Quadrature of the lunes
A lune is the area left when part of a circle is cut off by another circle. Can you work out the area?
Age
16 to 18
Challenge level
Problem
A lune is the area left when part of a circle is cut off by another circle, as in the following problems. It is called a lune because it looks a bit like the moon.
- In the following figure, two semicircles have been drawn, one on the side $AB$ of the triangle, and the other on the side $AC$ of the triangle (with centre $O$). What is the area of the blue (shaded) lune which is bounded by the two semicircles?
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As a bonus, can you construct a square on the diagram with the same area as the blue lune, using only a straight edge (ruler) and compasses? This is called the quadrature (making into a square) of the lune.
- In the following figure, three semicircles have been drawn, one on each of the sides of the right-angled $6$-$8$-$10$ triangle. What is the total area of the two coloured (shaded) lunes in the drawing?
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This is an Underground Mathematics resource.
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
Student Solutions
Michael, from Exeter Mathematics School, explained how to do question 1.
To find the area of the lune, we can find the area of the semicircle with diameter $AB$, and subtract from this the difference between the areas of sector $OAB$ an
d triangle $OAB$.
Using Pythagoras' theorem on $AOB$, we have $AB = \sqrt{2^2+2^2} = 2 \sqrt{2}$, so the radius of the outer circle of the lune is $\sqrt{2}$.
He also demonstrates how to construct the quadrature of the lune:
The quadrature of the lune must have area $2$, so must have side lengths $\sqrt{2}$.
Let $M_1$ and $M_2$ be the midpoints of $AB$ and $AC$ respectively. Then the line $OM_1$ bisects $AB$ at right angles, and likewise for $OM_2$ and $BC$, since they are chords of the circle.
This is then a square, as the angles are all right angles (we know that $M_1BM_2$ is a right angle because it is subtended in a semicircle) and $M_1B = M_2B = \sqrt{2}$.
This square has the same area as the lune, and therefore is the quadrature of the lune.
Kristian, from Maidstone Grammar School, was able to use the same method to answer question 2. Here is his solution:
Let $D$ be the midpoint of $AC$, this is distance $5$ from $A$, $B$ and $C$.
The angle $B\hat{C}A$ can be calculated using trigonometry, to be $B\hat{C}A = \mathrm{cos}^{-1}\left(\frac{8}{10}\right) = 36.86...^\circ$.
Then, using the circle theorem that the angle subtended at the centre is twice that subtended at the circumference, $B\hat{D}A = 2B\hat{C}A = 73.73...^\circ$.
The triangle $ADB$ has area $\frac{1}{2} \times 5 \times 5 \times \mathrm{sin}\left( 73.73... \right) = 12$, using the formula for the area of a triangle.
The segment between $A$ and $B$ has area $16.08... - 12 = 4.08...$.
Then, the semicircle with diameter $AB$ has area $\frac{1}{2} \times \pi \times 3^2 = 14.13...$.
This means: $$\text{Area of Lune} = \text{Area of Semicircle} - \text{Area of Segment} = 14.13... - 4.08... = 10.049... $$
The triangle $CDB$ has area $\frac{1}{2} \times 5 \times 5 \times \mathrm{sin}\left( 106.26... \right) = 12$, using the formula for the area of a triangle.
The segment between $B$ and $C$ has area $23.18... - 12 = 11.18...$.
Then, the semicircle with diameter $BC$ has area $\frac{1}{2} \times \pi \times 4^2 = 25.13...$.
This means: $$\text{Area of Lune} = \text{Area of Semicircle} - \text{Area of Segment} = 25.13... - 11.18... = 13.95... $$
Therefore, the total lune area is: $$\text{Total Lune Area} = \text{Blue Lune Area} + \text{Green Lune Area} = 10.049... + 13.95... = 24$$
Joe, from Leventhorpe School, found the result in a very nice way, which explains why the area that Kristian found was an integer. Joe was able to show that the area of the two lunes adds up to give the area of the original triangle. Here is his working:
If a right angled triangle has sides $a$, $b$ and $c$, where $c$ is the hypotenuse, Pythagoras' Theorem states that $a^2+b^2=c^2$. This can be applied to the areas of squares constructed on the sides, but this also applies to any shape, as long as those constructed on the
three sides are similar. Suppose we do this with semicircles. In the diagram, the areas of semicircles A and B add to give that of semicircle C.
The triangle formed in this diagram is certainly right-angled, as it is subtended in a semi-circle.
Rotating the triangle by $180^\circ$ in the circle, the segments labelled as $x$ are the same, as are those labelled as $y$.
