Area, Volume, and Coordinate Geometry

Geometry on the SAT

Geometry and Trigonometry accounts for 15% of SAT Math — roughly 4 questions per module. The SAT provides a reference sheet with key formulas, but you still need to know when and how to apply them in multi-step problems.

Key Formulas (on the SAT reference sheet)

Shape	Formula
Circle area	$A = \pi r^2$
Circle circumference	$C = 2\pi r$
Rectangle area	$A = lw$
Triangle area	$A = \frac{1}{2}bh$
Pythagorean theorem	$a^2 + b^2 = c^2$
Cylinder volume	$V = \pi r^2 h$
Sphere volume	$V = \frac{4}{3}\pi r^3$
Cone volume	$V = \frac{1}{3}\pi r^2 h$
Rectangular prism volume	$V = lwh$

Coordinate Geometry Essentials

These formulas are NOT on the reference sheet — memorize them:

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	Slopes are negative reciprocals ($m_1 \cdot m_2 = -1$)

SAT Application: Multi-Step Problems

The SAT rarely asks for a single formula application. A typical question might combine area and algebra:

A rectangular garden's length is 3 feet more than twice its width. If the area is 90 square feet, what is the width?

Setup: $w(2w + 3) = 90 \Rightarrow 2w^2 + 3w - 90 = 0$.

Factor: $(2w + 15)(w - 6) = 0$, so $w = 6$ feet (reject $-15/2$ since width is positive).

This is really an Advanced Math question disguised as geometry — the SAT loves cross-domain problems.