This puzzle looked daunting at first but turned out to be easier than it looked. The tweet solution may be a bit too terse to be followed easily, though.

I’ve labeled five points, and the sizes of the three squares: The largest square, touching the circle at point , has size ; the medium square, touching at , has size , and the smallest square, touching at , has size . I added a copy of the smallest square, touching the circle at , and drew in the chord .

The blue triangle is right, with legs and , so its area is . The blue triangle is also right, with legs and ; its area is .

By the intersecting chords theorem, the product of lengths and is equal to the product of and . But , so ; that means .

So the red triangle area is which is four times the area of the blue triangle. That area is 5, so the area of the red triangle is 20.

Sometimes, though, you just need to have a picture:

It’s obvious the red triangles’ height is but the blue triangles were less obvious to me. I noted that if you drop a perpendicular from the top vertex of the bottom blue triangle to the bottom line, the resulting right triangle has legs in a ratio of which sum to , so the height is . Another approach another poster mentioned is to note that a diagonal of the square is divided into three equal parts by the slanting lines and so the vertical projection of one of those parts has length .

Either way, the red and blue triangles both have base = so a red triangle has area , a blue triangle has area , and four of each add up to . Then the octagon’s area is the square’s () minus the triangles’ () which equals .

]]>

**20.
The Tiger-gon**

**What fraction is shaded?** The hexagon is regular, with equally spaced dots around its perimeter.

**Spoiler!**

Divide the hexagon into triangles:

The top triangle shares a base with triangle , and both have their third vertex at the same perpendicular distance to that base, so the top triangle has the same area as triangle .

The middle triangle shares a base with triangle , and its third vertex has four times the perpendicular distance to that base, so the middle triangle has four times the area of triangle .

Similarly the bottom triangle has three times the area of one of the equilateral triangles.

The shaded area is then equal in area to eight equilateral triangles, and the hexagon consists of 24 such triangles, so the shaded area is of the hexagon.

]]>

**19.
Fly the Flags**

Squares of the same colour are the same size. **What’s the total shaded area?**

**Spoiler!**

The blue squares touch at the center and their edges are at 45° from the big squares’, so they lie on diagonals which we might as well draw:

at which point it’s obvious the red squares are the size, the area of the big square. Of course the diagonals of the blue squares are half the sides of the big square so they are each the area. Total area is .

]]>**18.
The Tumble Dryer**

**What fraction of the big square is shaded?**

Spoiler!

The squares look chaotic but three of them have parallel edges and two contact the circle at two corners; that means the perpendicular bisectors of the edges connecting those corners intersect the circle’s center, and since the squares’ edges are parallel the centers of the circle and those two squares must be colinear. Therefore, if these are unit squares, the contact points are the corners of a 3 by 1 rectangle inscribed inside the circle.

Then the diameter of the circle is the diagonal of that rectangle, whose length is . Which also is the length of a side of the big square.

The big square’s area is and the squares add up to , so the answer is .

]]>**17.
Just One Fact**

**What’s the area of the square?**

**Spoiler!**

In contrast to the last one, this one borders on trivial. Add some lines:

AC is a diameter, so triangle ADC is right and triangles ABD and DBC are similar. AB is twice BD so BD is twice BC which is 1, so BD is 2, AB is 4, and the square’s area is 16.

]]>**16.
Going, Going, ‘gon**

Six identical squares and a smaller rectangle are fitted into this regular hexagon. **What fraction of the hexagon do they cover?**

**Spoiler!**

Is there any way to do this besides the hard way? I looked for a dissection solution but came up with nothing.

Look at just one twelfth of the hexagon, a right triangle, :

and are two sides of a square. and are two line segments I added. The central rectangle isn’t shown here because it differs in the different triangles around the hexagon.

If is defined to be length 1 and then and . But and , so

is the correct solution

Then the height of the triangle is and its area is , and the hexagon’s full area is .

The sides of the squares are , so the six squares have area .

The central rectangle has area .

Then the squares plus rectangle are , which is of the hexagon’s area.

]]>

**15.
Jewel Cutters**

Four equilateral triangles are arranged around a square which has area 12. **What’s the shaded area?**

**Spoiler!**

Draw a couple of diagonal lines:

**These divide the four triangles into four pairs of right triangles. Four such right triangles can be arranged around the inside of the center square:****and you’re left with a tilted square in red. What’s its area?**

Well, clearly a side of the red square is equal to the difference between the long and short legs of a right triangle.

But also clearly, half the diagonal of the original central square is also the difference between the long and short legs of a right triangle!

That means the area of the red square is half the area of the original central square, meaning four right triangles also are half that area, or all eight right triangles are equal in area to the central square: 12.

]]>**14.
Green vs. Blue**

Is more of this design green or blue (and by how much)?

Spoiler!

I was hoping I’d find a clever way to make the answer drop right out, but nope, I ended up mucking about with lengths of sides of similar triangles. Right triangles fortunately, and you don’t need any hypotenuses or angles, so that’s something.

Add some lines and label vertices:

and I’ve shown some congruent angles.

No line segment lengths are given, so let’s use symbols: define , the last coming from the fact triangles ADG and FHG have to be congruent. Define .

What we have to find is the difference between the blue area, which is , and the green area, which is . But and are congruent, so we can cross them off and we just need .

is similar to , and , so . The area of is .

is , so the area of triangle is .

so the area of triangle is . Then the area of triangle is .

So

.

Independent of ! Mysteriously. I wish I could find a clever way to see that.

Anyway, the area of triangle is . Mysteriously the same as! So the answer is, the blue area is larger by 5.

]]>**13.**

**Isosceles I Saw**

All 4 triangles are isosceles. **What’s the angle?**

Spoiler!

I have a feeling there is a simpler solution but this is all I’ve got…

The four isosceles triangles are: BCE, CEA, ADC, and BAC.

If the base of triangle BCE is and its two other sides are , the base of triangle BAC is and the two other sides are . They are similar, so , with solution . is the cosine of angle CBE; that angle then is .

As a consequence of the inscribed angle theorem, angles CBE and ADC are supplementary, so the latter is . I didn’t use the fact ADC is isosceles; in fact as long as D is on the circle between A and C, the angle remains the same.

]]>Spike in the Hive

Two of the regular hexagons are identical; the third has area 10. What’s the area of the red triangle?

Spoiler!

First, what’s the area of the big hexagons?

There’s a little equilateral triangle in the middle between the hexagons, and its equilateralness means a big hex side is equal to a little hex side plus a triangle side (, from the upper big hex), but also a big hex side is two little hex sides minus a triangle side (, from the lower big hex), and that means and . So the area of a big hex is .

Now, the red triangle area: It’s half base times height, so it’s twice the area of a triangle with the same base and half the height, or twice the area of the same triangle sheared parallel to its base:

And that last triangle clearly is 1/3 the area of the hexagon it’s inscribed in. So

]]>