Geometry
The Layout and Design
Geometry is the layout and design of GRE math, dealing with shapes, sizes, and spatial relationships. This section covers approximately 20% of Quantitative Reasoning questions.
- Lines, angles, and parallel/perpendicular lines
- Triangles (including special right triangles)
- Quadrilaterals (squares, rectangles, parallelograms, trapezoids)
- Circles, arcs, and sectors
- Polygons and their properties
- 3D figures (volume and surface area)
Polygon Interior Angles
Interior Angle Sum Formula
The sum of the interior angles of an n-sided polygon is:
(n - 2) × 180°
For a regular polygon (all sides and angles equal), each interior angle measures: (n - 2) × 180° / n
Example:
Find the measure of each interior angle in a regular octagon (8 sides):
Solution:
- Sum of interior angles: (8 - 2) × 180° = 6 × 180° = 1,080°
- Each angle in a regular octagon: 1,080° ÷ 8
- Answer: 135°
Common Polygons
Triangle: 3 sides, sum = 180°
Quadrilateral: 4 sides, sum = 360°
Pentagon: 5 sides, sum = 540°
Hexagon: 6 sides, sum = 720°
Octagon: 8 sides, sum = 1,080°
Decagon: 10 sides, sum = 1,440°
Special Right Triangles
30°-60°-90° Triangle
The GRE frequently tests the 30°-60°-90° triangle, where the side lengths are in the ratio:
1 : √3 : 2
(opposite 30° : opposite 60° : hypotenuse)
Example:
In a 30°-60°-90° triangle with hypotenuse 10:
- Hypotenuse = 2x, so 2x = 10, thus x = 5
- Side opposite 30° = x = 5
- Side opposite 60° = x√3 = 5√3
- Answer: Sides are 5, 5√3, and 10
45°-45°-90° Triangle (Isosceles Right Triangle)
Side lengths are in the ratio:
1 : 1 : √2
(leg : leg : hypotenuse)
Pythagorean Theorem:
a2 + b2 = c2
Common Pythagorean triples to memorize: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
Circles and Arcs
Key Circle Formulas
Circumference:
C = 2πr = πd
Area:
A = πr2
Arc Length Formula
The length of an arc is proportional to the degree measure of its central angle relative to 360°:
Arc Length = (θ/360°) × 2πr
where θ is the central angle in degrees
Example:
In a circle with radius 5, find the arc length for a 50° central angle:
Solution:
- Circumference = 2πr = 2π(5) = 10π
- Arc length = (50/360) × 10π
- Arc length = (5/36) × 10π
- Arc length = (50π)/36 ≈ 4.36
- Answer: ≈ 4.4
Sector Area
The area of a sector (pie slice) is:
Sector Area = (θ/360°) × πr2
Triangle Formulas
Area
A = (1/2)bh
base × height ÷ 2
Perimeter
P = a + b + c
sum of all sides
Triangle Inequality:
The sum of any two sides must be greater than the third side. Also, the largest side is opposite the largest angle.
Quadrilaterals
Rectangle
Opposite sides equal, all angles 90°
Area = length × width
Perimeter = 2(l + w)
Square
All sides equal, all angles 90°
Area = side2
Perimeter = 4 × side
Parallelogram
Opposite sides parallel and equal
Area = base × height
(height is perpendicular to base)
Trapezoid
One pair of parallel sides
Area = (1/2)(b1+b2)h
(average of bases × height)
3D Figures (Solids)
Rectangular Solid (Box)
Volume = length × width × height
Surface Area = 2(lw + lh + wh)
Cube
Volume = side3
Surface Area = 6 × side2
Cylinder
Volume = πr2h
Surface Area = 2πr2 + 2πrh
(two circles + lateral surface)
Sphere
Volume = (4/3)πr3
Surface Area = 4πr2
Ready to Practice Geometry?
Apply these geometric concepts with targeted practice questions.