CH 17: Geometry
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17.1: Lines, Rays, and Line Segments
A line can be viewed as a geometric figure connecting two points and extending indefinitely from each point while
keeping the line's straightness. Thus, all lines are straight; they have no curvature.
If a line extends from only one of its points, then it is called a ray or half-line. If a line ends at both points, that is,
it doesn't extend in either direction, it is called a line segment or simply a segment. These points would then be
called end points.
17.1.1: Naming Lines
17.2: Angles
An angle is a union of two rays sharing a common endpoint. This common end point is called the vertex of the
angle. The two rays are called the sides of the angle. Angles are measured in degrees; degrees basically denote
the span between an angle's two sides.
17.2.1: Classification of Angles by Their Measures
An acute angle is an angle whose measure is more than 0° but less than 90°.
A right angle is an angle whose measure is exactly 90°. As you can see, if the angle is a right angle, we draw a
small square at the vertex to signify that it is a right angle.
An obtuse angle is an angle whose measure is more than 90° but less than 180°.
A straight angle is an angle whose measure is exactly 180°.
17.2.2: Naming Angles
Usually, you will see three points on an angle: one indicates the vertex, another designates a point on the ray
forming one side of the angle, and the third designates a point on the ray forming the other side of the angle.
Subsequently, we can name the angle by these three points.
For example, the angle shown on diagram 2 below is called angle ABC or angle CBA, often denoted by ∠ABC or
∠CBA. Notice that thevertex is the middle of the three points and must be in the middle of the angle’sdesignation.
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17.2.3: Other Things
17.2.4: Intersecting Lines
When n lines intersect through a common point, the sum of all the angles created by those n lines at that point is
360 degrees.
17.2.5: Perpendicular Lines
When intersecting lines are perpendicular, each angle formed between those lines is 90°. Another way of saying
this is that two straight lines are said to be perpendicular when they intersect and form four right angles.
Remember that a right angle measures 90°.
17.2.6: Supplementary Angles
Angles are supplementary if their measures add up to 180°. Another way of saying this is that supplemental angles
form a straight angle.
17.2.7: Parallel Lines Intersected by a Transversal
A transversal is simply a line that passes through two or more lines at different points.
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17.2.8: Vertical Angles Are Equal and Corresponding Angles Are Equal
Vertical angles are angles that are diagonally oriented to each other.
17.2.9: Supplementary Angles Sum to 180 Degrees
A pair of supplementary angles can always be “pasted” together to form a straight line.
Acute angles are those that measure less than 90°.
Obtuse angles are those that measure more than 90° but less than 180°.
When parallel lines are cut by a non-perpendicular transversal, both obtuse and acute angles are necessarily created.
Any one of the obtuse angles (a, d, e, and h) and any one of the acute angles (b, c, f,and g) contained in such a figure
will be supplementary, that is, they will add up to180°, a straight line.
17.3: Polygons
A polygon is a closed two-dimensional geometric shape that is composed solely of straight line segments.
Triangles, squares, rectangles, and hexagons are all examples of polygons.
When a polygon has four sides, it’s called a quadrilateral. Parallelograms, squares, and rectangles are all
quadrilaterals. When a polygon has three sides, it’s called a triangle.
17.4: Triangles
A triangle is a three-sided polygon and, for the GMAT, one of the most important geometric shapes to understand.
17.4.1: Interior Angles of a Triangle
The measures of the three interior angles of any triangle add up to 180°.
17.4.2: Relationship Between Angles and Sides Within a Triangle
The largest angle is always opposite the longest side of the triangle; the smallest angle is opposite the shortest side of
the triangle. In addition, equal sides will always be opposite equal angles.
17.4.3: The Exterior Angles of a Triangle
An exterior angle is the angle created by one side of a triangle and the extension of an adjacent side.
In the diagram below, angles x, y, and z are exterior angles. Taking one exterior angle per vertex, the sum of any
triangle’s exterior angles is 360°. Consider the figure below. The three interior angles are 50°, 85°, and
45°,respectively. Notice that when we take one exterior angle per vertex, the three exterior angles add up to 360°.
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Given any polygon, when taking one exterior angle at each vertex, the sum ofthe measures of the exterior
angles will always equal 360°.
