Tag Archives: plane geometry

Two and a Half Circles

Here is a problem from the 2022 Math Calendar.

“Two small circles of radius 4 are inscribed in a large semicircle as shown.  Find the radius of the large semicircle.”

As before, recall that all the answers are integer days of the month.

As seemed to be implied by the original Math Calendar diagram, I made explicit that the upper circle was tangent to the midpoint of the chord.  Otherwise, the problem is insufficiently constrained.

See Two and a Half Circles

A Divine Language

I have just finished reading a most remarkable book by Alec Wilkinson, called A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age.  I had read an essay of his in the New Yorker that turned out to be essentially excerpts from the book.  I was so impressed with his descriptions of mathematics and intrigued by the premise of a mature adult in his 60s revisiting the nightmare of his high school experience with mathematics that I was eager to see if the book was as good as the essay.  It was, and more.

The book is difficult to categorize—it is not primarily a history of mathematics, as suggested by Amazon. But it is fascinating on several levels.  There is the issue of a mature perspective revisiting a period of one’s youth; the challenges of teaching a novice mathematics, especially a novice who has a strong antagonism for the subject; and insights into why someone would want to learn a subject that can be of no “use” to them in life, especially their later years.

Wilkinson has a strong philosophical urge; he wanted to understand the role of mathematics in human knowledge and the perspective it brought to life.  He was constantly asking the big questions:  is mathematics discovered or invented, what is the balance between nature and nurture, why does mathematics seem to describe the world so well,  what is the link between memorization and understanding, how do you come to understand anything?

See A Divine Language

Missing Interval Puzzle

Henk Reuling posted a deceptively simple-looking geometric problem on Twitter.

“I found this old one cleaning up my ‘downloads’ [source unknown] I haven’t been able to solve it, so help!

According to the given information in the figure, what is the length of the missing interval on the diagonal of the square?”

See the Missing Interval Puzzle.

15 Degree Triangle Puzzle

This math problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) is a bit more challenging.

“In the diagram, AB represents the diameter, C lies on the circumference of the circle, and you are given that

(Area of Circle) / (Area of Triangle) = 2π.

Prove that the two smaller angles in the triangle are exactly 15° and 75° respectively.”

See the 15 Degree Triangle Puzzle

Turning Wheels Puzzle

This is a thoughtful little problem from Posamentier’s and Lehmann’s Mathematical Curiosities.

“We have nine wheels touching each other with diameters successively increasing by 1 cm. Beginning with 1 cm as the smallest circle, and 9 cm for the largest circle, how many degrees does the largest circle turn when the smallest circle turns by 90°?”

See the Turning Wheels Puzzle

Playing with Triangles

Here is another elegant Quantum math magazine Brainteaser from the imaginative V. Proizvolov.

“Two isosceles right triangles are placed one on the other so that the vertices of each of their right angles lie on the hypotenuse of the other triangle (see the figure at left). Their other four vertices form a quadrilateral. Prove that its area is divided in half by the segment joining the right angles. (V. Proizvolov)”

See Playing with Triangles

Octagonal Area Problem

Here is another problem from the Polish Mathematical Olympiads published in 1960.

“95. In a parallelogram of given area S each vertex has been connected with the mid-points of the opposite two sides.  In this manner the parallelogram has been cut into parts, one of them being an octagon.  Find the area of that octagon.”

See the Octagonal Area Problem