mozaMedical by Mozaik Education

The area of rectangles

Example

How many tiles should you buy to cover your bathroom floor if \latex{ 5 } tiles fit along one side and \latex{ 3 } along the other?

Solution

Draw the solution:

\latex{ 5 } tiles can be put in one row along one of the sides of the bathroom. There are \latex{ 3 } rows, so you must buy \latex{5 \times 3 = 15} tiles.

The area of a square with sides that are one unit long is one unit of area. The area of a rectangle can be determined by covering it with squares whose sides are one unit long.

Calculating the area of a rectangle

The sides of a rectangle are \latex{ 3 } \latex{ cm } and \latex{ 5 } \latex{ cm }. How can you calculate its area?

Unit area: the area of a square whose sides measure \latex{ 1 } \latex{ cm } (\latex{ 1 } \latex{ cm^{2} }).
\latex{ 5 } squares with \latex{ 1 } \latex{ cm } long sides fit along the \latex{ 5 } \latex{ cm } long side of the rectangle.
\latex{ 3 } squares with \latex{ 1 } \latex{ cm } long sides fit along the \latex{ 3 } \latex{ cm } long side of the rectangle.
The area of the rectangle can be covered with \latex{5\times3} squares with \latex{ 1 } \latex{ cm } long sides.

The area of a rectangle with \latex{ 5 } \latex{ cm } and \latex{ 3 } \latex{ cm } sides is \latex{A = 3 \times 5\;cm^{2}= 15\;cm^{2}}.

The area of a rectangle is the product of the lengths of two adjacent sides. The area of a rectangle with sides \latex{a} and \latex{b} is:

\latex{A = a \times b}.

The area of the square

Every square is a rectangle, so the previous method can also be used here. A square with

\latex{ 3 } \latex{ cm } long sides can be covered by \latex{3\times3} squares with \latex{ 1 } \latex{ cm } long sides, so the area of the square is:
\latex{A = 3 \times 3\;cm^{2} = 9\;cm^{2}}.

The area of a square is the length of the sides multiplied by the same number. The area of a square with sides \latex{a} is:

\latex{A = a \times a}.

The area of a right-angled triangle

The diagonal of a rectangle divides it into two congruent right-angled triangles. The area of one of these triangles is half the area of the rectangle.

Exercises

{{exercise_number}}. Measure the area of your school desk. Use the cover of your maths book as the unit area.

{{exercise_number}}. Estimate the area of

a) a picture;
b) your desk;
c) your room;
d) your classroom.

Then, measure and calculate to check your estimates.

{{exercise_number}}. Calculate the area of a rectangular sheet of paper if one side is \latex{ 21 } \latex{ cm } and the other is

\latex{ 15 } \latex{ cm } long.

{{exercise_number}}. What is the area of the rectangle if the length of the adjacent sides is

a) \latex{ 8 } \latex{ cm } and \latex{ 5 } \latex{ cm };

b) \latex{ 4 } \latex{ m } and \latex{ 35 } \latex{ dm };

c) \latex{\frac{4}{5}} \latex{ dm } and \latex{ 12 } \latex{ cm };

d) \latex{4\frac{7}{10}} \latex{ m } and \latex{ 5 } \latex{ m ?}

{{exercise_number}}. How many \latex{ square } \latex{ metres } is a room that is square shaped with sides that are \latex{ 4 } \latex{ m } long?

{{exercise_number}}. What is the area of the rectangular plot if one side is \latex{ 18 } \latex{ m } and the other side is longer by \latex{ 6 } \latex{ m ?}

{{exercise_number}}. The sides of the square-shaped plant box in the middle of a town are \latex{ 6 } and a half \latex{ metres } long. How many flowers were planted in this box if one plant occupies one \latex{ square } \latex{ decimetre? }

{{exercise_number}}. What is the area of the rectangle if the difference between the adjacent sides is \latex{ 3 } \latex{ cm } and the shorter side is \latex{ 10 } \latex{ cm } long?

{{exercise_number}}. The difference between the sides of a rectangle is \latex{ 5 } \latex{ cm }, and one of the sides is \latex{ 10 } \latex{ cm } long. What is the area of the rectangle?