17.4.4: An Exterior Angle of a Triangle Is Equal to the Sum of the Two Remote Interior Angles
Consider the figure below. Angle a is an exterior angle of the triangle. Angles x, y,and z are interior angles.
In addition, angles x and y are remote, or on the opposite side of the triangle, from angle a.
Thus, angles x and y are termed remote interior angles relative to angle a.
The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
That is, x + y = a.
17.4.5: The Area of a Triangle
The area of any triangle equals one-half of the product of the base of the triangle multiplied by the height of the
triangle.
The base of a triangle and the height of a triangle are always perpendicular.
17.4.6: Calculating the Height of a Triangle Using an Altitude of the Triangle
The base of the triangle must be perpendicular to the height. To determine the height, we must sometimes look to
lines outside of the triangle itself.
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17.4.7: The Triangle Inequality Theorem
In any triangle, the sum of the lengths of any two sides of the triangle is greater than the length of the third side, and
the difference of lengths of any two sides of the triangle is less than the length of the third side.
17.4.8: Types of Triangles
We can classify a triangle by the relationship between its sides or by the relationship between its angles.
If we use the sides as a basis for classification, we have scalene, isosceles, and equilateral triangles.
A scalene triangle is one in which all three sides are of different lengths.Correspondingly, all three angles would be of
different measures as well.
An isosceles triangle is one in which exactly two sides are of the same length and two angles are of the same measure.
An equilateral triangle is one in which all three sides are of the same length and all three angles are of the same
measure.
If we use the angles as a basis for classification, we have acute, obtuse, and right triangles.
An acute triangle is one in which each of its angles measures less than 90 degrees. An acute triangle can be scalene,
isosceles, or equilateral.
An obtuse triangle is one in which one angle is greater than 90 degrees but less than 180 degrees. An obtuse triangle
can be scalene or isosceles.
A right triangle is one in which one angle measures exactly 90 degrees. A right triangle can be scalene or isosceles.
On the GMAT, right triangles are by far the most important type of triangle to master because they are commonly
tested.
17.4.9: The Pythagorean Theorem
It is important to remember that in a right triangle, the hypotenuse, or the side opposite the right angle, will always
be the longest side of the triangle, because the right angle is the largest angle of the triangle (recall that the longest
side of a triangle is always opposite the largest angle of the triangle). The other, shorter sides of a right triangle are
sometimes referred to as the legs of a triangle.
17.4.10: The Converse of the Pythagorean Theorem
Given any triangle with sides A, B, and C, if C2 = A2 + B2 , then the angle opposite side C must measure 90°, and
thus the triangle must be a right triangle.
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17.4.11: Pythagorean Triples
17.4.12: The 3 4 5 Right Triangle
3-4-5 Right Triangle: When the sides of a triangle are in the ratio of 3:4:5, it is a3-4-5 right triangle.
Thus, triangles with the following side lengths areconsidered to be 3-4-5 right triangles: {3, 4, 5}, {6,
8, 10}, {9, 12, 15}, etc. Be on the lookout for larger multiples of the 3-4-5 right triangle.
As long as the ratioof the sides is 3:4:5, with 5 being the hypotenuse, the triangle is a 3-4-5 right
triangle.
17.4.13: The 5 12 13 Right Triangle
5-12-13 Right Triangle: When the sides of a triangle are in the ratio of 5:12:13, it is a 5-12-13 right triangle.
Thus, triangles with the following side lengths are considered to be 5-12-13 right triangles: {5, 12, 13}, {10, 24, 26},
{15, 36, 39}, etc.
Be on the lookout for Pythagorean triples: 3-4-5 and 5-12-13.
17.4.14: The Isosceles Right Triangle (45 45 90 Right Triangle)
An isosceles triangle has two equal sides and two equal angles. In an isosceles right triangle, the two equal angles
each measure 45 degrees; the third angle measures 90 degrees. This triangle is also referred to as a 45-45-90 right
triangle.
The area of an isosceles right triangle can be found using the formula area = l2/2 , where l is the length of one of the
equal sides.
17.4.15: The Ratio of the Sides of a 45 45 90 Right Triangle
The sides of a 45-45-90 right triangle are in a set ratio of x: x:x 2 , where x
hypotenuse and x represents the length of each ofthe shorter legs.
represents the length of the
17.4.16: The Area of a 45 45 90 Right Triangle Is One Half of the Area of a Square
The diagonals of a square cut the square into two 45-45-90 right triangles. The area of each of these triangles is
half of the area of the square that they form.
