Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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?

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Diagonals for Area

Stage: 4 Challenge Level:

Tom and James from Queen Mary's Grammar School, Walsall and Shu Cao from Oxford High School gave this solution for a convex quadrilateral. Can you see how to adapt the solution for the case of the arrow shaped quadrilateral?

Suppose there is a convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ cross each other at $O$. The angle between $AO$ and $BO$ is $\theta$ degrees, the angle between $DO$ and $CO$ is the same. The angle between $AO$ and $DO$ is $180-\theta$ degrees, the angle between $BO$ and $CO$ is the same.

The area of triangle $AOB$ is ${1\over 2}AO\times BO \sin \theta$.
The area of triangle $AOD$ is ${1\over 2}AO\times DO \sin (180-\theta)={1\over 2}AO\times DO \sin \theta$.
The area of triangle $DOC$ is ${1\over 2}DO\times CO \sin \theta$.
The area of triangle $BOC$ is ${1\over 2}BO\times CO \sin (180-\theta)={1\over 2}BO\times CO \sin \theta$.
The area of the quadrilateral is the sum of these four triangles.

\eqalign{ {\rm Area}&={1\over 2}AO\times BO \sin \theta+{1\over 2}AO\times DO \sin \theta+{1\over 2}DO\times CO \sin \theta+{1\over 2}BO\times CO \sin \theta \cr &= {1\over 2}[AO(BO+DO) + CO(DO+ BO)]\sin \theta \cr &={1\over 2}(AO\times BD+ CO\times BD)\sin \theta \cr &={1\over 2}AC\times BD \sin \theta }

So we have proved that for a convex quadrilateral the area of the quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.