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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

U in a Pentagon

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Each interior angle of a regular pentagon is $108^\circ$, so $\angle SRQ=108^\circ$. As $SR=QR$, the triangle is isosceles with $\angle RQS=\angle RSQ = 36^\circ$. Similarly, $\angle SRT= \angle STR = 36^\circ$. So $\angle SUR=(180-2\times36)^\circ=108^\circ$. From the symmetry of the figure, $\angle PUR=\angle PUS= (360^\circ - 108^\circ)/2 = 126^\circ$.

This problem is taken from the UKMT Mathematical Challenges.
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