Page 3 Mathematics Achievement Study Guide for the ISEE


Two Dimensional Shapes

Two dimensional shapes, such as polygons and the conic sections, are shapes on flat surfaces and are the subjects in plane geometry. Know the properties of these shapes and be familiar with the formulas for solving their areas and determining their other dimensions.

Here are some important formulas:


where A is the area, and r is the radius

where C is the circumference, and r is the radius

Learn to manipulate the formulas when the other elements, such as r, are the unknown value.

where A is the area, b is the base and h is the height or altitude

Practice solving problems that involve sum of interior angles, Pythagorean triples, unknown sides, and unknown angles.

Pythagoras’ Theorem:

where a and b are the sides of a right triangle, and c is the hypotenuse

Rectangles, Squares, Other Polygons:

Perimeter = sum of the lengths of the sides of a polygon
Area of a rectangle = length × width
Area of a square = side × side = side

Three Dimensional Shapes

Three dimensional shapes are solids, which can either be polyhedra (solids with flat surfaces, such as cubes, rectangular prisms, and pyramids) or non-polyhedra (solids with any or all of its surfaces are not flat, such as cones, spheres, and cylinders). Questions on these shapes will mostly be about their capacities or volume. Here are some of the common formulas for volume:


V =

Rectangular Prism:

V = length × width × height


V = (side)


V =


V =


V =

Relationships Between Two Shapes

Two shapes are said to be similar if they have the same shape and their measurements (length, width, radius) are proportional. Shapes that undergo transformation by resizing or dilation remain similar to their initial shape.

Congruent shapes have the same shape and measurement, but may have different orientation or position. Shapes that undergo transformation by translation, rotation or reflection remain congruent to their initial shape.

Proving Theorems

It’s not likely that the ISEE will ask you to prove theorems, but it is important that you are familiar with the steps involved in proving theorems. A question may ask you to identify the next step in a designed proving procedure, or the appropriate definition, postulate, or theorems to use as reason for a given statement.

There are at least four types of proofs – the two-column proof, the paragraph proof, flow proof and transformational proof. Two-column proofs are the most common, and are also called T-proofs or ledger proofs. The first column shows the statements; the second column shows the reasons. The statements are steps listed chronologically and culminate in the desired conclusion. Reasoning uses “if-then” logic in putting forth the justification or support for the statements. These are based on definitions, postulates, and theorems or properties.

Basic Trigonometric Relationships

This section emphasizes the importance of Pythagoras’ Theorem, and expands the subject to trigonometric functions and trigonometric identities, which govern right-angled triangles.

These are the trigonometric functions:

Sine function: Sin θ =

Cosine function: Cos θ =

Tangent function: Tan θ =

There will also be quite a number of trigonometric identities that you need to familiarize yourself with, but these are some of the most useful:

Csc θ =

Sec θ =

Cot θ =

Sin θ =

Cos θ =

Tan θ =

For triangles that are not right-angled, the relationships given above will not apply, so supplement your readings on triangle identities.

Cartesian Coordinate System

The Cartesian Coordinate System is composed of perpendicular axes x and y which intersect at the origin (0, 0). The vertical axis y extends from negative infinity to positive infinity; the horizontal axis extends from negative infinity to positive infinity. The two axes create a plane of four quadrants: I, II, III and IV. Points are designated as (x, y) or (x, f(x)), which also indicate their x and y location relative to the axes and the point of origin.

This system makes it easy to visualize a function. Given point (x, f(x)), you may plot the point by finding its location on the Cartesian plane. Successively plotting points with different input and output values for (x, f(x)) will result in a collection of points which approximates the visual appearance of the function.

The xy plane may also be extended to include the z-axis, which makes the Cartesian Coordinate System a 3-dimensional space.

Other Coordinate Systems

Coordinate systems other than the Cartesian may be used in some questions, such as the polar and navigational coordinate systems. Polar coordinates are also referred to as spherical or spherical polar coordinates. A point is located using its radial distance r, azimuthal angle θ, and polar angle ϕ.

These formulas are useful for converting Cartesian coordinates to polar, or the other way around:

And these, if 3-dimensional Cartesian coordinates are used:

Read further on navigational coordinates system, which is often referred to as geographic coordinate system. To locate any point on earth, these geodetic coordinates are used: latitude ϕ, longitude λ, and elevation or height h.


Transforming shapes may change their size, position or orientation but will not affect their shape and proportion.


Translation – to transform an object by sliding it to the left or right of its original position. For example, translating the triangle in the illustration 2 spaces to the left will change the coordinates of its vertices to: (0, 7), (0, 3) and (3, 3).

Reflection – to flip an object about a line of symmetry to create its mirror-image. Reflecting the illustrated triangle about y = -1 will create its mirror-image in Quadrant IV with vertices at points: (2, -5), (2, -9) and (5, -5).

Rotation – to turn an object around a point. Rotating the triangle around (0, 0) at 90-degrees counter-clockwise will move the triangle to Quadrant II resting on its longer side and the coordinates of its vertices changed to: (-6, 3), (-2, 3) and (-2, 6).

