# Numerically Equal

I want to draw a square in which the perimeter is numerically equal to the area.

Of course, the perimeter will be measured in units of length, for example, centimetres (cm) while the area will be measured in square units, for example, square centimetres (cm$^2$).

What size square will I need to draw?

What about drawing a rectangle that is twice as long as it is wide which still has a perimeter numerically equal to its area?

Can They Be Equal? offers a suitable extension to this problem.

You could try drawing squares on squared paper.

How will you know you don't miss out the square that works?

The solutions that arrived on our desk
for Numerically Equal all had the same answer, but slightly
different ways of finding it. **Jack** of Tattingstone Primary School sketched the stages of
his thinking.

Chris used addition to help him with the perimeter calculation:

$4cm+4cm+4cm+4cm = 16cm$Whereas, **Sam** of St Margaret's
Primary School in Newcastle-under-Lyme, changed this to
multiplication:

$4cm$ x $4$ (sides) $= 16cm$

Does this measurement of 4cm work for
the area? According to **Annice** and
**Grace** in
Yarm Primary School, and **Thomas** it does!
Backing them up with their answers were **Jade and Marion** both
of Tattingstone Primary. Great explanations came from both
girls.

**Asher** had the same idea as a **Franco** of Hazelwood
School, London. Franco solved this "within a few minutes by
thinking of square numbers and dividing them by 4". He
hit upon a 4cm square as one possible answer but remains convinced
it is not the only one and has gone to do further investigations on
his own! Good for you Franco, let us know of any other solutions
your investigations reveal.

There was a second challenge here,
finding a rectangle that is twice as long as it is wide and that
has an area and perimeter of 18
units. **Daniel**
, **Marion** and
**Jade** (all of Tattingstone School) had the same strategy that
worked very well for each of them. Each drew a rectangle then drew
the same size rectangle attached to it and calculated the
area. **Jack** shows us a similar way to Marion and Jade's and how he
can prove his answer.

**Christopher** and **James**
both explained in words and numbers rather
than diagrams:

The perimeter will be $6+3+6+3$ which equals $18cm$.

The area is $6$ x $3$ which equals $18 cm^2$.