SAT · SAT Math domains
Additional Topics in Math
Angle relationships, including parallel lines cut by a transversal.
Triangle angle relationships and properties.
Area and perimeter of plane figures.
Right triangles, the Pythagorean theorem and distance.
Circumference, area, arcs, sectors and circle equations.
Similarity, congruence and scale factor.
Volume of prisms, cylinders, cones, pyramids and spheres.
Reasoning from given information: conditionals, counterexamples and justified steps.
Angle relationships when parallel lines are cut by a transversal.
Midpoint, distance and slope in the coordinate plane; classifying figures from coordinates.
Congruence criteria for triangles and corresponding parts.
Polygon angle sums and the properties of parallelograms, rectangles, rhombi, squares, trapezoids and kites.
Central and inscribed angles, chords, tangents and secants.
The equation of a circle in the coordinate plane, including completing the square.
The 45-45-90 and 30-60-90 triangles and exact side lengths.
Scale factors of similar figures and solids; area and volume of composite figures.
Right-triangle trigonometry: sine, cosine and tangent ratios.
Radian measure and conversion between radians and degrees.
Trigonometric values of angles beyond the right triangle.
A note on this mapping: The SAT reaches the unit circle only lightly — enough for radian measure and the sine/cosine of a standard angle, not the full precalculus treatment.
The Pythagorean identity and complementary-angle relationships.
A note on this mapping: Identities appear rarely on the SAT, and almost always as sin²+cos²=1 or sin(x) = cos(90° − x).
Arithmetic with complex numbers and the quadratics whose solutions are not real.
Amplitude, period and midline of a sine or cosine graph.
Angles with a given sine or cosine, read off the unit circle.
Exact trigonometric values and identities.
A note on this mapping: The SAT tests the Pythagorean identity and the unit circle; the half-angle and product formulas appear only as exact-value reasoning.
Finding an angle from a trigonometric ratio.
Complex numbers, their modulus and their arithmetic.
A note on this mapping: The SAT tests complex arithmetic; polar form and De Moivre are beyond it.
The 26 skills that cover it
- AnglesMeasure and classify angles, and use complementary, supplementary, vertical and parallel-line relationships to find unknown angles.Read the lesson →
- TrianglesClassify triangles, use the 180-degree angle sum, and apply the triangle inequality and exterior angle theorem.Read the lesson →
- Perimeter & AreaCompute perimeter and area for rectangles, triangles, parallelograms, trapezoids and composite figures.Read the lesson →
- Pythagorean TheoremRelate the three sides of a right triangle with a^2 + b^2 = c^2, and use it to find distances.Read the lesson →
- CirclesUse radius, diameter, circumference and area, and work with arcs and sectors.Read the lesson →
- Similar TrianglesRecognise similar figures, write proportions between corresponding sides, and use them to find unknown lengths.Read the lesson →
- Volume & Surface AreaCompute volume and surface area of prisms, cylinders, pyramids, cones and spheres.Read the lesson →
- Logic & ProofRead a conditional statement, write its converse, inverse and contrapositive, test a claim with a counterexample, and give the reason for every step of a proof.Read the lesson →
- Parallel Lines & TransversalsName the angle pairs a transversal makes with two lines, use them to find unknown angles when the lines are parallel, and use them backwards to prove that two lines are parallel.Read the lesson →
- Coordinate GeometryUse the distance, midpoint and slope formulas to measure and classify figures drawn on the coordinate plane, and to prove that lines are parallel or perpendicular.Read the lesson →
- Triangle CongruenceDecide when two triangles are congruent from the SSS, SAS, ASA, AAS and HL criteria, match corresponding parts, and supply the missing statement or reason in a congruence proof.Read the lesson →
- Polygons & QuadrilateralsFind the angle sums of any polygon, use the properties of parallelograms, rectangles, rhombi, squares, trapezoids and kites, and classify a quadrilateral from what is known about it.Read the lesson →
- Angles & Segments in CirclesRelate the angles and segments inside and around a circle: central and inscribed angles, chords, tangents and secants, and the products of the segments they cut.Read the lesson →
- Equations of CirclesWrite the equation of a circle from its centre and radius, read the centre and radius back from an equation, complete the square to find them, and test whether a point lies on the circle.Read the lesson →
- Special Right TrianglesUse the 45-45-90 and 30-60-90 side ratios to find exact side lengths written with radicals, and recognise the two triangles inside squares, equilateral triangles and hexagons.Read the lesson →
