Downside 1

I learn this was a PSAT query, which is a standardized take a look at given to US college students aged about 15 or 16 who almost certainly haven’t taken a course in trigonometry.

If every of the three semicircles beneath has an space equal to 72, and is the midpoint of , what’s the space of the shaded area?

Downside 2

That is tailored from the 2020 AMC 10B, downside 14.

An everyday hexagon with facet size 2 has semicircles constructed in its inside of every facet. What’s the shaded space contained in the hexagon not lined by the semicircles?

Downside 3

Because of Andrei for the suggestion!

A sq. has a facet size of seven. On every nook, 1 / 4 circle with radius 3 is constructed with the nook as its middle. And on the midpoint of every facet, a semicircle with radius 2 is constructed with the midpoint as its middle. What’s the distinction between the orange shaded areas and the blue shaded areas?

As standard, watch the video for options.

3 Puzzles About Shaded Areas And Circles

Or preserve studying.
.
.

“All might be properly should you use your thoughts in your selections, and thoughts solely your selections.” Since 2007, I’ve devoted my life to sharing the enjoyment of sport idea and arithmetic. MindYourDecisions now has over 1,000 free articles with no advertisements due to neighborhood assist! Assist out and get early entry to posts with a pledge on Patreon.

.
.

.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
P
U
Z
Z
L
E
.
.
.
.
Reply To PSAT Query Three Semicircles

(Just about all posts are transcribed shortly after I make the movies for them–please let me know if there are any typos/errors and I’ll appropriate them, thanks).

Thanks Erik for alerting me of typos!

Downside 1

There’s a remarkably simple technique to calculate the world!

Since Q is the midpoint of the diameter PR, it’s the middle of the left semicircle. Because the three semicircles have the identical space, all three are congruent. The 2 semicircles on the appropriate go via Q, so their intersections with semicircle Q might be congruent and have the identical space. Thus the 2 inexperienced areas are equal in space and can cancel out within the calculation.

Thus the world of the claw is the same as the world of a 20 diploma sector. Since every semicircle has 180 levels, and an space of 72, the reply is:

space claw
= space 20 diploma sector
= 72(20/180)
= 8

That’s it!

Downside 2

Divide the hexagon right into a grid of equilateral triangles of facet size 1, as proven. The shaded space is then 6 instances the world of a single blue slice.

However every blue slice is the same as the world of two equilateral triangles minus a round sector with a central angle equal to 1. Each the triangles have a facet of 1 and the circle has a radius of 1.

A single blue slice has space equal to:

2(s²√3)/4 – πr²(60/360)
= 2(√3)/4 – π/6
= (3√3 – π)/6

This space instances 6 provides the shaded space as 3√3 – π ≈ 2.05.

Downside 3

At first I had tried to calculate the world of the overlapping area. However it is a far too difficult computation. So I assumed there should be one other technique.

Let every of the orange areas be equal to a, the overlapping areas be b, and the blue areas be c.

We wish to calculate:

4a – 4c

However we do know:

a + 2b = space quarter circle
c + 2b = space semicircle

Multiplying every equation by 4 and subtracting provides:

4a – 4c
= 4(space quarter circle) – 4(space semicircle)
= 4(π3²/4) – 4(π2²/2)
= π

And like magic we now have discovered the reply!

I obtained a pleasant electronic mail from Atanas who has developed code in 3 languages if you wish to simulate this end result your self! Test it out on Github: https://github.com/batinkov/simulations/tree/principal/2_puzzles_about_shuffling_cards

Sources

Downside 1
Math StackExchange
https://math.stackexchange.com/questions/2334447/area-of-intersection-of-three-circles

Downside 2
AoPS
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10B_Problems/Problem_14

MY BOOKS

If you are going to buy via these hyperlinks, I could also be compensated for purchases made on Amazon. As an Amazon Affiliate I earn from qualifying purchases. This doesn’t have an effect on the worth you pay.

E book rankings are from January 2023.

(US and worldwide hyperlinks)
https://mindyourdecisions.com/weblog/my-books

Thoughts Your Selections is a compilation of 5 books:

(1) The Pleasure of Recreation Concept: An Introduction to Strategic Pondering
(2) 40 Paradoxes in Logic, Likelihood, and Recreation Concept
(3) The Irrationality Phantasm: How To Make Good Selections And Overcome Bias
(4) The Finest Psychological Math Tips
(5) Multiply Numbers By Drawing Strains

The Pleasure of Recreation Concept exhibits how you should use math to out-think your competitors. (rated 4.3/5 stars on 290 evaluations)

40 Paradoxes in Logic, Likelihood, and Recreation Concept comprises thought-provoking and counter-intuitive outcomes. (rated 4.2/5 stars on 54 evaluations)

The Irrationality Phantasm: How To Make Good Selections And Overcome Bias is a handbook that explains the various methods we’re biased about decision-making and provides methods to make sensible selections. (rated 4.1/5 stars on 33 evaluations)

The Finest Psychological Math Tips teaches how one can seem like a math genius by fixing issues in your head (rated 4.3/5 stars on 116 evaluations)

Multiply Numbers By Drawing Strains This e book is a reference information for my video that has over 1 million views on a geometrical technique to multiply numbers. (rated 4.4/5 stars on 37 evaluations)

Thoughts Your Puzzles is a set of the three “Math Puzzles” books, volumes 1, 2, and three. The puzzles subjects embody the mathematical topics together with geometry, likelihood, logic, and sport idea.

Math Puzzles Quantity 1 options basic mind teasers and riddles with full options for issues in counting, geometry, likelihood, and sport idea. Quantity 1 is rated 4.4/5 stars on 112 evaluations.

Math Puzzles Quantity 2 is a sequel e book with extra nice issues. (rated 4.2/5 stars on 33 evaluations)

Math Puzzles Quantity 3 is the third within the collection. (rated 4.2/5 stars on 29 evaluations)

KINDLE UNLIMITED

Academics and college students all over the world usually electronic mail me concerning the books. Since training can have such a huge effect, I attempt to make the ebooks out there as broadly as attainable at as low a worth as attainable.

Presently you’ll be able to learn most of my ebooks via Amazon’s “Kindle Limitless” program. Included within the subscription you’re going to get entry to thousands and thousands of ebooks. You do not want a Kindle machine: you’ll be able to set up the Kindle app on any smartphone/pill/laptop/and many others. I’ve compiled hyperlinks to applications in some nations beneath. Please examine your native Amazon web site for availability and program phrases.

US, record of my books (US)
UK, record of my books (UK)
Canada, e book outcomes (CA)
Germany, record of my books (DE)
France, record of my books (FR)
India, record of my books (IN)
Australia, e book outcomes (AU)
Italy, record of my books (IT)
Spain, record of my books (ES)
Japan, record of my books (JP)
Brazil, e book outcomes (BR)
Mexico, e book outcomes (MX)

MERCHANDISE

Seize a mug, tshirt, and extra on the official website for merchandise: Thoughts Your Selections at Teespring.