Subtracting this from the equality of areas established above, this gives that $A + B = C$ in the diagram to the right. Since rotating does not change the area, this says that the original triangle had the same area as the two lunes.
Joe was then able to use this to establish that the total area of the lunes in question 2 was $24$, the area of the triangle. He also adapted this to question 1:
The two lunes in the diagram have the same area, as the right angled triangle is isosceles. The triangle has area $\frac{1}{2} \times 2 \times 4 = 4$, so each lune has area $2$.
Thank you and well done to everyone who contributed their solutions to this problem
To find the area of the lune, we can find the area of the semicircle with diameter $AB$, and subtract from this the difference between the areas of sector $OAB$ an
Image
Using Pythagoras' theorem on $AOB$, we have $AB = \sqrt{2^2+2^2} = 2 \sqrt{2}$, so the radius of the outer circle of the lune is $\sqrt{2}$.
- Semicircle area: $\frac{\pi r^2}{2} = \frac{\pi \left( \sqrt{2} \right)^2}{2} = \pi $
- Sector area: $\frac{\pi r^2}{4} = \frac{\pi \left( 2 \right) ^ 2}{4} = \pi $
- Triangle area: $\frac{1}{2}bh = \frac{1}{2} \times 2 \times 2 = 2$
He also demonstrates how to construct the quadrature of the lune:
Image
Let $M_1$ and $M_2$ be the midpoints of $AB$ and $AC$ respectively. Then the line $OM_1$ bisects $AB$ at right angles, and likewise for $OM_2$ and $BC$, since they are chords of the circle.
This is then a square, as the angles are all right angles (we know that $M_1BM_2$ is a right angle because it is subtended in a semicircle) and $M_1B = M_2B = \sqrt{2}$.
This square has the same area as the lune, and therefore is the quadrature of the lune.
Kristian, from Maidstone Grammar School, was able to use the same method to answer question 2. Here is his solution:
Image
The angle $B\hat{C}A$ can be calculated using trigonometry, to be $B\hat{C}A = \mathrm{cos}^{-1}\left(\frac{8}{10}\right) = 36.86...^\circ$.
Then, using the circle theorem that the angle subtended at the centre is twice that subtended at the circumference, $B\hat{D}A = 2B\hat{C}A = 73.73...^\circ$.
Blue Lune
The sector $ADB$ has area $\frac{A\hat{D}B}{360}\pi r^2 = \frac{73.73...}{360} \times \pi \times 5^2 = 16.08...$.The triangle $ADB$ has area $\frac{1}{2} \times 5 \times 5 \times \mathrm{sin}\left( 73.73... \right) = 12$, using the formula for the area of a triangle.
The segment between $A$ and $B$ has area $16.08... - 12 = 4.08...$.
Then, the semicircle with diameter $AB$ has area $\frac{1}{2} \times \pi \times 3^2 = 14.13...$.
This means: $$\text{Area of Lune} = \text{Area of Semicircle} - \text{Area of Segment} = 14.13... - 4.08... = 10.049... $$
Green Lune
The sector $CDB$ has area $\frac{C\hat{D}B}{360}\pi r^2 = \frac{106.26...}{360} \times \pi \times 5^2 = 23.18...$.The triangle $CDB$ has area $\frac{1}{2} \times 5 \times 5 \times \mathrm{sin}\left( 106.26... \right) = 12$, using the formula for the area of a triangle.
The segment between $B$ and $C$ has area $23.18... - 12 = 11.18...$.
Then, the semicircle with diameter $BC$ has area $\frac{1}{2} \times \pi \times 4^2 = 25.13...$.
This means: $$\text{Area of Lune} = \text{Area of Semicircle} - \text{Area of Segment} = 25.13... - 11.18... = 13.95... $$
Therefore, the total lune area is: $$\text{Total Lune Area} = \text{Blue Lune Area} + \text{Green Lune Area} = 10.049... + 13.95... = 24$$
Joe, from Leventhorpe School, found the result in a very nice way, which explains why the area that Kristian found was an integer. Joe was able to show that the area of the two lunes adds up to give the area of the original triangle. Here is his working:
Image
Image
Image
Subtracting this from the equality of areas established above, this gives that $A + B = C$ in the diagram to the right. Since rotating does not change the area, this says that the original triangle had the same area as the two lunes.
Joe was then able to use this to establish that the total area of the lunes in question 2 was $24$, the area of the triangle. He also adapted this to question 1:
Image
Thank you and well done to everyone who contributed their solutions to this problem