- Similar Solids & Composite FiguresScale the area and volume of similar figures by the square and cube of the scale factor, find the area of regular polygons, and measure composite figures and cross-sections of solids.Read the lesson →
- Right Triangle TrigonometryUse sine, cosine and tangent as ratios of sides in a right triangle to find unknown sides and angles.Read the lesson →
- Radians & DegreesMeasure angles by arc length instead of by 360ths of a turn, and convert between radians and degrees.Read the lesson →
- The Unit CircleExtend sine and cosine to all angles using coordinates on the unit circle, and read exact values off it.Read the lesson →
- Trigonometric IdentitiesUse the Pythagorean and quotient identities to simplify expressions and verify equations.Read the lesson →
- Complex NumbersExtend the number system with i = sqrt(-1): simplify powers of i, add, subtract, multiply and divide complex numbers, and solve the quadratics whose discriminant is negative.Read the lesson →
- Graphs of Sine & CosineUnroll the unit circle into a wave: read the amplitude, period, midline and phase shift of y = a sin(b(x - h)) + k off its equation and its graph, and write the equation of a wave you are shown.Read the lesson →
- Trigonometric EquationsFind every angle that makes a trigonometric equation true: isolate the ratio, read the reference angle off the unit circle, place it in the right quadrants, and use an identity or a substitution when the equation is quadratic.Read the lesson →
- Half-Angle & Product IdentitiesFinish the identity toolkit: the half-angle formulas and their signs, the power-reducing formulas, product-to-sum and sum-to-product, tan(t/2), and verifying an identity with all of them.Read the lesson →
- Inverse Trigonometric FunctionsGo from a ratio back to an angle, carefully: exact values of arcsin, arccos and arctan, the restricted ranges that make them functions, compositions such as sin(arccos x), calculator values, and the angle a situation asks for.Read the lesson →
- The Complex Plane & Polar FormSee a complex number as a point and an arrow: plot a + bi, read its modulus as a distance and its argument as an angle, convert between rectangular and polar form, multiply and divide in polar form, raise to a power with De Moivre, and find every nth root.Read the lesson →
What a student should be able to do
- Classify angles as acute, right, obtuse or straight.
- Use complementary and supplementary relationships to find an angle.
- Apply vertical angle and linear pair relationships.
- Find angles formed when a transversal crosses parallel lines.
- Classify triangles by sides and by angles.
- Find a missing angle using the 180-degree angle sum.
- Apply the exterior angle theorem.
- Decide whether three lengths can form a triangle.
- Find the perimeter and area of rectangles and triangles.
- Use the area formulas for parallelograms and trapezoids.
- Decompose a composite figure to find its area.
- Work backwards from an area to a missing side length.
- Find the hypotenuse given two legs.
- Find a leg given the hypotenuse and the other leg.
- Decide whether a triangle is right from its side lengths.
- Apply the theorem to ladders, diagonals and distance problems.
- Convert between radius and diameter.
- Compute circumference and area of a circle.
- Find an arc length and a sector area.
- Work backwards from circumference or area to the radius.
- Identify similar triangles from angle relationships.
- Write and solve a proportion between corresponding sides, including the side-splitter, midsegment and angle-bisector theorems.
- Use a scale factor to find perimeters and areas.
- Solve indirect measurement problems such as shadow heights.
- Find the volume of a prism and a cylinder.
- Find the surface area of a prism and a cylinder using nets.
- Apply the one-third factor for pyramids and cones.
- Use volume in capacity and packing word problems.
- Identify the hypothesis and conclusion of a conditional statement.
- Write the converse, inverse and contrapositive and decide which are true.
- Use a counterexample, the law of detachment and the law of syllogism to test reasoning.
- Justify each step of an algebraic or geometric proof with a definition, property, postulate or theorem.
- Name corresponding, alternate interior, alternate exterior and co-interior angle pairs.
- Find unknown angles when a transversal crosses two parallel lines.
- Decide from one angle pair whether two lines are parallel.
- Solve for a variable in an algebraic angle relationship and relate perpendicular lines to right angles.
- Find the midpoint of a segment and the distance between two points.
- Use slopes to decide whether two segments are parallel, perpendicular or neither.
- Find the point that divides a segment in a given ratio.
- Classify a triangle or quadrilateral from the coordinates of its vertices and find its perimeter or area.