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17.4.17: The 30 60 90 Right Triangle
17.4.18: The Equilateral Triangle
17.4.19: Cutting an Equilateral Triangle in Half Forms Two 30 60 90 Triangles
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17.4.20: Similar Triangles
Triangles are similar if
(1) the three angles of one triangle are the same measure as the three angles of another triangle,
(2) the three pairs of corresponding sides have lengths in the same ratio, or
(3) an angle of one triangle is the same measure as an angle of another triangle and the sides surrounding these
angles are in the same ratio.
It should be noted that if two corresponding pairs of angles of the two triangles are equal, then the third pair of
angles of the two triangles must also be equal, since the sum of the three angles of a triangle is always 180
degrees.
17.5: Quadrilaterals
Quadrilaterals are four-sided polygons: rectangles, squares, parallelograms,rhombuses, and trapezoids.
All squares are rectangles, and all rectangles are parallelograms.
17.6: The Parallelogram
A parallelogram is a quadrilateral that has two pairs of parallel sides.
Everyparallelogram has the following features: opposite sides are equal in length,opposite angles are equal in
measure, the diagonals bisect each other, and each diagonal divides the parallelogram into two congruent triangles.
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17.6.1: Finding the Area of a Parallelogram
17.7: Rectangle
A rectangle is any quadrilateral with four right angles. The rectangle shares all of the properties of the
parallelogram already covered.
The area of a rectangle is length × width. The perimeter of a rectangle is found by adding together all four
sides. Perimeter = 2l + 2w, where l = length and w =width.
17.7.1: The Longest Line Segment of a Rectangle
The longest line segment that can be drawn within a rectangle is a diagonal.
In addition, each diagonal divides the rectangle into two congruent right triangles, which have the same area.
The area of each of these congruent triangles is half the area of the rectangle.
First, the angles formed by the intersection of the diagonals are not always the same measure. The
measurement of these angles depends on the proportion of length to width of the rectangle.
Second, the diagonals of a rectangle do not intersect at 90 degree angles unless the rectangle is also a square.
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17.8: Squares
A square has four equal sides and four equal angles. Each angle measures 90 degrees.
Since a square is simply a special case of the rectangle in which all four sides are of equal length and a rectangle
is a special case of the parallelogram, the square possesses all the properties belonging to those two figures.
17.8.1: Area of a Square
Area of a Square = side2
17.8.2: Perimeter of a Square
Perimeter of a Square = 4 × side
17.9: The Maximum Area of a Rectangle
Given a rectangle with a fixed perimeter, the rectangle with the maximum area is a square.
Consider a square and a rectangle (one that is not a square) with equal perimeters. Without more
information, can we tell which has a greater area? We can.
A square will always have an area greater than that of a rectangle with an equal perimeter.
17.10: The Minimum Perimeter of a Rectangle
Now, let's consider rectangles of equal areas. Of all the possible differing dimensions that would equal such a
rectangle, the square will have the minimum perimeter.
Given a rectangle with a fixed area, the rectangle with the minimum perimeter is a square.
17.11: Trapezoids
A trapezoid is a quadrilateral in which one pair of opposite sides are parallel but the other pair of opposite sides are
not parallel.
If the two non parallel sides are equal in length, the trapezoid is referred to as an isosceles trapezoid.
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17.11.1: Area of a Trapezoid
Notice that when an altitude is drawn from point A down to point E, a right triangle(AED) is created.
17.12: The Interior Angles of a Polygon
The sum of the interior angles of a polygon = (n – 2) × 180, where n = the number of sides in the polygon.
When all of the interior angles of a polygon are of equal measure, all of the sides of the polygon are equal in length
as well, and we call the polygon a regular polygon.
In a regular polygon, the degree measure of any one of the interior angles can easily be calculated.
The measure of any one interior angle in a regular polygon = 180(n−2)/n , where n is the number of sides of the
polygon.
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17.13: Hexagons
A hexagon is a polygon that has six sides and six vertices. A hexagon is referred to as a regular hexagon when all
of its sides are of equal length and all of its interior angles are of equal measure.
The sum of the interior angles of any hexagon is 720°. Any one interior angle of a regular hexagon
measures 120°.
