You may also like

problem icon

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

problem icon

Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

problem icon

Equilateral Areas

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Areas of Parallelograms

Age 14 to 16 Challenge Level:

Here are two parallelograms, defined by the vectors $\mathbf{p}$ and $\mathbf{q}$. Can you find their areas?

a) $\mathbf{p}=\left(\begin{array}{c}3\\ 0\end{array}\right), \mathbf{q}=\left(\begin{array}{c}5 \\ 2\end{array}\right)$
 

b) $\mathbf{p}=\left(\begin{array}{c}3 \\ 2\end{array}\right), \mathbf{q}=\left(\begin{array}{c}0 \\ 4\end{array}\right)$

Choose different vectors $\mathbf{p}$ and $\mathbf{q}$, where one vector is parallel to an axis, and find the areas of the corresponding parallelograms.
Can you discover a quick way of doing this?

Here are two more parallelograms, again defined by vectors $\mathbf{p}$ and $\mathbf{q}$. This time, neither $\mathbf{p}$ nor $\mathbf{q}$ is parallel to an axis.
Can you find the areas of these parallelograms?

c) $\mathbf{p}=\left(\begin{array}{c}4 \\ 1\end{array}\right), \mathbf{q}=\left(\begin{array}{c}3 \\ 3\end{array}\right)$

 
 

d) $\mathbf{p}=\left(\begin{array}{c}2 \\ 4\end{array}\right), \mathbf{q}=\left(\begin{array}{c}-1 \\ 3\end{array}\right)$
 
 


Choose some other vectors p and q, where neither p nor q is parallel to an axis.

Can you find a quick way of working out the areas of the corresponding parallelograms?

Can you find the area of the parallelogram defined by the vectors $\mathbf{p}=\left(\begin{array}{c}a \\ b\end{array}\right)$ and $\mathbf{q}=\left(\begin{array}{c}c \\ d\end{array}\right)$?

If you have found a rule, does it ever give you negative areas?
If so, can you predict which vector pairs have this effect?