GRE Geometry

GRE

Triangle Area

Triangle Area

The area of a triangle is calculated using the formula A = 1/2 * base * height, where the base is any side of the triangle and the height is the perpendicular distance from the base to the opposite vertex.

Isosceles Right Triangle Angles

45,45,90

Isosceles Right Triangle Side Proportions

(45-45-90) has sides in a ratio of x:x:x√2

A 30-60-90 triangle has sides in a ratio of

A 30-60-90 triangle has sides in a ratio of x:x√3:2x, with the 1x side opposite the 30 degree angle.

Pythagorean Theorem

For a right triangle with legs a and b and hypotenuse c: 𝑐2=𝑎2+𝑏2c2=a2+b2. This is called the Pythagorean Theorem

Right Triangle longest side

A right triangle has a right angle (a 90 degree angle); the side opposite the right angle is called the hypotenuse, and is always the longest side.

Pythagorean triples

Each side of certain right triangles are integers. These sets of numbers are called Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. A multiple of a Pythagorean triple is a Pythagorean triple (e.g., 6-8-10)

Triangle Longest side compared to other two sides

The length of the longest side can never be greater than the sum of the two other sides

Triangle shortest side compared to other two sides

The length of the shortest side can never be less than the positive difference of the other two sides

Circle Area

Area=πr2

Circumference

Circumference=2πr

Arc

A circle has 360 degrees. An arc is the portion of the circumference of a circle in x degrees of the circle

A fraction of the circumference of a circle is called an arc. To find the degree measure of an arc, look at the central angle

Arc Length

Arc Length=𝑥/360*2πr

Area of sector

Area of sector=𝑥/360*π𝑟2

Chord

A chord is a line segment between two points on a circle. A chord that passes through the middle of the circle is a diameter

Inscribed angles with the same chord/arc

If two inscribed angles hold the same chord, the two inscribed angles are equal

Inscribed angles holding chords/arcs of equal length are equal.

Degrees of an inscribed angle holding the diameter

An inscribed angle holding the diameter is a right angle (90 degrees)

Polygon total degrees formula

180(n − 2)

Cube formulas

• Cube volume=𝑠3

• Cube surface area=6𝑠2