Semicircular design
Weekly Problem 9 - 2016
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?
Problem
Image
![Semicircular Design Semicircular Design](/sites/default/files/styles/large/public/thumbnails/content-id-11615-Weekly%2525202016%252520-%2525209%252520Diagram.png?itok=7Q6dQJgl)
The diagram to the right shows a logo made from semi-circular arcs, each with a radius of $2$cm, $4$cm or $8$cm.
What fraction of the logo is shaded?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $\frac14$
Using areas of semicircles
Shaded part:
Image
![Semicircular Design Semicircular Design](/sites/default/files/styles/large/public/thumbnails/content-id-11615-semicircular%252520design%2525201%252520corr.png?itok=NR0bamQ2)
Image
![Semicircular Design Semicircular Design](/sites/default/files/styles/large/public/thumbnails/content-id-11615-semicircular%252520design%2525202%252520corr.png?itok=RG6JrrG3)
Semicircle radius $4$ cm has area $\frac12\pi\times4^2 = 8\pi$ cm$^2$
Whole design:
Image
![Semicircular Design Semicircular Design](/sites/default/files/styles/large/public/thumbnails/content-id-11615-semicircular%252520design%2525203%252520corr.png?itok=DXP5rGAQ)
Image
![Semicircular Design Semicircular Design](/sites/default/files/styles/large/public/thumbnails/content-id-11615-semicircular%252520design%2525204%252520corr.png?itok=HHD9WMZM)
Semicircle radius $8$ cm has area $\frac12\pi\times8^2 = 32\pi$ cm$^2$
$\frac{8\pi}{32\pi} = \frac{1}{4}$ of the shape is shaded.
Using scale factors
Shaded part:
Image
![Semicircular Design Semicircular Design](/sites/default/files/styles/large/public/thumbnails/content-id-11615-semicircular%252520design%2525201%252520corr.png?itok=NR0bamQ2)
Image
![Semicircular Design Semicircular Design](/sites/default/files/styles/large/public/thumbnails/content-id-11615-semicircular%252520design%252520meth%2525202%2525201.png?itok=sbcw0LLb)
Whole design:
Image
![Semicircular Design Semicircular Design](/sites/default/files/styles/large/public/thumbnails/content-id-11615-semicircular%252520design%252520meth%2525202%2525202.png?itok=4hjzefb_)
Shaded part is similar to whole design, length scale factor 2
Length scale factor 2 $\Rightarrow$ area scale factor 4
$\frac14$ of the shape is shaded