim: any quadrangle

area of the quadrangle

pole czworokąta

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wymowa w języku angielskim: area of the quadrangle

deltoid

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wymowa w języku angielskim: deltoid

diagonal of the quadrangle

przekątna czworokąta

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wymowa w języku angielskim: diagonal of the quadrangle

parallelogram

równoległobok

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wymowa w języku angielskim: parallelogram

rectangle

prostokąt

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wymowa w języku angielskim: rectangle

square

kwadrat

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wymowa w języku angielskim: square

trapezoid

trapez

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wymowa w języku angielskim: trapezoid

Keywords

altitude of the rhombusaltitude of the rhombusaltitude of the rhombus

angle between the diagonals of the quadrangleangle between the diagonals of the quadrangle

any quadrangleany quadrangleany quadrangle

area of the quadranglearea of the quadranglearea of the quadrangle

diagonal of the quadranglediagonal of the quadranglediagonal of the quadrangle

Scenariusz

Exercises

Glossary

Keywords