Geometry in GRE Quant is quite crucial where time is always a crisis. Every math student knows that Geometry is a scoring chapter. A little focus on this chapter will boost up your score exponentially.
Let’s go with an overview of GRE quant. Math in GRE has 2 sections with each section containing 20 questions each and the total time given is only 35 minutes.
In general, 2 types of questions appear in GRE quant.
- Quantitative comparison
- Problem solving
In addition to this, the test does have numeric entry problems in which the student have to enter their own answers.
Fact is that all know that GRE quantitative reasoning syllabus is up to the 10th grade. But the presentation of the problem in the test is different. That is how GRE measures the reasoning abilities of the test takers.
GRE quant comprises both arithmetic and geometry. Today we feel pleasure to give you ‘the complete guide to geometry in GRE quant’ in brief. This is a pocket notes for reference at a glance.
GRE Geometry Concepts and Formulas
GEOMETRY, as we all know is easy to understand but crisis generally arises when you solve a problem. Because, most of the students have a misconception that GRE quant geometry can be solved just by eyeballing. But it’s not true.
To become successful in geometry, we have some set strategies. Now, let’s discuss one after other.
Whatever the calculation it is, if it is geometry, then there are 3 rules to your success.
- Application (this regards practice) and you are ready for geometry now.
GRE Geometry Concept: Lines and Angles
As said; definitions first.
GRE Geometry: Lines
- A straight line with now end points is called a line.
- A ray originates from a point and extends in only one direction.
- A line – segment is a part of line. Mostly line segments are used in geometrical constructions (in diagrams as well).
- There are 3 major pairs of lines they are; Intersecting lines, perpendicular lines and parallel lines.
GRE Geometry: Angles
- Two lines intersecting at a point subtends an ANGLE on one side and that point is called VERTEX. As a line has no ends, two intersecting lines subtend 4 angles in total.
- Acute angle limits from 0 to 90 degrees.
- Right angle is 90 degrees.
- An obtuse angle limits from 90 to 180 degrees.
- Straight angle is 180 degrees.
- The angle from 180 to 360 is called reflex angle.
- When two lines meet at a point, pairs of opposite angles are equal.
- Two angles are supplementary if their angle sum is 180 degrees.
X +y =180 => x = 180 – y => y = 180 – x
- Two angles are complimentary if their angle sum is 90 degrees.
X +y =90 => x = 90 – y => y = 90 – x
- Perpendicular lines meet at right angles.
- Two parallel lines have an equal slope.
- The shortest distance from a point to a line is always a perpendicular.
- When parallel lines are intersected by a transversal, there are three major angle relations;
The Transverse Theory:
- Alternate interior angles are equal
- Corresponding angles are equal
- Interior angles on the same side of the transversal are supplementary.
GRE Geometry Concept: Triangles
‘Triangles’ is a really vast subject to deal with. Once you reach the level of Pythagoras theorem, it takes a leap to another subject. Let’s start with definitions.
Any closed 2 dimensional figure with 3 sides is called Triangle. It has 3 sides and 3 angles. There are 3 basic classifications based on sides.
- If all sides of a triangle are equal, then it is an Equilateral triangle. Consequently all angles are same and equal to 60 degrees.
- If 2 sides of a triangle are equal, then it is an Isosceles triangle. Consequently 2 angles subtended by the equal sides on the base are equal
- If all the three sides are different, then it is a Scalene triangle. It has three different angles.
If a triangle has one right angle, then it is said to be a Right Angled Triangle. One of the angles = 90 degrees and the longest side is called ‘hypotenuse’ opposite to the right angle.
The sum of all the 3 angles of a triangle => A + B + C = 180
An altitude of a triangle is the line joining the base perpendicularly from the opposite angle.
The altitude of an equilateral triangle and an isosceles triangle bisect the angle and the base too.
For a right angled triangle with right angle any side on the base, altitude is measured as the length of the perpendicular.
In some cases like in the case of an Obtuse angled triangles, the base should be extended in order to draw altitude.
Altitude is also called as height of the triangle.
Area of the triangle is ½ bh. [b is base and h is height of triangle]
In every triangle, longer side is opposite the larger angle and smaller side opposite to the smaller angle.
This theorem implies right angled triangles. It is also called ‘Right Hypotenuse Theorem’. It states that the sum of individual squares of base and height is equal to the square of hypotenuse of a right angled triangle.
Pythagorean Triples are the numbers which most often represent the sides of a right angled triangle.
Example: 3,4,5 and 5,12,13. The largest number in each set represents the hypotenuse of the right angled triangle.
Triangle congruency is typically based on proportion and correspondence in dimensions. Congruent triangles are also called similar or identical triangles.
[ Also Read: GRE Math Tips and Tricks ]
Triangles different in size but with identical dimensions are congruent. We have four properties about triangle congruency. When you have two triangles to compare the congruency;
- Side angle side property (SAS) in which 2 sides and 1 angle match.
