Lines, Triangles, Circles, and Solids

What GRE Geometry Actually Tests

GRE Geometry is entirely Euclidean plane and solid geometry — no trigonometry, no proofs. The recurring skill is choosing the one relationship that unlocks a figure: an angle sum, a similarity ratio, the Pythagorean theorem, or a circle formula.

Lines and Angles

• Angles on a straight line sum to $180°$ (supplementary); angles around a point sum to $360°$.
• When two parallel lines are cut by a transversal: corresponding angles are equal, alternate interior angles are equal, and same-side (co-interior) interior angles are supplementary (sum to $180°$).
• Vertical angles (opposite angles at an intersection) are always equal.

Triangles

Property	Statement
Angle sum	Interior angles sum to $180°$
Exterior angle	Equals the sum of the two remote interior angles
Pythagorean theorem	In a right triangle, $a^2 + b^2 = c^2$ (hypotenuse $c$)
Common triples	3-4-5, 5-12-13, 8-15-17, and their multiples
Special right triangles	45-45-90 sides $1 : 1 : \sqrt2$; 30-60-90 sides $1 : \sqrt3 : 2$
Area	$\dfrac{1}{2} \times \text{base} \times \text{height}$
Similarity	Equal angles ⇒ proportional sides

Polygons

• Sum of interior angles of an $n$-gon: $(n - 2) \times 180°$.
• Each interior angle of a regular $n$-gon: $\dfrac{(n - 2) \times 180°}{n}$. A regular hexagon: $\frac{4 \times 180}{6} = 120°$; a regular pentagon: $108°$.
• Rectangle area $= lw$; parallelogram area $= bh$; trapezoid area $= \frac{1}{2}(b_1 + b_2)h$.

Circles

Quantity	Formula
Circumference	$C = 2\pi r = \pi d$
Area	$A = \pi r^2$
Arc length (central angle $\theta°$)	$\dfrac{\theta}{360} \times 2\pi r$
Sector area	$\dfrac{\theta}{360} \times \pi r^2$

If the area is $36\pi$, then $r^2 = 36$, so $r = 6$ and $C = 2\pi(6) = 12\pi$. Learn to move between area, radius, and circumference in one step.

Three-Dimensional Figures

Solid	Volume	Surface Area
Rectangular solid	$lwh$	$2(lw + lh + wh)$
Cube (edge $s$)	$s^3$	$6s^2$
Cylinder	$\pi r^2 h$	$2\pi r^2 + 2\pi r h$
Sphere	$\dfrac{4}{3}\pi r^3$	$4\pi r^2$

A cylinder with radius 3 and height 10 has volume $\pi (3)^2 (10) = 90\pi$.