Inscribed in a Circle
Inscribed in a Circle printable sheet
The area of a square inscribed in a circle with a unit radius is, satisfyingly, $2$.
What is the area of a regular hexagon inscribed in a circle with a unit radius?
What is the area of an equilateral triangle inscribed in a circle with a unit radius?
Can you use the fact that a regular hexagon comprises equilateral triangles?
Can you see a connection between the area of the equilateral triangle and the hexagon?
Imagine drawing a straight line from each vertex of the hexagon to the centre of the circle.
The hexagon is now split into six identical triangles.
The triangles are equilaterals. How do we know this?
The angle at the centre is ${360^o \over 6}=60^o$ and the two edges next to this vertex are radii so have length $1$. Because these two edges have the same length the angles at the remaining two vertices must be equal. Angles in a triangle add up to $180^o$ so all the angles must be $60^o$.
The area of a single triangle is
$$\mbox{Area of triangle} = {1\over 2} \times \mbox{height} \times \mbox{base}.$$
Consider one of the triangles making up the hexagon. Dropping a perpendicular from one of the points to the opposite edge and using Pythagoras' Theorem gives that the height of the triangle is ${\sqrt 3\over 2}$.
Therefore the area of one of the triangles is
$$\mbox{Area of triangle} = {1\over 2} \times {\sqrt 3\over 2} \times 1={\sqrt 3\over 4}.$$
And the area of the hexagon is $6$ times this, which is ${3 \sqrt 3\over 2}$.
To find the area of the equilateral triangle, draw it on the same diagram as the first part. Notice that each edge of the large triangle cuts two of the smaller triangles and divides their area in half. Therefore the area of the equilateral triangle in the unit circle is half that of the hexagon, which is ${3\sqrt 3\over 4}$.
Why do this problem?
This problem encourages students to use a range of ideas including symmetry and Pythagoras’ Theorem whilst developing their geometrical problem-solving skills.
The areas of the shapes presented in this problem can be found without trigonometry, so can be used as a challenging problem for students who have not yet used trigonometry. Alternatively, Pythagoras’ Theorem and trigonometry could be compared as different methods, with trigonometry offering a method which is easier to generalise.
Possible approach
Show the first image of the square inscribed inside the circle and ask the students to confirm in pairs that the area is two. Can they find more than one way of doing it? Share their ideas with the group. Make sure that students see the method of splitting the square into 4 triangles (and rearranging into two smaller squares), splitting the square into 2 triangles (and rearraning into a larger triangle that is half of a larger square) and finding the side length using Pythagoras’ Theorem. If appropriate, you could also make sure they see a method which uses trigonometry,
Allow students time to work in groups or pairs to try and split up the hexagon in a similar way. What do they know about the triangles they split the hexagon into? Can they find the heights of those triangles? Can Pythagoras’ Theorem help them? Can trigonometry?
For the triangle case, most students will be familiar at this stage with the splitting approach, but they will find that it yields insufficient information to use Pythagoras’ Theorem alone. Students either need to use trigonometry or the result from the hexagon. Encourage them to think about what is the same and what is different between the triangle and the hexagon. Can the triangle fit into the hexagon just like the hexagon fits into the circle?
Have a whole class discussion to allow students to share their methods and their reasoning. You might want to get students to practice explaining their methods to their partners before presenting them to the class. Try to get as many different approaches as possible.
If you still have time, you could list the areas by number of sides: triangle, square, hexagon. What do they think the area of an inscribed pentagon might be? Or an octagon? Will the areas keep getting bigger? Is there a maximum area that the area of the inscribed polygons won’t ever exceed? The limit is the area of the circle and this is an ancient way of estimating it.
Key questions
How can we split the shape into smaller shapes?
What information have we got, and what else can we find?
What information do we need to find the area of the shape/the smaller shapes?
Is there more than one way of doing it?
Possible support
As a starter, you could present students with some isoceles triangles and ask them to find their areas. This will introduce the idea of using Pythagoras’ Theorem for triangles that are not right-angled by splitting them into triangles that are.
You might also want to remind students about the properties of circles and triangles. You could present them with some triangles in circles and ask what they know about the triangles: are they equilateral, isosceles, right-angled, scalene?
Possible extension
Which other inscribed polygons do students think they can find the area of? Challenge them to choose and find the areas for themselves. Without trigonometry, the area of an inscribed octagon can be found with the help of the square, similar to the triangle and the hexagon.
If they are using trigonometry, then they can find the area of any polygon. Challenge them to find a formula for the area of an n-sided polygon.