17.13.1: The Area of a Regular Hexagon
17.13.2: A Regular Hexagon Can Be Divided Into Six Equilateral Triangles
Let’s look at the regular hexagon with the three diagonals drawn through the center again. It follows that every line
segment in this figure is of equal length. In other words, AB = BC = CD =DE = EF = FA = AG = BG = CG = DG =
EG = FG.
A
B
G
F
E
C
D
A regular hexagon can be divided into six equilateral triangles.
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17.14: The Exterior Angles of Any Polygon Sum to 360°
The measures of the exterior angles of any polygon, when taking one exterior angle at each vertex, always add up to
360°.
Remember that an exterior angle of a polygon is an angle formed by any side of the polygon and the extension of its
adjacent side.
17.15: Comparing Polygons
At some point on the GMAT, you will be provided with a comparison between two polygons and asked to
determine some parameter (length, width, height, area, etc.) of one or both of the polygons.
For example, “The length of a side of a certain rectangle is two inches longer than the side of a certain square.” In
this scenario, we can let the side of the square be x.
Then, the length of the rectangle’s side would be x + 2. Notice that these questions are simply algebraic word
problems involving geometrical shapes.
17.16: Circles
Consider the diagram of the circle below. Its center is at point O. The total angle measurement of a circle is 360°.
A chord is a line segment that connects any two points on the circle. If a chord passes through the center of the
circle, it’s called a diameter of the circle. Thus, a diameter is the longest chord that can be drawn within a circle.
A radius is simply half of a diameter; it is a line segment drawn from the center of the circle to a point on the
circle.
The circumference of a circle can be expressed as either C = 2πr or C = πd.
The area of a circle = πr2 .
17.16.1: Three Equivalent Circle Ratios
Within any circle, three particularly important ratios can be equated with each other.
To better understand these ratios, let’s first consider the concepts of central angle, arc length, and sector area,
since each concept plays a part in the circle ratios.
A central angle is any angle at the center of the circle that is formed by two radii.
Notice below that angle x is the central angle since point O is the center of the circle and line segments OA and
OB are both radii.
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A central angle is any angle at the center of the circle that is formed by two radii.
Notice below that angle x is the central angle since point O is the center of the circle and line segments OA and OB
are both radii.
An arc is a portion of the circumference of a circle. Notice below that the section of the circumference from point
A to point B is an arc of the circle. It follows then that the arc length is the distance along that portion of the
circumference. We could say that arc AB is the arc inscribed by the central angle x.
Arc length is measured by a unit of length such as the inch or centimeter. However, the arc itself can be
measured in degrees. The measure of the arc in degrees is equal to the measure of the central angle that intercepts
it. For example, if central angle AOB is 60°, then arc AB is also 60°.
A sector of a circle is the region of a circle that is defined by two radii and their intercepted arc.
Notice that the entire region from point A to point O to point B and back to point A along the circumference is a
sector created by angle x. The sector area is then just the area of that region.
Now that we have that basic working terminology covered, we can discuss the three-part circle ratio.
In any circle, the following three-part ratio is always true:
Perimeter of sector AOB is AO+OB+arc lenght of AB.
If points A and B are two points on a circle and arc AB is not a semicircle, arc AB refers to the shorter portion of
the circumference between A and B. This shorter portion is also known as the minor arc.
The longer portion of the circumference between A and B is known as the major arc but is usually referred as arc
ACB, C representing some third point on this longer portion of the circumference.
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17.17: Inscribed Angles
An inscribed angle is an angle formed by two chords of a circle with the vertex of the angle on the circumference.
In the figure below, angle ABC is an inscribed angle, whereas angle AOC is a central angle.
The degree measure of an inscribed angle equals half of the degree measure of the arc that it intercepts.
Conversely, when an inscribed angle shares the same endpoints on the circumference as the central angle, the
degree measure of the central angle is twice the degree measure of the inscribed angle.
The degree measure of an inscribed angle is equal to half of the degree measure of the arc that it intercepts.
Conversely, when an inscribed angle shares the same endpoints as the central angle, the degree measure of
the central angle is twice the degree measure of the inscribed angle.
17.17.1: Triangles Inscribed in a Circle
A triangle is inscribed in a circle when all three of the triangle’s vertices lie on the circle’s circumference.
When a triangle is inscribed in a circle, if one side of the triangle is also the diameter of the circle, then the
triangle is a right triangle. Consider the circle below whose center is at point O.
