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Triangles in a Square

Age 11 to 14 Challenge Level:

We had lots of great solutions to this problem. Thank you to all of you who wrote in!

Julian from the British School Manila, Ahrus and Ben from Dixons Trinity Academy, and Kira from Wycombe High School all managed to find the area of each of the coloured triangles on the $5 \times 5$ grid. Here's what Kira did: 

How to calculate the area of each triangle:
When the triangles are drawn on the dotty paper they are surrounded by $3$
right-angled triangles. The total area of all three right-angled triangles
must be subtracted from the area of the square ($25$ squares) to give the
area of the triangle.

Red triangle is $10.5$ squares
Yellow triangle is $10.5$ squares
Blue triangle is $11.5$ squares
Green triangle is $11$ squares
Purple triangle is $10.5$ squares

Therefore the blue triangle has the largest area.

Jurmana, Haidi, and Nour from the Continental School Cairo all found the largest and smallest triangles they could make on a $5 \times 5$ grid:

The largest area I found was $12.5$ cm$^2$ when the corners of the triangle are at $(0,0)$, $(0,5)$ and $(5,5)$.
The smallest area I found was $2.5$ cm$^2$.

Here's how Julian found a general formula for the area of a triangle with vertices at $(5,5)$, $(0,y)$, and $(x,0)$:

Area of the top left white triangle:
base: $5$
height: $5-y$
area: $\frac{5(5-y)}{2}$

Area of the bottom left white triangle:
base: $x$
height: $y$
area: $\frac{xy}{2}$

Area of the bottom right white triangle:
base: $5-x$
height: $5$
area: $\frac{5(5-x)}{2}$

Since the area of the colored triangle is $25$-(the area of the three
triangles covering the white space), we can add the three areas and
subtract it from $25$.
This will give us an area of $\frac{5x+5y-xy}{2}$.

Victor from Dulwich College Seoul found this formula and used it to find all the possible areas of triangles on a $5 \times 5$ grid:

The possible areas are $2.5cm^2$, $4.5cm^2$, $5cm^2$, $6.5cm^2$, $7.5cm^2$, $8cm^2$, $8.5cm^2$, $9.5cm^2$,$10cm^2$, $10.5cm^2$, $11cm^2$, $11.5cm^2$, $12cm^2$, $12.5cm^2$ which is a total of $14$ areas. 

Kira thought about triangles on bigger grids:

When a triangle is drawn in a $6$ by $6$ grid the area can be represented by
$3x + 3y – 0.5xy$.

When a triangle is drawn in a $7$ by $7$ grid the area can be represented by
$3.5x + 3.5y – 0.5xy$.

This shows that there is a pattern for finding the area of the triangle.  $x$
and $y$ are multiplied by half of the number of squares up or down.

Let $n$ represent size of grid eg $7$ by $7$ grid is when $n=7$:

Area of triangle in square = $0.5nx + 0.5 ny - 0.5xy$ squares.