Weekly Problem 51

7146 /www/nrich/html/content/id/7146/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The diagram shows a triangle and three circles whose centres are at the vertices of the triangle. The area of the triangle is $80 cm^2$ and each of the circles has radius $2cm$. What is the area, in $cm^2$, of the shaded area? <mdo:image alt="" height="209" src="51%202010.PNG" width="405"></mdo:image> If you liked this problem, <a href="http://nrich.maths.org/637&part=">here is an NRICH task</a> which challenges you to use similar mathematical ideas. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The three angles of the triangle add to $180^{\circ}$, so the combined area of the three sectors of the circles that are inside the triangle add up to half a circle with area: $\frac{1}{2} \times \pi \times \times 2^2 = \frac {4\pi}{2}= 2\pi$. So the grey area is $(80- 2\pi) cm^2$. <mdo:image width="405" height="209" alt="" src="51%202010.PNG"></mdo:image> </mdoxml> 2 3 0 0 1 1 0 Weekly Problem 51 - 2010 Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area? 2D Geometry, Shape and Space Shape, space & measures - generally 2D Geometry, Shape and Space Angle properties of shapes 2D Geometry, Shape and Space Circles Secondary Mapping Document Area and volume US