Geometry and Trigonometry on the ACT

ACT Geometry: Broader Than the SAT

The ACT tests geometry more heavily than the SAT and includes several topics the SAT omits entirely. Expect 12–15 geometry questions and 2–4 trigonometry questions per test.

Plane Geometry Essentials

Shape	Key Formulas
Triangle	Area = $\frac{1}{2}bh$; angles sum to $180°$
Rectangle	Area = $lw$; Perimeter = $2l + 2w$
Circle	Area = $\pi r^2$; Circumference = $2\pi r$; Arc length = $\frac{\theta}{360°} \cdot 2\pi r$
Trapezoid	Area = $\frac{1}{2}(b_1 + b_2)h$
Regular polygon ($n$ sides)	Interior angle = $\frac{(n-2) \cdot 180°}{n}$

Similar Triangles

If two triangles are similar (same angles), their corresponding sides are proportional: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

The ratio of their areas is the square of the side ratio. If sides are in ratio $2:3$, areas are in ratio $4:9$.

Coordinate Geometry

Concept	Formula
Distance	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint	$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Slope	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Parallel lines	Same slope: $m_1 = m_2$
Perpendicular lines	Negative reciprocal slopes: $m_1 \cdot m_2 = -1$

Trigonometry — Right Triangle (SOH-CAH-TOA)

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Trigonometry — Oblique Triangles (ACT-Specific)

These formulas work for ANY triangle, not just right triangles:

Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Use when you know: an angle and its opposite side + one more piece.

Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$

Use when you know: two sides and the included angle, or all three sides.

Unit Circle Values (Memorize These)

Angle	$\sin$	$\cos$	$\tan$
$0°$	$0$	$1$	$0$
$30°$	$1/2$	$\sqrt{3}/2$	$\sqrt{3}/3$
$45°$	$\sqrt{2}/2$	$\sqrt{2}/2$	$1$
$60°$	$\sqrt{3}/2$	$1/2$	$\sqrt{3}$
$90°$	$1$	$0$	undefined

Matrices (ACT-Specific — Not on SAT)

A matrix is a rectangular array of numbers. The ACT tests:

• Addition/Subtraction: Add corresponding entries (matrices must be same size)
• Scalar multiplication: Multiply every entry by the scalar
• Matrix multiplication: Row-by-column dot products (only when columns of first = rows of second)
• 2×2 Determinant: $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

Complex Numbers (ACT-Specific — Not on SAT)

A complex number has the form $a + bi$, where $i = \sqrt{-1}$.

Powers of $i$ cycle every 4:

$i^1$	$i^2$	$i^3$	$i^4$
$i$	$-1$	$-i$	$1$

To find $i^n$: divide $n$ by 4 and use the remainder. $i^{27} = i^3 = -i$ (since $27 \div 4 = 6$ remainder $3$).

Operations: Treat $i$ like a variable, but replace $i^2$ with $-1$. $(3 + 2i)(1 - 4i) = 3 - 12i + 2i - 8i^2 = 3 - 10i + 8 = 11 - 10i$