- Angle side angle property (ASA) in which 2 angles and 1 side match.
- Side side side property (SSS) in which all the 3 sides match.
- Angle angle angle property (AAA) in which all the 3 angles match.
- Right hypotenuse side (RHS) in which any side and hypotenuse of a right angled triangles match.
GRE Geometry Concept: Quadrilateral
A quadrilateral is a closed figure with 4 straight lines. The total sum of all interior angles of a quadrilateral is 360 degrees.
Every 2D closed figure with four sides is a Quadrilateral. Therefore square, rectangle, parallelogram, rhombus, trapezium are quadrilaterals. Let us move along their basic properties. The surface area of a quadrilateral is the product of length and breadth. The perimeter is the sum of all the sides.
Quadrilaterals are classified into 5 different types according to their properties.
All sides are equal. Each angle equals to 90 degrees. Diagonals bisect each other at right angles.
Area = side * side
Perimeter = 4 * side
Opposite sides are equal. Each angle equals to 90 degrees. Diagonals bisect each other but not at right angles.
Area = length * breadth
Perimeter = 2* (l + b)
Opposite sides are equal. Opposite angles are equal. Diagonals bisect each other but not at right angles.
Area = length * breadth
Perimeter = 2* (l + b)
All sides are equal in length. Opposite sides are parallel to each other. Diagonals bisect perpendicularly.
Adjacent angles are supplementary (angle 1 + angle 2 = 180).
Vertex angles of a rhombus are not equal to 90 degrees.
Suppose ‘a’ and ‘b’ are the lengths of diagonals, then
Area of the Rhombus = (a * b)/ 2
Perimeter = 4 *length of side.
[ Also read: How to Study for GRE Math ]
Only the base and its opposite side are parallel to each other. Sides, angles or diagonals are not congruent.
Area of trapezium: ½ * height of the trapezium * (length of the base + length of the side opposite to base)
It is the average of the two base times the height.
A = 1/2 [L + L2] * h
Perimeter = sum of the lengths of all the sides = L + L1 + L2 +L3
GRE Geometry Concept: CIRCLES
A circle is defined as a set of points equidistant from a center on a plane surface. A circle’s perimeter is called the ‘circumference of the circle’.
Now let’s talk about circle’s terminology
Radius: The radius of a circle the line segment joining the center to any point on the circle.
Chord: The chord is the line segment joining any two points on the circle.
Diameter: The diameter of a circle is the line segment joining any two points of the circle passing through the center.
The diameter is also referred as 2 * radius. It is the longest chord of the circle.
Diameter = 2 * radius => radius = diameter / 2
Arc: A part of circumference is the arc of the circle.
Sector: An area bounded by the circumference (arc) at an angle at the center (as the vertex) is called sector.
SECANT is a line passing through the circle intersecting at 2 points.
TANGENT is a line passing through the circle intersecting at only 1 point.
Circumference of the circle is the measurement of the boundary edge of the circle.
C = 2 π r = π d where π = 222/7 or 3.14 approximately
Area of the circle
A = π r2 where π = 222/7 or 3.14 approximately
‘r’ can be replaced by ‘d/2’
So A = πd2/4
GRE Geometry Formulae: ARC and Sector Formulae
If ‘θ’ is the angle subtended by the arc or a sector, then
Length of the arc = (θ/360) * circumference = (θ/360) * 2 π r
Area of the sector = (θ/360) * area = (θ/360) * π r2
GRE Geometry Concept: 3D Geometry
3D geometry mainly deals with the figures like Cube, Cuboid, Cylinder, Cone, and Hemi – Sphere.
There are two main calculations of 3D figures and they are total surface area and volume. Lateral surface area is the sum of all surface areas except the top and the base.
A cuboid is a volume covered by 6 rectangles where all the rectangles are called as the faces of the cuboid.
The vertices are the point of intersection of 3 edges. There are 8 vertices.
The edge of the cuboid is the line segment joining the adjacent vertices. There are a total of 12 edges.
All angles at the vertices are 90 degrees.
Diagonals bisect each other.
If ‘l’ is the length, ‘w’ is the width and ‘h’ is the height; then
GRE Geometry Formulas: Cuboid Formulas
Total surface area = 2(lw + wh + hl)
Lateral surface area = 2h (l+w)
Volume of cuboids’ = l * w * h
Length of the longest Diagonal = √ (l² + w² + h²)
The cube is also a cuboid with measurements of all faces being squares. It has 6 faces, 8 vertices and 12 edges. All angles at the vertices are equal to 90. The length, breadth and height of a cube are same.
If ‘l’ is length, ‘b’ is breadth and ‘h’ is the height of the cube. But for a cube l = b = h = x
GRE Geometry Formulas: Cube Formula
Volume = x³
Total surface area = 6x2
Lateral surface area = 4x2
Length of the diagonal of cube = √3x
A cylinder is a 3 dimensional solid with 2 circular and parallel faces connected by a curved surface. The major dimensions of a cylinder are radius of base and height.