Notice that since line segment XZ is a diameter of the circle and is also a side of the triangle, triangle XYZ must
be a right triangle, with angle XYZ measuring 90 degrees.
17.17.2: Using a Triangle to Determine the Length of an Arc
In the earlier section on inscribed angles, you learned that if an inscribed angle shares the same endpoints on the
circle as the central angle, the degree measure of the central angle is twice the degree measure of the inscribed
angle. We’re going to take that rule further here. Consider the figure below. Triangle XYZ is inscribed in the circle
whose center is at point O.
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It should be clear that 30° is not a central angle, but rather an inscribed angle within the triangle.
However, by applying our rule of inscribed angles, we can quickly determine the central angle.
Notice that by drawing a radius from point O to point Y,we’ve created angle XOY.
Because angle XOY is a central angle and because it shares the same endpoints on the circle as inscribed angle XZY,
the measure of angle XOY must be twice the measure of angle XZY. Thus, angle XOY measures 60°.
Since XOY is a central angle, 60° could be used within the circle ratio.
17.17.3: The Legs of an Isosceles Triangle Can Be the Radii of a Circle
Recognizing that the legs (two equal sides) of an isosceles triangle are also the radii of a circle can be a powerful
problem-solving skill.
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17.17.4: Equilateral Triangles Inscribed in Circles
When an equilateral triangle is inscribed in a circle, the center of the triangle coincides with the center of the circle.
A line segment drawn from the center of the triangle to an interior angle of the triangle would not only be a radius of
the circle but would also bisect that angle (recall that an interior angle of an equilateral trianglemeasures 60 degrees).
Furthermore, dropping a perpendicular line segment fromthe common center to the base of the triangle creates a
30-60-90 right triangle.
In the diagram below, OC is a radius that bisects the 60° angle, so angle OCB = 30°,and OB is perpendicular to CB, so
angle OBC = 90°.
As a consequence, triangle OBC is a 30-60-90 right triangle.
Recall that if we know the length of one of the triangle’s sides, we can easily calculate the radius using the known
relationships of the 30-60-90 right triangle.
17.17.5: An Equilateral Triangle Inscribed in a Circle Breaks the Circle Up Into Three Arcs of Equal Length
When an equilateral triangle is inscribed in a circle, the triangle divides the circumference into three arcs of equal
lengths. Consider the figure below.
If triangle ABC is an equilateral triangle inscribed in the circle, it must be true that the length of arc AB = the length
of arc BC = the length of arc CA.
Therefore, if the length of any of these arcs is known, the length of the circumference is easily derived.
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17.17.6: Circles Inscribed in Equilateral Triangles
A circle is inscribed in an equilateral triangle when each side of the
equilateral triangle is tangent to the circle, the circle is the largest circle that
can fit inside the equilateral triangle, and the two figures touch one another
at exactly three points.
It’s important to notice that each point at which the circle touches the
triangle is the midpoint of a side of the triangle.
In the diagram below, circle O is inscribed in equilateral triangle RST.
Notice that by drawing line segment OZ from the center of the circle to the
base of the triangle, and by drawing line segments OT and OS, we’ve
created two 30-60-90 right triangles.
In addition, notice that line segment OZ is a radius of the circle.
17.17.7: Squares Inscribed in Circles
A square is inscribed in a circle when all of the square’s
vertices lie on the circumference of the circle.
Notice that the square’s diagonals are both diameters of
the circle.
Therefore, if we know the length of one of the square’s
diagonals, we also know the length of the diameter of the
circle (and vice versa).
17.17.8: Circles Inscribed in Squares
A circle is inscribed in a square when each side of the square
is tangent to the circle,the circle is the largest circle that can
fit inside the square, and the two figures touch one another
at exactly four points.
Each point at which the circle touches the square is the
midpoint of a side of the square.
Furthermore, the diameter of the circle has the same length
as a side of the square.
Notice in the figure below that the length of the diameter of
the circle is equal to the length of a side of the square.
17.17.9: Triangles Inscribed in a Square
When a triangle is inscribed in a square, one side of the
triangle will coincide with one side of the square.
When this occurs, the side of the triangle and the side of the
square are of equal lengths, and the vertex of the triangle
opposite the coincident sides touches the opposite side of
the square.
