Geometry 1 — triangles, circles and the facts to memorise

Thirteen questions across the smallest Math domain — five to seven questions on the real test, and a short enough list of facts that it is genuinely finishable.

The point of this set is the distinction that decides how much work this domain needs: what the reference sheet in the app gives you, and what it does not. It gives you areas, volumes and the two special right triangles. It does not give you similarity, the circle theorems, the definitions of sine, cosine and tangent, or the equation of a circle — so those are the ones worth knowing cold, and they are what most of these questions are about.

Similarity and scaling. Two similar triangles with sides in the ratio 1:3 have areas in the ratio 1:9, because area is two dimensions and volume is three. Double a sphere's radius and the volume is multiplied by eight. Every wrong answer to a scaling question is one of the other two factors, so knowing which power applies is the whole question.

The other similarity shape is the one with a line drawn parallel to a side: if $DE$ is parallel to $BC$, the small triangle and the whole triangle are similar, and the proportion is $AD\mathrm{/}AB=DE\mathrm{/}BC$. Note $AB$, not $DB$ — setting the proportion up against the wrong length is the standard error here and it is a distractor every time.

Right triangles and trigonometry. Pythagoras, the 3-4-5 and 5-12-13 families worth recognising on sight, and sine, cosine and tangent as ratios of sides. Given $\tan \theta =3\mathrm{/}4$ you can reconstruct the whole triangle — opposite 3, adjacent 4, hypotenuse 5 — and read off any other ratio, which is faster than any calculator.

And the identity that pays for itself: $\sin \theta =\cos (90\mathrm{°}-\theta )$. Sine and cosine of complementary angles are equal. The SAT asks this far more often than its share of the syllabus suggests, and it is a one-second question once you know it.

Circles. Arc length and sector area, in degrees and in radians — $π$ radians is 180°, arc length is $r\theta$ and sector area is $\frac{1}{2}{r}^2\theta$ with $θ$ in radians. And the equation of a circle, $(x-h{)}^2+(y-k{)}^2=r^2$, where the two things that catch people are both here: the sign inside the bracket is the opposite of the coordinate's sign, so a centre at $y=-2$ appears as $(y+2)$; and the right-hand side is the radius squared.

One thing these materials cannot do. The real test draws figures. Everything here is described in words, with every length and angle given — which is a fair test of the reasoning and an unfair one of your ability to read a diagram. Draw each of these yourself before you answer, because drawing and labelling is the habit that fixes most errors in this domain, and you will want it on the day.

A closing note on time: this is 15% of the Math section. If your Algebra or Advanced Math is shaky, those are worth five times as much between them.

• Apply the scaling rule: multiply a linear dimension by k and area goes up by k², volume by k³
• Set up the correct proportion between similar triangles, using the whole side and not the part
• Reconstruct a right triangle from one trigonometric ratio and read off any other
• Use sin θ = cos(90° − θ) to answer a complement question instantly
• Work with the 30-60-90 and 45-45-90 triangles without deriving them
• Compute arc length and sector area in both degrees and radians
• Write the equation of a circle from its centre and radius, with the signs and the squaring right
• Know which facts the test's reference sheet supplies and which you have to bring yourself

In the xy-plane, a circle has its centre at (3, −2) and a radius of 5. Which equation represents this circle? — The standard form is (x − h)² + (y − k)² = r². Here h = 3, so the first bracket is (x − 3); k = −2, so the second is (y − (−2)), which is (y + 2); and the right-hand side is 5² = 25. Copying the coordinates into the brackets with their own signs describes the reflected circle, and putting 5 rather than 25 on the right describes a circle of radius √5.