The mighty Australian 50c coin

Geometry Level pending

The twelve-sided Australian fifty-cent coin is the third-highest denomination coin of the Australian dollar and the largest in terms of size in circulation. Interestingly, it has a dodecagonal shape and its diameter is roughly $32\ \rm mm$ .

Assuming that its diameter is exactly $32\ \rm mm$ , find the area of a face of the coin.

768 \ \rm mm^2

868\ \rm mm^2

668\ \rm mm^2

968\ \rm mm^2

2 solutions

Hongqi Wang
Apr 4, 2021

split the dodecagon into 6 same "diamonds", then 2 diagonals are perpendicular and both are 16mm of each diamond. So the area of each diamond is $\dfrac 12 \times16^2 = 128$ and the area of the dodecagon is $6 \times 128 = 768$

Ethan Mandelez
Mar 31, 2021

Split the dodecagon into twelve isosceles triangles, with $2$ equal sides of $16$ mm and the angle between them is $30$ degrees $(\frac{360}{12} = 30)$ . The area of the face can be calculated as follows:

$A = 12 \times \dfrac {1} {2} \times 16^{2} \times sin(30)$

$A = 768$ mm squared.

The mighty Australian 50c coin

2 solutions

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