Curvy areas

6468 /www/nrich/html/content/id/6468/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Take a look at the image below: <div><mdo:image src="Curvy1.png"></mdo:image></div> Can you see how the image was created? Try to recreate it using a ruler and compasses. Here are two images created in a similar way. <mdo:image src="Curvy2.png"></mdo:image> <mdo:image src="Curvy3.png"></mdo:image> Can you work out the proportion of the 3-colour, 4-colour and 5-colour circles which is shaded red? Can you make any generalisations? Can you prove your ideas? Extension What about the proportion which is shaded orange? Yellow? ... Can you make any generalisations? Can you prove your ideas? <a href="http://nrich.maths.org/8001">Click here for a poster of this problem</a>.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> This problem offers the opportunity to practise calculating areas of semicircles and working in terms of $\pi$ and leads to a surprising result that invites students to generalise. <h3>Possible approach</h3> <div>Begin by showing one of the three diagrams on <a class="powerpoint" href="/content/id/6468/Curvy%20areas.ppt">these slides</a> for a short while, then hide it. "Think about the image you just saw. Can you make a sketch of it? Can you describe to your partner how it was drawn?"</div> Show the image again so that students can compare their first impression with the actual image. Recreating the image offers students a good opportunity for practising constructions with a pair of compasses. "Now that we can see how the images were created, talk to your partner and see if you can come up with a method for working out the shaded areas." Give students a little time to discuss, and then bring the class together to share their suggested methods. Then ask students to use these methods to work out all the areas for each curvy pattern. Students might need prompting to choose a letter or value to represent the radius of the smallest semi-circle in order to work out the areas. There is scope for some discussion here about the merits of assigning a unit length rather than using a variable, and why this is 'allowed' in a question asking about proportion. Surprisingly, each coloured region on a diagram is the same proportion of the total area. Once students have found that result, they could work on a pattern with more regions to see if the same result follows. To prove the general case that all regions have equal area is quite challenging. A more accessible question to work on first of all is to show that the first region (coloured red in our diagrams) is always $\frac{1}{n}$ of the total area, for a shape with $n$ regions. The diagram below builds a similar pattern from rectangles and could be used to develop appropriate arguments, without needing to include $\pi$ in calculations. <div style="text-align: center;"> <mdo:image alt="" height="200" src="pepsisqu.png" width="200"></mdo:image></div> <h3>Key questions</h3> <div>Where are the centres of semicircles in the diagrams?</div> <div>What is the relationship between the different radii?</div> <div>For a circle with $n$ different coloured sections, how would you work out the red area?</div> <h3>Possible extension</h3> Prove algebraically that all sections have equal area for a circle with $n$ sections <h3>Possible support</h3> The problem <a href="http://nrich.maths.org/2161&part=">An Unusual Shape</a> provides practice in calculating the areas of semicircles. The problem <a href="http://nrich.maths.org/809&part=">Blue and White</a> investigates sequences based on the areas of circles. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Suppose the radius of the smallest semicircle is $x$. What are the radii of the other semicircles? What areas can you work out, in terms of $x$ and $\pi$?</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Herschel, from the European School of Varese, sent us this solution: First, notice that the picture is made of two halves, with semicircles of various sizes in each half of the picture. Each coloured shape is made from a larger semicircle with a smaller one cut out, and the "band" (coloured curvy shape) created this way is joined to one in the other half of the picture, making the wavy shapes. To work out the areas of the various coloured bands, we must work out the area of each half of the shape. For simplicity, we'll take the smallest semicircles (the red and blue ones at either side of the picture) to have an area of 1 unit, so the succesively bigger semicircles have areas of $2^2, 3^2, 4^2 \text{ and }5^2$ (including the cut-out region). Let's take the example of the green band. To work out the area of the top half, we will take away the area of the yellow semicircle (including the cut-out region) from the green semicircle, so the area of the top of the green band is $4^2-3^2 = 16 - 9 = 7$ units of area. Likewise, we work out the area of the bottom half to be $2^2 - 1^2 = 4 - 1 = 3$. The total area of the green shape is $7+3=10$ units (where, if you recall, 1 unit is the area of the top half of the red shape). Now we'll generalise: we'll take a circle where the number of coloured bands is $x$ and we'll pick the $n^{th}$ band from the left (so in the previous example, $x=5$ and our for our green band, $n=4$). So, we need to separately calculate the top and bottom halves. The area of the top half will be $A_1 = n^2 - (n-1)^2$ (like the green area was $4^2 - 3^2$). The area of the bottom half will be $A_2 = (x+1-n)^2 - (x-n)^2$ (that makes the green area $2^2 - 1^2$). Add the two areas together and we get $A = A_1 + A_2 = [n^2 - (n-1)^2] + [(x+1-n)^2 - (x-n)^2]$ Now it's just a case of working through the algebra: Expand the brackets: $A = [n^2 -n^2 +2n -1] + [x^2 +x -xn +x +1 -n -xn -n +n^2 - (x^2 -2xn +n^2)]$ Regrouping the like terms and cancelling out, we end up with: $A = [2n-1] + [x^2 +2x -2xn +1 -2n +n^2 - x^2 +2xn -n^2]$ Cancelling the remaining parts we finally get: $A = 2x$ As you can see, the $n$ terms have been eliminated - all the bands will have an area of $2x$, where $x$ is the number of coloured wavy shapes. This agrees nicely with the area of the green shape we worked out above - the area was 10, and indeed, the number of colours in the example image is 5, so $2x=2 \times 5 = 10$. So, to conclude, the area of each of the "waves" in a shape such as this is $2 \times$ [the number of wavy shapes]$\times$ [the area of the smallest semicircle]. All the waves in a given image will have the same area. Throughout this process, we have defined 1 unit to be the area of the smallest semicircle. We could well have decided to define 1 unit to be the radius of the smallest semicircle; indeed, this would appear to be more obvious a unit to use. However, we would then have had to use the formula $A=\pi r^2$, which would unnecessarily clutter and complicate the already tedious algebra, only to get a similar answer. This answer would in fact be $x \times \pi$ square units, since $2x$ semicircles is $x$ whole circles, and as mentioned, the area of a circle = $\pi r^2$, so a radius of 1 means an area of $\pi$. Regardless of which units we choose, the fact remains that all the bands in a circle have exactly the same area, which is $\frac{1}{x}$ of the area of the complete circle (thus the green band in the example image takes up exactly 1/5 of the total area of the circle, as do the red, orange, yellow and blue bands). </mdoxml> 2 3 0 0 0 1 0 Curvy areas Have a go at creating these images based on circles. What do you notice about the areas of the different sections? 2D Geometry, Shape and Space Straight edge & compass constructions 2D Geometry, Shape and Space Circles Measures and Mensuration Area Using, Applying and Reasoning about Mathematics Making and proving conjectures Information and Communications Technology smartphone ajk44 live for solution Secondary Mapping Document MD Perimeter, Area and Volume US Secondary Mapping Document DisplayCabinet