You may also like

problem icon

Right Time

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?

problem icon

Isosceles

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

problem icon

Linkage

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Fraction of a Square

Age 11 to 14 Short Challenge Level:

Drawing more little triangles
The length DE, which is the base of the triangle, fits 4 times along the side DC. So 4 copies of the triangle fit with their bases along DC:
 
The white unshaded triangles are congruent to the grey shaded triangles, so they have the same area. There are 4 shaded triangles and 4 unshaded triangles, so there are 8 triangles altogether.

So the area of each triangle is $\frac18$ of the area of the square.



Finding the area of the square and the triangle
Suppose the square has side length $1$, so that its area is $1$. We can do this because we don't need to know what the area of the triangle is as a number, we just want to know it as a fraction of the area of the square. And finding numbers as fractions of $1$ is easy - for example, $\frac12$ of $1$ is just $\frac12$.

Then the height of the triangle is $1$, since it is the side length of the square.

$4\times=1$. So the length DE, which is the base of the triangle, is $\frac14$

So the area of the triangle is $\frac12\times\frac14\times1=\frac12\times\frac14=\frac18$

And $\frac18$ as a fraction of $1$ is just $\frac18$. So the area of the triangle is $\frac18$ of the area of the square.
You can find more short problems, arranged by curriculum topic, in our short problems collection.