SAT Math: Geometry and Trigonometry (Concepts, Skills, and Worked Problems)

Area

The amount of 2D space a figure covers; measured in square units.

Volume

The amount of 3D space a solid holds; measured in cubic units.

Surface Area

The total area covering the outside of a solid; the amount of material needed to wrap it.

Decomposition

A strategy for finding a composite area by splitting a complex figure into non-overlapping simple shapes and adding or subtracting their areas.

Scale Factor

If all lengths change by a factor of $k$, area changes by $k^2$ and volume changes by $k^3$.

Perimeter

The total distance around a figure; unlike area, it uses linear units.

Area of a Triangle

A = 1/2 bh, where the height must be perpendicular to the chosen base.

Area of a Trapezoid

A = 1/2 (b1 + b2)h, where b1 and b2 are the parallel bases and h is the perpendicular height.

Rectangular Prism

A box-shaped solid with volume $V = lwh$ and surface area $SA = 2(lw + lh + wh)$.

Cylinder

A solid with volume $V = \text{pi} r^2 h$ and surface area $SA = 2\text{pi} r^2 + 2\text{pi} rh$.

Vertical Angles

Opposite angles formed by two intersecting lines; they are equal.

Linear Pair

Two adjacent angles that form a straight line and sum to $180$ degrees.

Parallel Line Angle Relationships

When a transversal cuts parallel lines, corresponding angles and alternate interior angles are equal, while same-side interior angles are supplementary.

Exterior Angle Theorem

In a triangle, an exterior angle equals the sum of the two remote interior angles.

Similar Triangles

Triangles with equal corresponding angles and proportional corresponding sides; their side lengths are related by a scale factor.

Slope

A measure of steepness given by $m = \frac{y_2 - y_1}{x_2 - x_1}$; parallel lines have equal slopes and perpendicular lines have slopes whose product is $-1$ when both are defined.

Distance Formula

d = $\sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)$, a formula derived from the Pythagorean theorem.

Pythagorean Theorem

For a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse opposite the $90$-degree angle.

45-45-90 Triangle

A special right triangle with equal legs and a hypotenuse equal to leg times $\sqrt{2}$.

30-60-90 Triangle

A special right triangle with side ratio $1 : \sqrt{3} : 2$ for short leg, long leg, and hypotenuse.

SOH-CAH-TOA

A mnemonic for right-triangle trig: $\sin = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan = \frac{\text{opposite}}{\text{adjacent}}$.

Circle Measurements

For a circle of radius $r$, circumference is $C = 2\text{pi} r$ and area is $A = \text{pi} r^2$.

Tangent Line

A line that touches a circle at exactly one point; a radius to the point of tangency is perpendicular to the tangent, and tangent segments from the same external point are equal.

Inscribed Angle

An angle with its vertex on the circle; if it intercepts the same arc as a central angle, its measure is half the central angle.

Circle Equation

A circle centered at $(h, k)$ with radius $r$ has equation $(x - h)^2 + (y - k)^2 = r^2$.