Mastering Geometry for GRE GMAT

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, angles, polygons, circles, surfaces, solids, and higher dimensional analogs. It is one of the segments of GRE/GMAT quantitative section which is to be structured to adept by means of sensible practices to command over the stipulated ideas, techniques and problems where the base of basic concepts is urged to be trained.  

 

Circle

Figure 1: Circle; https://sites.google.com/a/friscoisd.org/greenemath

• A circle is equal to 360 degrees.
• The circumference is the length around a circle (i.e., the perimeter of a circle).
• The radius (plural: radii) is the length from the middle of a circle to any point on the edge of a circle.
• The diameter is the length of any straight line cutting a circle in half (and passing through the center point).
• The radius is half the length of the diameter (and the diameter is twice the length of the radius). So, 2r=d; π (pi) can be rounded to 3.14.
• An arc is all points between two points on the edge of a circle.
• A sector is the (shaded) region enclosed by an arc and two radii.
• A central angle is an angle created by the intersection of two radii at the center of a circle.  
• Area of a Circle= π r²
• Circumference of a Circle  = 2 π r
• Arc Length= (x/360) ×2 π r
• Area of Sector= (x/360) ×π r²

 

Rules of Angles

Figure 2: Types of Angle; Source: http://psn.virtualnerd.com

 

• A straight line is equal to 180 degrees.
• An acute angle is an angle less than 90 degrees.
• An obtuse angle is an angle between 90 and 180 degrees.
• A right angle is equal to 90 degrees and is usually signified by a small square.
• The symbol ∠ is often used to denote an angle. For example angle A=∠A.
• Two lines that intersect to create 90-degree angles are perpendicular to each other.

Figure 3: Two Parallel line which never intersects; source: http://etc.usf.edu

• Two lines that never intersect and have the same slope are parallel to each other.

Figure 4: Two intersecting lines; source: https://math.stackexchange.com

• When two lines intersect, four angles are created:
• opposite angles are equal in measure (∠a=∠c, ∠b=∠d).
• adjacent angles add up to 180 degrees (∠a+∠d=180, ∠d+∠c=180, ∠c+∠b=180, ∠b+∠a=180).

 

Figure 5: Two parallel lines are cut by a transversal; source: http://www.platinumgmat.com

 

When two parallel lines are cut by a transversal (i.e., a third line intersects the two parallel lines), a number of relationships exist between the resulting angles.

• Alternate Interior Angles Are Equal: K = L; O = J
• Alternate Exterior Angles Are Equal: H = M; N = I
• Corresponding Angles Are Equal: K = N; J = M; H = O; I = L
• Non-Alternate Interior Angles Are Supplementary: L + J = 180; K + O = 180

It is important to note that if two lines cut by a transversal have any of the above properties, then the two lines must be parallel. For example, if alternate interior angles are equal, then the two lines cut by a transversal must be parallel.

Squares

Figure 6: Square; http://en.citizendium.org/wiki

Perimeter : P=4s; s = side

Area: A=s²

Diagonal:  d = s √2

Rectangles

Figure 7: Rectangle; www.symbolab.com

• Opposite sides are equal in length and parallel to each other.
• All interior angles are right angles.
• The perimeter is the total length of all sides.
• Perimeter: P=2l+2w; l = length, w = width
• Area: A=lw 

Trapezoids

Figure 8: Trapezoids; www.ck12.org

• Area of Trapezoid =  × (Parallel side 1+ Parallel side 2) × Height = × (b₁+b₂) × h
• Opposite bases are parallel to each other.

Parallelogram

• Opposite sides are equal in length and parallel to each other.
• Opposite angles are equal.

Figure 9: Paralleogram; https://en.wikipedia.org/wiki

• Area : A= bh; b=base, h= height ;
• A parallelogram can be rearranged into a rectangle with the same area.

Triangles

Equilateral triangle

Figure 10: Equilateral triangle; http://mathworld.wolfram.com

An equilateral triangle is a triangle with all three sides of equal length, corresponding to what could also be known as a "regular" triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides equal. An equilateral triangle also has three equal 60° angles. The altitude h of an equilateral triangle is  h = a sin60° = ½ (√3 a); where a is the side length, so the area is  A = ½ (ah) =  (√3 a²)

Isosceles triangle

Figure 11: Isosceles triangle; http://mathworld.wolfram.com

The height of the isosceles triangle illustrated above can be found from the Pythagorean theorem as

\(h = \sqrt{b^2-1\over 4a^2}\)

 

Right triangle

A right triangle is a triangle with an angle of 90 degree (\(\pi\over2\)  radians). The sides a, b, and c of such a triangle satisfy the Pythagorean theorem: a²+b²=c²; where the largest side is conventionally denoted c and is called the hypotenuse. The other two sides of lengths a and b are called legs.

Figure 12: Right triangle; http://mathworld.wolfram.com

• All interior angles of any triangle add up to 180 degrees.

Pythagorean Theorem

• This theorem only applies to right triangles i.e. triangles with a 90-degree angle.
• c² = a² + b² where a and b are the sides and c is the hypotenuse.
• a and b are the two shorter sides, or “legs,” and c is the hypotenuse (the longest side of a right triangle).
• Certain triangle-side combinations (a:b:c), called Pythagorean triples, are easy & useful to memorize: [3:4:5] , [5:12:13] , [8:15:17]

Polygons

• The sum of the angles in a polygon:  (n-2) × 180° where n = number of sides
• Average degrees per side = \(x = {180^0\times{n-2\over n}}\)

Three dimensional shape

• Volume of a Cylinder= r²πh  , r= radius, h= height
• Surface Area of Cylinder = 2πr² + 2πrh
• Volume of rectangular solid = lwh , w=width
• Surface area of rectangular solid = 2(l×w)+2(l×h)+2(w×h)
• Volume of circular cylinder = πr²h
• Surface of circular cylinder = 2πr² + 2πrh

Distance

• Distance Formula : d = √[(y₂ - y₁)² + ( x₂ - x₁)²]
• Finding the Midpoint: when the endpoints are (x₁, y₁) and (x₂, y₂), the midpoint is: ((x₁+x₂)/2), ((y₁+y₂)/2)

Cube

• If each side of a cube is s, then the volume= s³  and the area = 6s²