The ‘height’ of the cylinder is the distance between 2 circular faces.
Usually ‘h’ is the height of the cylinder and ‘r’ is the radius of the cylinder.
GRE Geometry Formulas: Cylinder Formula
Volume of a cylinder = π r2h
Lateral surface area = 2π rh
A cylinder and a prism are similar because they have the same cross section everywhere.
A cone is a 3 dimensional shape usually with a circular base and the shape gradually narrows to one point. A cone is much similar to a pyramid but a pyramid has a triangular base. A cone has no faces except one circular base. It has no edges and there is only 1 vertex. That is called apex.
The distance between the centre of the base and the vertex (apex) is called the height of the cone(not in the case of oblique cone).
The cone has 3 important readings (measurements). They are;
- The height (the perpendicular distance from the apex)
- The radius (radius of the circular base) and
- The slant height (the distance between the apex and any point on the circumference of the circular base).
These 3 measurements make a right – angled triangle. Therefore, Pythagoras theorem can be applied.
GRE Geometry Formulas: Cone Formula
The slant height l = √(r2+h2)
Volume of a cone = ⅓ πr2h
Total surface area = πrl + πr2 => πr being a common factor, TSA of a cone = πr(l + r)
A sphere is a unique 3D figure with no vertices and no edges. It has neither faces and nor it is flat. A sphere has only 1 notable measurement. That is its radius ‘r’.
GRE Geometry Formulas: Sphere Formula
Volume of a sphere = (4/3) * π * r3
Surface area of a sphere = 4 * π * r2
A hemisphere is simply half the Sphere. It has one circular edge. It has only one face and no vertex.
The main measurement is the ‘radius’ or ‘r’.
GRE Geometry Formulas: Hemi-Sphere Formula
The curved surface area of hemisphere: 2 * π * r2
Total surface area of the hemisphere: curved surface area + base circle area =
= 2 * π * r2 + π * r2 = 3 * π * r2
Volume of a hemisphere: (2/3) * π * r3
Area of a hollow hemisphere: Outer hemisphere area – Inner hemisphere area
GRE Geometry Concept: Co–ordinate Geometry
Calculations in the co – ordinate geometry are done on co – ordinate axis. The key element of the co – ordinate axis is number line.
The numbers are in increasing order from left to right where numbers right to Zero are positive and numbers on the left are negative.
Now, draw a line perpendicular to the number line intersecting at point ‘0’. The numbers above ‘0’are positive and numbers below are negative.
The horizontal line is said to be the X – axis and the vertical line is Y – axis. Hence, these are co – ordinate axis. Their intersection point is called origin (0, 0).
Now, think about the co–ordinate axis and not the number line;
On the X – axis, the numbers right to origin and are positive and increase further right and the numbers left to origin are negative and decrease further left
On the Y – axis, the numbers above origin are positive and increase further above and the numbers below are negative and decrease further.
Point on the Co–ordinate plane:
A point on the co – ordinate plane is represented with an ordered pair (x, y) where ‘x’ is the abscissa and ‘y’ is the ordinate. They are co – ordinates collectively. The coordinates represent the readings on the axis correspondingly.
Quadrant: The ‘X’ and ‘Y’ axes divide the co – ordinate plane in to 4 quadrants counting counter clock – wise from the 1st quadrant
Note: If x ≠ y, then (x, y) and (y, x) represent different points on the plane.
GRE Geometry Formulas: Co-ordinate Geometry Formula
Distance Formula: The distance formula on co – ordinate plane is extracted from Pythagorean Theorem. If (x, y) and (a, b) are 2 points on the plane then according to Pythagorean Theorem, then
Distance d2 = (x – a)2 + (y − b)2
Theorem therefore, d = √ (x – a)2 + (y − b)2
Mid – point formula: Suppose (x, y) and (a, b) are 2 points on the plane;
The mid – point between the 2 points is given by M = (x + a/2, y + b/2)
It is the average of the corresponding coordinates of the 2 points.
Slope Formula: The inclination of a line is measured by the slope formula. The vertical change in inclination is called Rise and the horizontal change is called Run. Slope is the ratio of the vertical change to the horizontal change.
Let (x, y) and (a, b) are two points on the plane, then the slope is the rise over the run and is given by;
m = y – b/x – a
Note: The line equation on a co – ordinate plane can be given by slope intercept form as
Y = mx + b [where ‘m’ is the slope and ‘b’ is the y – intercept]
Point slope form = (y – y1) = m(x – x1)
Areas and Perimeters on a Co–ordinate plane:
Here comes the most important point to remember is that a co – ordinate plane with 2 axes is a 2 dimensional plane.
If you are asked to find the area or perimeter on a co – ordinate plane, then you have to divide the figure in to rectangles and triangles and use the properties to understand them. Later applying proper formulae of 2D geometry, you will be able to solve the problem with no effort.
So, this is a pocket notes with all the basic postulates of GRE quant math portion. We wish you all the best!!!