Dilation – to proportionately increase or decrease the size of a shape. This triangle is a 3-4-5 triangle, and dilating it must result to the same proportion.


The triangle described above under the topic of Reflection is symmetrical to its reflected image. Each point on the reflected triangle will be at the same distance as its matching point in the original triangle from the line of symmetry, only in the opposite direction.

A shape may also be described as symmetrical if its parts can be flipped or folded and every point along its edges perfectly overlay the other edges, such as a square is symmetrical if folded in halves or fourths. A square has 4 lines of symmetry running through its center – 1 vertical, 1 horizontal, and 2 diagonals.


Visualization in geometry is aided by sketching and graphing on the Cartesian plane, as explained in the previous headings in the Cartesian Coordinate System, Two- and Three Dimensional Shapes, Transformation and Symmetry.

A good knowledge base on functions, the approximate graphs of parent functions, locating vertices, lines of symmetry and asymptotes, maxima and minima, intercepts and slopes will make graphing and visualizing easier.

Spatial Reasoning

Learning geometry starts with the concept of the point and the most basic shapes from lines to flat surfaces and solids. Later in the course, geometry becomes a medium to visualize algebra and other mathematical concepts. The geometry of shapes in a three-dimensional space helps the student “see” and make spatial visualization.

Spatial visualization is an aspect of spatial reasoning that allows you to predict what happens to shapes, such as after transformations. Another aspect of spatial reasoning is spatial orientation, your ability to perform spatial visualization from different perspectives or view objects from different points.

Geometric Modeling

Modeling aids visualization and learning. Geometric models and vertex-edge graphs are just two examples of these visuals.

Geometry and spatial reasoning interfaces with digital technology and uses computer software, such as CAD, to model shapes. Geometric modeling software generates visual images of shapes, and scales or transforms them based on their mathematical and geometric description.

A vertex-edge graph, on the other hand, is a visual diagram that makes a concrete representation of a real situation or problem. You will see circles or dots (called the vertices) in the diagram which are connected by lines (called the edge). Vertices are described as “degree-n” where n represents the number of lines connected to the vertex.


Appropriate Measurement Scales and Units

There are two major measurement systems – the Metric System and the US Standard System. You must do adequate practice using both systems until you are able to convert measurements from one system to another with ease.

Applying and converting units within the Metric System are much easier compared to the US Standard. But since you use US Standard in your daily life, it will not be too difficult. Here are some of the commonly-used units for measuring lengths, area, volume or capacity, time and mass, and their conversion equivalents:

Conversion units will be provided in the questions with values in US Standard units, so you don’t need to worry about memorizing. It is important, however, that you are familiar with how units are converted, including unit analysis which is discussed below. Invest an adequate amount of time to practice converting units of time (years, months, days, hours, minutes and seconds) and temperature (Fahrenheit, Celsius and other scales).

Formulas for Geometric Measurement

Formulas will usually be provided in test questions. For practice, refer to them under several topics in Geometry. Remember that questions will not always ask for the Area or Volume. In fact, most questions will ask for the other elements or variables in the formula, so be sure that you know how to manipulate formulas to your advantage.

Precision, Accuracy, and Approximate Error

Accuracy indicates how close a measurement is to the true value. Precision indicates how a measurement is close to or consistent with other measurements. This means that measurements can be precise but inaccurate. Inaccuracies in measurement, e.g. length or width, can greatly affect computations derived from these measurements, e.g. area or volume.

Measuring tools have built-in errors or biases, and these introduce deviations from the true value. To have more accurate measurements, you have to consider the approximate errors in measured values. An instrument that is “accurate to 10 millimeters” measures lengths at +5mm or -5mm error. This means that a measurement of 100mm may actually be any length from 95mm to 105mm.

Measurement Concepts

Any measurement can only be as accurate as the smallest calibrated unit of the measuring device. This is referred to as the limit of reading measurements. It would not be possible to accurately measure 24.8 cm using a scale with 1 cm as its smallest calibration.

So, a measurement of 25 cm using a device that is accurate to the nearest centimeter has a lower bound of 24.5 cm and upper bound of 25.5 cm. Any value between these bounds will be read as 25 cm.

Errors in measurement can be reduced by repeating measurements and averaging results or by the informal method of successive approximations. In successive approximations, you start by assuming a value for a variable and solving a difficult equation or formula using the assumed value and other known values. After that, you repeat the procedure using the result as the next assumed value for the variable. This is repeated until a constant value is obtained.

Unit Analysis

Unit analysis or dimensional analysis is a useful method for checking answers, especially those that involve measurement units and unit conversion. For instance, the speed of a car is given in the problem as 7 mph and you need to compute for the distance traveled in 2 hours. However, the answer choices are in kilometer units, you may use the unit analysis method:

Units that appear as numerator and denominator cancel out, leaving only the relevant unit or units – in this case, kilometer.