SAT Math Geometry and Trigonometry: Complete Study Notes

Area

A measure of how much 2D space a shape covers; expressed in square units (e.g., cm^2, in^2).

Volume

A measure of how much 3D space a solid occupies; expressed in cubic units (e.g., m^3, ft^3).

Surface Area

The total area of the outer faces of a 3D solid; found by adding the areas of all external faces (not internal ones).

Perimeter

The total length of the boundary of a 2D figure; expressed in linear units.

Composite Figure (Decomposition)

A complex figure that can be split into simpler shapes (rectangles, triangles, semicircles, etc.) so areas can be added/subtracted without overlap.

Height (in area formulas)

The perpendicular distance associated with a chosen base (for triangles, parallelograms, trapezoids); it is not necessarily a side length.

Area of a Parallelogram

Computed as base × height (A = bh), where height is perpendicular to the base.

Area of a Triangle

Computed as half the product of base and perpendicular height (A = 1/2 bh).

Area of a Trapezoid

Computed as half the sum of the parallel bases times the height (A = 1/2(b1 + b2)h).

Volume as (Cross-Section Area) × (Height)

A unifying idea: many solids (especially prisms/cylinders) have volume equal to the area of a base/cross-section times the height.

Cylinder Volume

V = πr^2h; interpreted as a circular base area (πr^2) times height.

Cone/Pyramid Volume

One-third the volume of the corresponding prism/cylinder with the same base and height (V = 1/3 Bh; for a cone, B = πr^2).

Sphere Volume

V = 4/3 πr^3; depends on the cube of the radius.

Scaling (Perimeter/Area/Volume)

If all lengths scale by k, then perimeter scales by k, area by k^2, and volume by k^3.

Vertical Angles

Opposite angles formed by intersecting lines; they are equal.

Linear Pair

Two adjacent angles that form a straight line; they sum to 180°.

Corresponding Angles

Angles in matching positions when a transversal cuts parallel lines; they are equal.

Same-Side Interior Angles

Interior angles on the same side of a transversal cutting parallel lines; they are supplementary (sum to 180°).

Slope

A measure of a line’s steepness: m = (y2 − y1)/(x2 − x1); parallel lines have equal slopes and perpendicular non-vertical lines have negative reciprocal slopes (m1m2 = −1).

Distance Formula

Distance between (x1,y1) and (x2,y2): d = √((x2 − x1)^2 + (y2 − y1)^2).

Similar Triangles

Triangles with equal corresponding angles and proportional corresponding side lengths; used to set up ratios via a scale factor.

Pythagorean Theorem

In a right triangle with legs a and b and hypotenuse c: a^2 + b^2 = c^2; c is opposite the 90° angle and is the longest side.

SOH-CAH-TOA

Right-triangle trig definitions: sin(θ)=opposite/hypotenuse, cos(θ)=adjacent/hypotenuse, tan(θ)=opposite/adjacent (all relative to angle θ).

Arc Length (Degrees)

Length of an arc: L = (θ/360)·2πr, where θ is the central angle in degrees.

Circle Standard Form (Coordinate Plane)

Equation of a circle with center (h,k) and radius r: (x − h)^2 + (y − k)^2 = r^2 (watch sign conventions).