Problem 1

Thanks to Rafael for this suggestion! This comes from the 2023 UKMT IMC Q21.

A right triangle has a semicircles with radii of 3 and 4 on its legs, as shown. The semicircles intersect along the hypotenuse. What is the shaded area?

Problem 2

Thanks to McKenzie from the UK for the suggestion! This problem is adapted from a 1987 STEP paper, which is a test used for admission into various universities including Cambridge and the University of Warwick.

Until the 1500s, Europe mostly used Roman numerals and tended to use strange fractions for units, being very slow to adopt the decimal numeral system that originated in India. The British pound was a legacy of the old system and there were 12 pence to a shilling and 20 shillings to a pound, making for 240 pence to a pound. In 1971, the British pound was officially decimalized to 100 pence per pound, necessitating various new coins like 1p, 2p, 5p, 10p, and 50p.

The new 50 pence coin was a challenge to design. For a coin to be successful, it should be recognizable by touch, sight, and weight. The coin also needed to roll like a circle because many parking meters and vending machine depended on that. The technical designers ultimately consulted mathematical textbooks and experimented with many shapes until they found a solution: a 7-sided Reuleaux polygon.

So now to the question. The 50 pence coin is made up of 7 equal arcs, where each arc is centered at the vertex opposite the midpoint of the arc.

Supposing the radius of each circle is 1, show that the area of the coin is:

π/2 – (7/2) tan(π/14)

As usual, watch the video for a solution.

Cambridge Entrance Exam – Area Of 50 Pence Coin

Or keep reading.
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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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Answer To Find Areas UKMT And 50 Pence Coin

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Problem 1

Each triangle leg is twice its semicircles radius. So the sides are 6 and 8. Now we can calculate the shaded area like this:

The shaded area is precisely the areas of the two semicircles minus the area of the right triangle.

shaded area
= a + c + e

two semicircles – triangle
a + b + c + c + d + e – (b + c + d)
= a + c + e

two semicircles – triangle
0.5π(3)² + 0.5π(4)² – 6(8)/2
= 25π/2 – 24
≈ 15.270

Problem 2

Construct a pizza “slice” from the center of the coin to the circular arc of one side. If we figure out the area of this slice, we can then multiply by 7 to find the area of the coin.

Let the slice have a central angle equal to a. The two triangles adjacent are congruent isosceles triangles, so let the vertex angles be b and the other angles be c. A single slice is 1/7 of the coin, so:

a = 2π/7

Similarly angles a and 2b span a circle, so

2b + a = 2π
2b + 2π/7 = 2π
b + 6π/7

The three angles of a triangle sum to π so we have:

2c + b = π
2c + 6π/7 = π
c + π/14

Construct a perpendicular from the center to the base of 1 in an isosceles triangle.

We now have:

slice
= circular sector area – 2(triangles)
= (1/2)(radius)²(angle) – 2((1/2) × base × height)
= (1/2)1²(2c) – 2((1/2) × 1 × height))
= (1/2)1²(π/7) – 2((1/2) × 1 × 1/2 × tan c)
= π/14 – (1/2)tan(π/14)

Multiplying this by 7 gives the area of the entire coin.

area coin = π/2 – (7/2)tan (π/14)

References

Problem 1 UKMT IMC, 2023, Q21
https://ukmt.org.uk/wp-content/uploads/2023/08/IMC_2023_Paper-1.pdf

Problem 2 references

Stephen Siklos, Advanced Problems in Mathematics: Preparing for University. Cambridge, UK: Open Book Publishers, 2015. CC BY 4.0
https://books.openbookpublishers.com/10.11647/obp.0075/Chapters/P6.html

50 Pence coin
https://en.wikipedia.org/wiki/Fifty_pence_(British_coin)

Bristol Post
https://www.bristolpost.co.uk/news/bristol-news/history-50p-bristol-invention-pocket-3595865

Reuleaux polygon
https://en.wikipedia.org/wiki/Reuleaux_polygon

CC By 3.0
https://en.wikipedia.org/wiki/Reuleaux_triangle#/media/File:Reuleaux_polygons.svg

How to pronounce Franz Reuleaux in German
https://youtu.be/jpoDzLhNeto?si=u1WU_O9UxfD7R4nC

Science behind 50p shape
https://www.royalmint.com/discover/the-50-year-anniversary-of-the-50-pence-piece/the-science-behind-the-shape-of-the-50p/

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