{{exercise_number}}. The sides of a rectangular carpet are \latex{ 2 } \latex{ metres } and \latex{ 3 } and a half \latex{ metres } long. How much does the carpet cost if the price is €\latex{ 30/ }\latex{ m^{2} ?}

{{exercise_number}}. What is the area of the rectangle if one of its sides is \latex{ 5 } \latex{ cm } and the adjacent side is shorter by \latex{\frac{1}{3}} \latex{ cm ?}

{{exercise_number}}. What is the area of the rectangle if one of its sides is \latex{ 2 } and a half \latex{ centimetres } and the difference between the adjacent sides is \latex{\frac{1}{2}} \latex{ cm ?}

{{exercise_number}}. How long are the sides of a square if its area is

a) \latex{ 16 } \latex{ cm^{2} };

b) \latex{ 64 } \latex{ m^{2} };

c) \latex{ 900 } \latex{ cm^{2} };

d) \latex{ 8,100 } \latex{ mm^{2} ?}

{{exercise_number}}. How long is the side of a rectangle with an area of \latex{ 48 } \latex{ cm^{2} } if the other side is

a) \latex{ 12 } \latex{ mm };

b) \latex{ 16 } \latex{ cm };

c) \latex{ 48 } \latex{ cm };

d) \latex{ 10 } \latex{ mm ?}

{{exercise_number}}. The area of a rectangular building site is \latex{ 782 } \latex{ m^{2} }, while the street frontage is \latex{ 23 } \latex{ m } long. How long should the fence be to surround the two sides and the back of the site?

{{exercise_number}}. The sides of a rectangle in \latex{ centimetres } are whole numbers; its perimeter is \latex{ 12 } \latex{ cm }. How many \latex{ square } \latex{ centimetres } is its area?

{{exercise_number}}. Which rectangles with a perimeter of \latex{ 20 } \latex{ cm } have the smallest and largest areas if the length of their sides in \latex{ centimetres } is a whole number?

{{exercise_number}}. The image shows the floor plan of a flat. Calculate the area of each room and that of the entire flat. (Do not take into account the thickness of the walls.) (→)

\latex{ 1 } \latex{ m }

toilet

bathroom

hallway

room

kitchen

{{exercise_number}}. What is the area of the square if its perimeter is

a) \latex{ 24 } \latex{ mm };

b) \latex{ 36 } \latex{ cm };

c) \latex{ 92 } \latex{ m };

d) \latex{ 256 } \latex{ m ?}

{{exercise_number}}. What is the perimeter of the square if its area is

a) \latex{ 25 } \latex{ mm^{2} };

b) \latex{ 49 } \latex{ cm^{2} };

c) \latex{ 100 } \latex{ m^{2} };

d) \latex{ 1 } \latex{ hectare? }

{{exercise_number}}. Calculate the area and perimeter of a rectangle with \latex{ 3 } \latex{ c m} and \latex{ 6 } \latex{ cm } sides. What do you notice? Is there a square whose area and perimeter are equal? If yes, how long are its sides?

{{exercise_number}}. Using three congruent rectangles, the rectangle shown in the image was assembled. What is the area of the large rectangle if its perimeter is \latex{ 40 } \latex{ cm ?} (→)

{{exercise_number}}. Tim designed the following flags for his button football team. How many \latex{ square } \latex{ centimetres } did he have to colour with each colour?

\latex{ 1 } \latex{ cm }

a)

b)

c)

d)

{{exercise_number}}. What fraction of the squares is coloured with red?

a)

b)

c)

d)

{{exercise_number}}. International football games are played on fields that are between \latex{ 64 } and \latex{ 75 } \latex{ m } wide and \latex{ 100–110 } \latex{ m } long. (→)

How many \latex{ square } \latex{ metres } is the difference between the area of the largest and the smallest football fields suitable for international games?

\latex{ 100-110 } \latex{ m }

\latex{ 64-75 } \latex{ m }

{{exercise_number}}. If the unit area is one square, determine the area of the following figures. Then, determine the rule and draw the next two members of the sequence.

a)

b)

{{exercise_number}}. Folding a square in half, you get a rectangle with a perimeter of \latex{ 12 } \latex{ cm }. What is the area of the square? (→)

{{exercise_number}}. How many rectangles are there whose sides in \latex{ metres } are whole numbers, one of their sides is \latex{ 100 } \latex{ m } long, and do not have longer sides than that?

{{exercise_number}}. How many rectangles are there whose sides in \latex{ metres } are whole numbers and whose perimeter is \latex{ 100 } \latex{ m ?}

Mathematics 5.

-