Foundations: Geometry and Trigonometry

Geometry and Trigonometry is the smallest domain on the SAT Math section, five to seven questions, and it is the domain where a reference sheet is one click away on every question. That combination tells you how to study it: learn the handful of facts that are not on the sheet, learn to see which formula a figure is asking for, and do not spend hours on theorems the test never touches.

The questions come in three families. Angle questions give you a figure with parallel lines or a triangle and ask for a missing angle; they are solved with four facts. Measurement questions ask for a length, area or volume, and the formulas are on the sheet, so the skill is choosing the right one and the right numbers to put in it. Right-triangle questions ask for a side or a trigonometric ratio, which is the Pythagorean theorem plus sine, cosine and tangent.

This lesson covers all three families at the level of a direct question. The Medium lesson adds similar triangles, special right triangles, arcs and sectors, and the equation of a circle; the Advanced lesson adds radians, trigonometric identities, circle theorems and the effect of scaling on area and volume.

Angles are measured in degrees, and a few facts generate all the angle questions at this level. A straight line is 180 degrees, so two angles that together form a straight line, called supplementary or a linear pair, add to 180. When two lines cross, the angles opposite each other, called vertical angles, are equal. A full turn around a point is 360 degrees. Two angles that add to 90 are complementary. When a line crosses two parallel lines, eight angles are formed, but only two different measures: every acute angle equals every other acute angle, every obtuse angle equals every other obtuse angle, and an acute plus an obtuse is 180. The named pairs, corresponding, alternate interior and alternate exterior, are all equal; co-interior angles, on the same side between the parallels, are supplementary.

The angles inside any triangle sum to 180 degrees. An exterior angle, formed by extending one side, equals the sum of the two interior angles it is not adjacent to. An isosceles triangle has two equal sides and the angles opposite them are equal; an equilateral triangle has three 60 degree angles. A right triangle has one 90 degree angle, so its other two angles are complementary. The interior angles of any polygon with n sides sum to 180(n - 2) degrees, so a quadrilateral's angles sum to 360 and a hexagon's to 720.

Perimeter is the distance around a figure: add the side lengths. Area measures the surface inside, in square units. The reference sheet gives the area of a rectangle as length times width, a triangle as one half the base times the height, and a circle as π r², with circumference 2 π r. The height of a triangle is the perpendicular distance from the base to the opposite vertex, not a slanted side, which matters in obtuse triangles where the height falls outside the figure. Not on the sheet: a parallelogram has area base times height, a trapezoid has area one half the sum of the parallel sides times the height, and a square with side s has area s² and diagonal s √2.

Volume measures space inside a solid, in cubic units, and the sheet gives all the common ones: a rectangular box is length times width times height, a cylinder is π r² h, a sphere is four thirds π r³, a cone is one third π r² h, and a pyramid is one third the base area times the height. The pattern to remember is that a cone and pyramid are one third of the cylinder and box with the same base and height. Surface area is not on the sheet and is rarely tested; when it is, add the areas of every face.

In a right triangle the side opposite the right angle is the hypotenuse, always the longest side, and the other two sides are the legs. The Pythagorean theorem says the legs a and b and the hypotenuse c satisfy a² + b² = c². Given any two sides, the third follows: add the squares of the legs to find the hypotenuse, or subtract the square of the known leg from the square of the hypotenuse to find the other leg. Whole-number solutions, the Pythagorean triples 3-4-5, 5-12-13, 8-15-17 and 7-24-25, along with their multiples, appear again and again because they keep answers whole; recognizing one saves the arithmetic. The distance between two points in the coordinate plane is the theorem in disguise: the horizontal and vertical changes are the legs.

Trigonometry on the SAT is, at this level, three ratios in a right triangle. Choose one acute angle. The hypotenuse is fixed; the leg across from the chosen angle is opposite and the leg touching it is adjacent. Then sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent, remembered as SOH CAH TOA. In a 3-4-5 triangle, for the angle opposite the side of length 3, sine is 3/5, cosine is 4/5 and tangent is 3/4; for the other acute angle the sine and cosine swap. That swap is a fact the test likes: the sine of one acute angle equals the cosine of the other, because the two angles are complementary. Given a ratio and one side, you can find another side by setting up the ratio as a proportion; given two sides, you can write any ratio directly.

The two special right triangles are on the reference sheet. A 45-45-90 triangle has legs of equal length x and hypotenuse x √2; a 30-60-90 triangle has a short leg x opposite the 30 degree angle, a long leg x √3 opposite the 60 degree angle, and a hypotenuse 2x. The Medium lesson uses them heavily; here it is enough to recognize them and to know the sheet has the ratios.

Using the diameter as the radius. Halve it first, then square; a circle of diameter 10 has area 25 π, not 100 π.

Calling a slanted side the height of a triangle. Height is the perpendicular distance to the base.

Swapping opposite and adjacent because you named them relative to the wrong angle. Put your finger on the angle in the question.

Adding the legs of a right triangle instead of adding their squares, or adding squares when you are looking for a leg rather than the hypotenuse.

Assuming an angle is a right angle, or two lines are parallel, because the figure looks that way. The SAT marks right angles with a square and states parallelism in words; if neither is there, it is not given.

Answering in the wrong unit: area in square units, volume in cubic units, and a mixed centimeters-and-meters figure needs one unit before any formula.

Spending time deriving a formula that is on the reference sheet, or trusting memory for one that is.