- Match corresponding sides and angles of congruent triangles from a congruence statement.
- Choose the criterion (SSS, SAS, ASA, AAS or HL) that proves two triangles congruent, or decide there is not enough information.
- Use CPCTC to find an unknown side or angle once triangles are congruent.
- Supply the missing statement or reason in a two-column congruence proof.
- Find the interior and exterior angle sums of a polygon and each angle of a regular one.
- Use the side, angle and diagonal properties of a parallelogram to find unknowns.
- Apply the special properties of rectangles, rhombi, squares, trapezoids and kites.
- Classify a quadrilateral from a list of its properties, choosing the most specific name.
- Find an inscribed angle from its arc or central angle, including angles in a semicircle.
- Use the properties of chords, tangents and the tangent-radius right angle.
- Find angles formed by chords, secants and tangents meeting inside or outside the circle.
- Use the chord, secant and tangent segment products to find an unknown length.
- Write the equation of a circle from its centre and radius, or from its centre and a point on it.
- Read the centre and radius from (x - h)2 + (y - k)2 = r2.
- Complete the square to rewrite x2 + y2 + ax + by + c = 0 in centre-radius form.
- Decide whether a point lies on, inside or outside a circle.
- Find the hypotenuse or a leg of a 45-45-90 triangle exactly.
- Find any side of a 30-60-90 triangle from any other side exactly.
- Recognise special right triangles inside squares, equilateral triangles and regular hexagons.
- Solve problems whose answers are exact lengths in simplest radical form.
- Relate the scale factor of similar solids to the ratio of their surface areas and volumes.
- Find the area of a regular polygon from its apothem and perimeter.
- Find the area or volume of a composite figure by splitting it into known pieces.
- Identify the cross-section of a solid and use density in a measurement problem.
- Identify the opposite, adjacent and hypotenuse sides for an angle.
- Use SOH-CAH-TOA to find a missing side.
- Use inverse trig functions to find a missing angle.
- Solve angle-of-elevation and depression problems.
- Convert degrees to radians and back.
- Explain why a full turn is 2*pi radians.
- Find arc length and sector area using radians.
- Recognise the common radian measures on sight.
- Give exact sine and cosine for the special angles.
- Use reference angles and quadrant signs.
- Relate coordinates on the unit circle to (cos t, sin t).
- Evaluate trig functions of angles beyond one full turn.
- Apply sin2 t + cos2 t = 1 and its rearrangements.
- Use the quotient and reciprocal identities.
- Simplify a trigonometric expression to a single function.
- Verify an identity step by step.
- Write the square root of a negative number in terms of i, and simplify a power of i.
- Add, subtract and multiply complex numbers, and write the result in the form a + bi.
- Divide complex numbers by multiplying by the conjugate of the denominator.
- Solve a quadratic equation with a negative discriminant and write its complex solutions.
- State the amplitude, period and midline of a sine or cosine function from its equation.
- Find the phase shift and the direction of a reflection.
- Write the equation of a sine or cosine wave from its features or its graph.
- Model a periodic situation, such as a tide or a wheel, with a sinusoidal function.
- Solve a basic trigonometric equation on a given interval.
- Solve an equation that needs factoring or a quadratic substitution.
- Use an identity to rewrite an equation in one ratio before solving.
- Solve an equation with a multiple angle, and write the general solution.
- Find exact values such as sin 15° and cos(π/8) with a half-angle formula, choosing the sign from the quadrant.
- Given one ratio of t and an interval for t, find sin(t/2), cos(t/2) or tan(t/2).
- Rewrite powers of sine and cosine with the power-reducing formulas, and convert between products and sums.
- Verify an identity or evaluate an expression using the half-angle, product-to-sum and sum-to-product formulas.
- Evaluate arcsin, arccos and arctan exactly for the special ratios, in radians and degrees.
- State the domain and range of each inverse trigonometric function and explain why arcsin(sin t) is not always t.
- Evaluate a composition such as sin(arccos(3/5)) or tan(arcsin x) exactly.
- Use an inverse trigonometric function to find the angle in an applied problem or an equation.
- Plot a complex number, and find the modulus and argument of a + bi.
- Convert between rectangular form a + bi and polar form r(cos θ + i sin θ), and between polar and rectangular coordinates of a point.
- Multiply and divide complex numbers in polar form, and use De Moivre's theorem to raise one to a power.
- Find the nth roots of a complex number and describe where they sit on the plane.