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17.17.10: Rectangles Inscribed in Circles
A rectangle is inscribed in a circle when all four of the rectangle’s vertices lie on the circumference of the circle.
When this occurs, a diagonal of the rectangle is also a diameter of the circle.
17.17.11: Regular Polygons Inscribed in Circles
We know that a regular polygon is a polygon whose sides are all equal and whose angles are all equal.
When a regular polygon is inscribed in a circle, the polygon divides the circumference of the circle into arcs of
equal length. For example, an equilateral triangle that is inscribed in a circle breaks the circle up into three arcs of
equal lengths, a square inscribed in a circle breaks the circle upinto four arcs of equal length, a regular hexagon
inscribed in a circle breaks the circle up into six arcs of equal length, and a regular octagon inscribed in a circle
breaks the circle up into eight arcs of equal length.
17.17.12: Inscribing a Square Within a Square
A square is inscribed in another when all four vertices of the inscribed square are touching the four respective
edges of the circumscribed square.
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When a square is inscribed in a square, there is an important relationship to see: the closer the vertex of the
inscribed square to a vertex of the circumscribed square, the larger the area of the inscribed square.
Similarly, the farther away the vertex of the inscribed square from a vertex of the circumscribed square, the smaller
the area of the inscribed square.
The area of an inscribed square will be smallest when the vertices of that square are located at the midpoints of the
respective edges of the circumscribed square.
Furthermore, the area of such an inscribed square will be half of the area of the circumscribed square.
17.17.13: Rectangles Inscribed in Semicircles
When a rectangle is inscribed in a semicircle, two of the
vertices of the rectangle (as well as the side determined
by these two vertices) are on the diameter of the
semicircle. The other two vertices of the rectangle are on
the semicircular arc.
When a rectangle is inscribed in a semicircle, a single
vertical line can bisect both the semicircle and the
rectangle.
When a rectangle is inscribed in a
semicircle and a radius is drawn from
avertex of the rectangle on the semicircular
arc to the center of the circle, a
righttriangle is created.
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17.17.14: Square Inscribed in a Semicircle
When a square is inscribed in a semicircle and
a radius is drawn from one of the vertices of
the square on the semicircular arc to the
center of the circle, a right triangle is created
in which the base of the triangle is equal to
half of the side of the square, the height is
equal to the side of the square, and the
hypotenuse is equal to the radius of the circle.
17.18: Shaded Regions
In general, when a geometrical figure has both shaded and unshaded regions,the area of the entire figure
– the area of the unshaded region = the area of the shaded region.
17.19: Rectangular Uniform Borders
A “uniform border” is a border of equal width that surrounds some object.
In general, the diagram we create for a uniform border question will display a smaller rectangle inside a larger
rectangle, and the border between the smaller and larger rectangles will be uniform (or of the same width).
In a uniform border question, if the length of the rectangle inside the border = L,the width of the
rectangle inside the border = W, and the border width = x, then the length of the border and inside
rectangle = L + 2x and the width of the border and inside rectangle = W + 2x.
Two formulas commonly used to solveuniform border questions are:
Area of rectangle and border = (W + 2x)(L + 2x)
Area of border alone = (W + 2x)(L + 2x) - LW
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17.20: The Area of a Circular Ring
In a two-circle system, the area of the outer ring can be calculated using
17.21: Solid Geometry
Solid geometry problems deal with three-dimensional shapes. The most important three-dimensional shapes to
know for the GMAT are the cube, the rectangular solid,and the right circular cylinder.
17.22: The Cube and the Rectangular Solid
A cube is a three-dimensional shape with six square sides, referred to as faces.
Because the six faces that bound the cube are each squares, the length of any edge of a cube equals the length of any
other edge. In addition, the area of any of the cube's faces equals that of any other face.
A rectangular solid (or a rectangular box) is a three-dimensional shape with six rectangular faces in which opposite
faces are of equal areas.
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17.22.1: Volume of a Cube or Rectangular Solid
The volume of a solid is the region enclosed within its borders. The volume of a cube= (edge)3.
The volume of a rectangular solid = length × width × height. Volume is expressed in cubic units, such as inches3,
meters3, and feet3.
17.22.2: The Longest Line Segment That Can Be Drawn Within a Rectangular Solid or Cube
Given any rectangular solid or cube, the longest line segment that can be drawn within the solid will be a diagonal
that goes from a corner of the box or cube, through the center of the box or cube, to the opposite corner.
