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4.04 Area


Areas of rectangles

The area of a rectangle is the size of the space within its boundary. This is the number of unit squares it takes to cover the shape. For a rectangle with length $5$5 centimetres and width $3$3 centimetres, how many squares of size $1$1 cm2 would it take to cover it?

If we divide the shape into a grid of $1$1 cm2, we can quickly see that it would take $3$3 rows of $5$5 squares to cover the shape, and so the area is $15$15 cm2. We can always take this approach of dividing a rectangle into a grid - and in doing so, we obtain the formula for the area of a rectangle:

Area of a rectangle

For rectangles:

$Area=\text{length}\times\text{Width}$Area=length×Width, this can be abbreviated to $A=L\times W$A=L×W


A square is a special rectangle where the length and width are equal, so we can use the following formula:

$A=L\times L$A=L×L, which simplifies to $A=L^2$A=L2


Note that units for area are length units squared. Don't forget to include units for your answers!


Practice questions

Question 1

Find the area of the rectangle shown.

A rectangle with a length marked 9 m with double ticks on each opposing side and a width labeled 6 m with single ticks on each opposing side.  Small squares are drawn at each corner, signifying that it is a right angle.

Question 2

Find the area of the attached figure.

A quadrilateral with all of its sides marked with a single tick mark, indicating that they are congruent. Each corners have a small square indicating right angles. One of the sides is labeled $11$11 cm, indicating its length.

Question 3

A kitchen floor is tiled with the tiles shown in the picture. If $30$30 tiles are needed to tile the floor, what is the total area of the floor? Give your answer in square centimetres.



Area of triangles

The area of a triangle is given by the formula $Area=\frac{1}{2}\text{base}\times\text{height}$Area=12base×height, which can be abbreviated to $A=\frac{1}{2}bh$A=12bh.

The height in this formula is referring to the height that is perpendicular to the base length. This may be one of the sides of the triangle (like in the right-angled triangle shown above), it may be within the triangle (like the second triangle above), or it may be outside of the triangle (like in the third triangle above).

Use the following applet to see a range of triangles and their related rectangles by moving the points and slider.


Worked example

Example 1

Find the area of the triangle pictured below.

Think: Identify the base and perpendicular height and use the formula. Here the base is $7$7 cm and the height perpendicular to this is $8$8 cm.



$A$A $=$= $\frac{1}{2}bh$12bh

Write the formula.

  $=$= $\frac{1}{2}\times7\times8$12×7×8 cm2

Substitute in known values and write down units.

  $=$= $\frac{1}{2}\times56$12×56 cm2


  $=$= $28$28 cm2



So the area of this triangle is $28$28 cm2.



The rule for the area of a triangle is:


This can be abbreviated to:


The base and height must be perpendicular, as shown in the diagrams below:




Practice question

Question 4

Find the area of the triangle with base length $10$10 m and perpendicular height $8$8 m shown below.


A triangle is depicted on the image. Horizontal broken line, indicating the measurement of the base of the triangle, is labeled with 10 m. Vertical broken line, indicating the measurement of height of the triangle, is labeled with 8 m. 




calculate areas of rectangles and triangles

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