Three Tears
Construct this design using only compasses
Age
14 to 16
Challenge level
Problem
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Using only compasses and a straight edge, construct this design so that each "tear" is $100 \; \text{cm}^2$.
So that your diagram is the right size you will need to calculate one length, to get you started.
Do this as a single calculation, then take that result as the initial setting for your compasses.
You can then create the design without further calculation or use of a ruler
(a protractor is not necessary for any part of the process).
Getting Started
Do you know the formula for the area of a whole circle?
Use that to find the correct radius for this circular design.
There are three equal "tears", what angle of rotation do you need for one "tear" to match with its neighbour?
What angle do you have in an equilateral triangle?
How does that help with the angle you need to make at the centre?
Three semicircles arc from the edge to the centre of the design.
Where is the centre for each semi-circular arc, and how will you locate each point accurately?
Student Solutions
Nice construction work from Ashley at Downham Market High School.
First we need the area of the whole circle using the formula $\pi r^2$ to produce $300 \; \text{cm}^2$
Then if $\pi r^2$ is $300$, $r^2$ is $95.49296586$
So $r$ is the square root of $95.49296586$ and has to be $9.8 \;\text{cm}$ to the nearest $\text{mm}$
Now draw the circle.
Keeping the radius fixed, move the point of the compasses to the circumference and mark off one radius length around the circumference.
Move the point of the compasses on to that mark and repeat until you have six marks equally spaced around the circle.
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Next join three of the marks to the centre to make thirds
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Measure half way along these lines to get the centre for the semi-circles that make the tear shapes.
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Finally rub out the working and add some colour.
Well done Ashley. Maybe someone can see how to find those mid-points by construction rather than measuring.
Teachers' Resources
This is initially an exercise in "de-construction" to enable
construction.
The first calculation may involve a little algebra to rearrange the familiar area formula or substitution and solving the equation.
Either way it is necessary to establish the radius which will give the correct area.
Marking off the six arcs around a circle circumference is an important formative experience.
The student needs to check that they understand how this "works", how the process amounts to six equilateral triangles.
The first calculation may involve a little algebra to rearrange the familiar area formula or substitution and solving the equation.
Either way it is necessary to establish the radius which will give the correct area.
Marking off the six arcs around a circle circumference is an important formative experience.
The student needs to check that they understand how this "works", how the process amounts to six equilateral triangles.