Ontario 09 Academic (MPM1D)
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Area of special quadrilaterals (review)
Lesson

We have already seen how to find the area of rectangles, triangles and parallelograms.  

There is another 2D shape whose area can be found by looking at the relationship the shape has with a rectangle.  

What's a trapezoid?

A trapezoid (sometimes called trapezoid) is a 2D shape with the specific geometric properties of 

  • 1 pair of opposite sides that are parallel

All of these are trapezoids

 

Area of a trapezoid

Let me show you something.

Move the point D anywhere you like to create a trapezoid and then slide the green slider to the right.

From the interactive you can see you to flip the trapezoid over and turn it into a parallelogram, and of course we already know that a parallelogram has the same area as a rectangle!

And guess what? Because we know  $\text{Area of a rectangle }=L\times W$Area of a rectangle =L×W, then we can work out the area of any trapezoid.

We tend to call the top side of the trapezoid  BASE 1 (a), and we call the bottom side BASE 2 (b) - because when we flip it both those sides becomes part of the base. We also need the HEIGHT of the trapezoid.  

This gives us the following formula:

Area of a Trapezoid

$\text{Area of a Trapezoid}=\frac{1}{2}\times\left(\text{Base 1 }+\text{Base 2 }\right)\times\text{Height }$Area of a Trapezoid=12×(Base 1 +Base 2 )×Height

$A=\frac{1}{2}\times\left(a+b\right)\times h$A=12×(a+b)×h

You might wonder why we divide this by two.  Well remember we flipped the trapezoid over to turn it into a rectangle, well that means that 2 trapezoids is the same as the parallelogram - but we really only want the area of 1 trapezoid.  

Lets do an example using our new rule

Example

Question: A new chocolate bar is to be made with the following dimensions,  the graphic artist needs to know the area of the trapezoid to begin working on a wrapping design.  Find the area.  

Think: I need to identify the base1, base2 and height.  I can see these on the diagram that is given.

Do:

$\text{Area of a Trapezoid}$Area of a Trapezoid  $=$=  $\frac{1}{2}\times\left(\text{Base 1 }+\text{Base 2}\right)\times\text{Height }$12×(Base 1 +Base 2)×Height
  $=$= $\frac{1}{2}\times\left(a+b\right)\times h$12×(a+b)×h
  $=$= $\frac{1}{2}\times\left(4+8\right)\times3$12×(4+8)×3
  $=$= $\frac{1}{2}\times12\times3$12×12×3
  $=$= $18$18 cm2

So now we can find the areas of rectangles, triangles (which are half of a rectangle), parallelograms (like a rectangle) and trapezoids (like half a rectangle).

More Worked Examples

QUESTION 1

Find the area of the trapezoid shown.

QUESTION 2

Find the value of $x$x if the area of the trapezoid shown is $65$65 cm2.

  1. Start by substituting the given values into the formula for the area of a trapezoid.

    $A=\frac{1}{2}\left(a+b\right)h$A=12(a+b)h

 

We have already seen how to find the area of rectangles, triangles and parallelograms, and trapezoids.    

There are just a few more 2D shapes whose area we need to be able to find. 

What's a kite?

A kite is a 2D shape with the specific geometric properties of 

  • 2 pairs of adjacent sides that are equal
  • 1 pair of equal angles

Of course the kite you fly around on a windy day is named after the geometric shape it looks like.

Kites can taken on many different shapes and sizes. Try moving points $A$A, $O$O and $D$D on this mathlet to make many kinds of kites.

 

Area of a Kite

Let me show you something. Slide the slider.  What shape is created?  Now try showing diagonals and then sliding the slider.  

From the interactive you'll notice that if you copy the inner triangles of the kite and rearrange them you can create - you guessed it a rectangle.  Of course, like the trapezoid, our original shape is only 1/2 of this rectangle. 

Because we know the $\text{Area of a rectangle }=L\times W$Area of a rectangle =L×W, then we can work out the area of any kite.

We tend to call the long diagonal $x$x and we call the short diagonal of the kite $y$y  These give us the length and width of the rectangle that the kite fits inside. 

Area of a Kite

$\text{Area of a Kite}=\frac{1}{2}\times\text{diagonal 1}\times\text{diagonal 2}$Area of a Kite=12×diagonal 1×diagonal 2

$A=\frac{1}{2}\times x\times y$A=12×x×y

Let's do an example using our new rule.

Example

Question: Find the area of this kite  

Think: I need to identify the long diagonal length and the short diagonal length.

Do:

$\text{Area of a Kite }$Area of a Kite $=$= $\frac{1}{2}\times x\times y$12×x×y
  $=$= $\frac{1}{2}\times4\times\left(2\times0.9\right)$12×4×(2×0.9)
  $=$= $\frac{1}{2}\times4\times1.8$12×4×1.8
  $=$= $3.6$3.6 mm2

 

So now we can find the areas of  rectanglestriangles  (which are half of a rectangle),  parallelograms  (like a rectangle),  trapezoids  (like half a rectangle) and kites (like half a rectangle).  

More Worked Examples

QUESTION 1

Find the area of the kite shown.

QUESTION 2

The area of a kite is $640$640 cm2 and one of the diagonals is $59$59 cm. If the length of the other diagonal is $y$y cm, what is the value of $y$y rounded to two decimal places?

 

We have already seen how to find the area of rectangles, triangles ,parallelograms, trapezoids and kites.    

There is just one more 2D shapes whose area we need to be able to find- a rhombus!

What's a RHOMBUS?

A rhombus is a 2D shape with the specific geometric properties of 

  • all sides equal in length
  • opposite sides parallel
  • opposite angles equal
  • diagonals bisect each other

You can play with this mathlet to make many kinds of rhombuses, it also shows that you only need 1 side length and 1 angle to create one.  

Area of a rhombus

Let me show you something;

From this interactive you can see that as you copy the inner triangles of the rhombus and place them accordingly you can create - you guessed it a rectangle.  Of course, like the trapezoid and kite, our original shape is only $\frac{1}{2}$12 of this rectangle. 

Because we know the $\text{Area of a rectangle }=L\times W$Area of a rectangle =L×W, then we can work out the area of any rhombus.

Lets call the diagonals $x$x and $y$y  These give us the length and width of the rectangle that the rhombus fits inside.

Area of a Rhombus

$\text{Area of a Rhombus }=\frac{1}{2}\times\text{diagonal 1}\times\text{diagonal 2}$Area of a Rhombus =12×diagonal 1×diagonal 2

$A=\frac{1}{2}\times x\times y$A=12×x×y

 

So now lets do an example using our new rule

Example

Question: A packing box with a square opening is squashed into the rhombus shown.  What is the area of the opening of the box?

Think: I need to be able to identify the two diagonals.

Do:

$\text{Area of a Rhombus }$Area of a Rhombus $=$= $\frac{1}{2}\times\text{diagonal 1 }\times\text{diagonal 2 }$12×diagonal 1 ×diagonal 2
  $=$= $\frac{1}{2}\times x\times y$12×x×y
  $=$= $\frac{1}{2}\times16\times4$12×16×4
  $=$= $32$32 cm2

 

 

So now we can find the areas of rectangles, triangles (which are half of a rectangle), parallelograms (like a rectangle), trapezoids (like half a rectangle), kites (like half a rectangle) and now rhombuses (like half a rectangle).

More Worked Examples

QUESTION 1

Find the shaded area shown in the figure.

 
 

 

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