Consider the rectangular solid below.
If l is the length of the box, w is the width of the box, and h is the height of thebox, the diagonal can be calculated
using the extended Pythagorean theorem: d2 = l2 + w2 + h2, where d is the diagonal of the box.
When the box is a cube, the diagonal can be calculated by d = s√3 , where s is the length of one edge of the cube.
17.22.3: Surface Area of a Rectangular Solid or Cube
The surface area of an object represents the combined areas of the outside regions of that object.
When thinking about surface area, it’s helpful to remember that an object’s surface area is the area that would have
to be painted to color the outside of the object completely. The surface area of a cube is the area of a face times the
number of faces. Therefore, the surface area of a cube = 6s2 , where s is the length of one side (or edge) of the cube.
Not all the faces of a rectangular solid are equal, so the surface area of a rectangular solid is a little more
complicated: surface area =2(LW) + 2(LH) + 2(HW), where L is the length, W is the width, and H is the height.
17.23: The Right Circular Cylinder
A right circular cylinder is a cylinder whose bases are circles that are perpendicular to the height of the cylinder.
The word “right” means the top and the bottom bases of the cylinder are aligned with each other.
If the bases are not aligned, the figure is termed an oblique cylinder. See diagrams above.
The word “circular” means the bases of the cylinder are circles. The bases of a cylinder can be ellipses. Such a
cylinder would be called an elliptic cylinder. See the diagram below.
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CH 17: Geometry
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As you can see, a right circular cylinder is just a special type of cylinder.
We might just as well have a right elliptic cylinder, an oblique circular cylinder, or an oblique elliptic cylinder.
However, when we say “cylinder,” we generally have a right circular cylinder in mind.
This is also the case on the test: if a problem on the GMAT simply says “cylinder,” it will always mean a right
circular cylinder.
17.23.1: Volume and Surface Area of a Right Circular Cylinder
The volume of a cylinder is the region contained within the figure and can be calculated by the formula
Volume = πr2h.
The surface area of a cylinder is the entire area of the outside of the cylinder, the region that would need to be
painted to completely change the color of the outside of the figure.
The surface area of any cylinder can be broken down into three sections: the two circular bases and the rolled
rectangle that forms the side wall of the cylinder.
The formula for this area is Surface area of a right circular cylinder = 2πr2 + 2πrh.
17.24: Volume and Rate
Because cylinders, cubes, and rectangular solids are capable of holding liquids,problems sometimes arise that
combine the principles of these solid figures with those of rates.
Remember that time = distance/rate.
When a liquid fills a container, the rate equation for time becomes time = volume of container/rate.
Notice that the time to fill the container is directly proportional to the volume of the container and inversely
proportional to the rate at which the container fills.
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CH 17: Geometry
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17.24.1: Common Volume Rate Traps
Be careful when handling volume-rate problems as there are a few common traps to be avoided.
Consider a rectangular box with a volume of 1,000 cm3.
If we knew that water was flowing into the box at a rate of 10 cm3 per minute, could we determine how high the
water in the box would be after 10 minutes?
Because many different combinations of length, width, and height dimensions can produce a volume of 1,000 cm3
, we cannot determine, based on volume alone, how high the water would be.
For example, the dimensions of the box could be length =1,000 cm, width = 1 cm, and height = 1 cm.
Or, the dimensions could be length = 100cm, width = 1 cm, and height = 10 cm.
Although both boxes have the same volume, the water levels, their heights, would rise at different rates even with a
constant flow rate.
A tall skinny box will allow the water to rise much faster than a short fat one. Thus, to calculate how quickly the
water would rise, the exact dimensions of the container must be known.
Another type of volume trap deals with placing objects into a box. Consider a box with a volume of 36 cm3.
Is it possible to determine the number of smaller boxes,each with a volume of 6 cm3, that would fit within the large
box? Not without more information.
Boxes of different dimensions will fit into each other differently,regardless of any similarity in volume.
Thus, to determine the number of smaller boxes that will completely fit within a larger box, we must know the
exact dimensions of both the larger box and the smaller box.
17.25: Ratios in Geometrical Figures
Because geometrical shapes lend themselves to questions that integrate ratios, it’s worth seeing how ratios can show
up in geometry problems.
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