Unit 20: Mixed Geometry

Whole take away hole

Almost every shaded-area puzzle is the same move: work out the whole thing, work out the bit that is missing, and subtract.

The hard part is never the arithmetic. It is noticing which shape is the whole one.

When the shaded region is a strange curve, that is your clue — nobody has a formula for it, so it must be a difference of two shapes that do.

Half an equilateral triangle

Cut an equilateral triangle of side 2 straight down the middle. You get a right-angled triangle with sides 1, sqrt(3) and 2, and angles 30°, 60°, 90°.

That triangle turns up constantly, so it is worth knowing by heart. The short side is always half the hypotenuse, and the other one is sqrt(3) times the short one.

Half a square

Cut a square of side 1 along its diagonal and you get the other famous triangle: sides 1, 1 and sqrt(2), with angles 45°, 45°, 90°.

So in any right-angled triangle with two equal sides, the long side is sqrt(2) times either short one.

Add them, then take the overlap off

Two shapes cover a region between them. Add both areas and the overlap has been counted twice, so subtract it once.

A 6 by 4 rectangle and a 5 by 4 rectangle overlapping in a 2 by 4 patch cover 24 + 20 - 8 = 36.

That is inclusion–exclusion from unit 27, arriving early and wearing a geometric hat.

The bit only one of them covers

Sometimes the question wants the part covered by the first shape only.

That is its own area minus the overlap — nothing more.

So of the 6 by 6 square above, 36 - 9 = 27 is covered by it alone. Drawing the two-circle picture from unit 27 makes this obvious even when the shapes do not.

Find the link between the two

Every "shape inside a shape" problem turns on one sentence connecting them.

Circle inside a square: diameter = side.
Square inside a circle: diagonal = diameter.
Circle inside an equilateral triangle: harder, but still one relationship.

Write that sentence down first. The rest is arithmetic you already have.

The ratio does not depend on the size

What fraction of a square does its inscribed circle cover?

For side s: the square is s^2 and the circle is pi (s/2)^2 = pi s^2/4.

The ratio is pi/4, about 0.785 — and the s^2 cancelled. Every square is 78.5% covered by its circle, whatever its size. That is worth more than any single answer.

Same height means the base decides

Two triangles with the same height have areas in the ratio of their bases. Nothing else matters.

So if a triangle’s base is split 2 : 3, the two pieces have areas in the ratio 2 : 3 as well.

That single fact turns a great many "what fraction is shaded?" problems into a one-line answer.

Similar shapes go by the square

When shapes are similar rather than sharing a height, areas go by the scale factor squared.

Sides in the ratio 2 : 3 means areas in the ratio 4 : 9.

So a small triangle sitting inside a big similar one, at half the size, takes only a quarter of the area — not half, which is the answer everybody expects.

A triangle has three bases

Any side of a triangle can be called the base — you just need the height that goes with it.

So the area can be worked out three different ways, and all three must agree.

That gives a powerful trick: if you know the area and one side, you can find the height onto that side, even when it is nowhere in the picture.

Counting the same thing twice

This is a habit worth keeping well beyond geometry: count one quantity in two different ways, then set the two answers equal.

It gave the height above. It gave the number of diagonals in unit 13. It will give the combinations formula in unit 25.

Whenever a problem looks stuck, ask what you could count twice.