GRE · Geometry · TOPIC LESSON
Connect shape definitions, angle totals and perpendicular heights to measurements that really follow.
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BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GRE General Test study; it does not reproduce official questions, certify a writing score or predict test content.
LEARN THROUGH ORIGINAL PRACTICE
All teaching passages, writing prompts, examples and exercises in this lesson are original. No official ETS questions, prompts or explanations are reproduced. Current structure and observed public sample metadata are separate from this original teaching; no official score or question probability is implied.
01 / UNDERSTAND & APPLY
All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.
A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.
A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.
02 / UNDERSTAND & APPLY
A polygon is a closed plane figure made of straight segments. Its perimeter is the sum of the boundary lengths. A simple polygon has no crossing sides. A convex polygon has every interior angle less than 180°; a concave polygon has at least one interior angle greater than 180°. A regular polygon has both equal sides and equal angles.
A quadrilateral has four sides and interior angles totaling 360°. A parallelogram has two pairs of parallel opposite sides. Opposite sides and opposite angles are equal, and consecutive angles are supplementary. A rectangle is a parallelogram with four right angles; a rhombus is a parallelogram with four equal sides; a square has both sets of properties.
A drawing that resembles a rectangle is not sufficient evidence of right angles. Likewise, a rhombus need not be a square. Shape names can overlap: every square is also a rectangle and a rhombus. Check the definition instead of treating names as mutually exclusive categories.
03 / UNDERSTAND & APPLY
A convex n-sided polygon can be divided from one vertex into n − 2 nonoverlapping triangles. Its interior-angle sum is therefore (n − 2)180°. For ordinary simple concave polygons, the same total holds through an appropriate triangulation. Use the total even if the individual angles are not equal.
In a regular n-gon each interior angle is (n − 2)180°/n. Dividing by n is justified only by the equal-angle condition. Equal side lengths alone do not guarantee equal angles for polygons with more than three sides.
ORIGINAL WORKED EXAMPLE
Four interior angles of a pentagon measure 94°, 108°, 121°, and 103°. Find the fifth.
Answer 114°.
04 / UNDERSTAND & APPLY
At each vertex of a convex polygon, extend one side and take the exterior turning angle in a consistent direction. The exterior angles sum to 360°. Each chosen exterior angle and its adjacent interior angle sum to 180°. Do not mix reflex angles, alternate turning directions, or multiple exterior angles at one vertex.
For a regular polygon, every exterior angle equals 360°/n. This is often the fastest route from an angle to the number of sides. A computed n must be a whole number at least 3; an incompatible value means the proposed regular polygon cannot exist.
ORIGINAL WORKED EXAMPLE
Each interior angle of a regular polygon is 156°. Find its number of sides.
Answer 15 sides.
05 / UNDERSTAND & APPLY
A rectangle with length l and width w has perimeter 2(l + w), area lw, and diagonal √(l² + w²). A square with side s has perimeter 4s, area s², and diagonal s√2. A diagonal contributes to neither the outer perimeter nor an extra area; it merely partitions the existing region.
A fixed perimeter gives a fixed sum l + w, not a fixed product lw. For positive l and w with sum S, the product is S²/4 − (l − w)²/4. Thus among rectangles of fixed perimeter, a square has the greatest area. This algebraic observation avoids calculus and explains why long thin rectangles cover less area.
ORIGINAL WORKED EXAMPLE
A rectangle has perimeter 46 cm, and its length is 5 cm greater than its width. Find its area.
Answer 126 cm².
06 / UNDERSTAND & APPLY
A parallelogram with base b and perpendicular height h has area bh. Imagine moving a triangular piece from one side to the other to create a rectangle: the base and altitude determine the area. The length of a sloping side generally exceeds the height and is used for perimeter, not automatically for area.
Opposite sides of a parallelogram are equal, so perimeter is 2(a + b), where a and b are adjacent side lengths. An altitude can meet an extension of the base, especially in a strongly slanted figure. The perpendicular distance between the parallel supporting lines remains the relevant height.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
A parallelogram has base 12 m, adjacent side 7 m, and height 5 m relative to that base. Find area and perimeter.
Answer Area 60 m²; perimeter 38 m.
07 / UNDERSTAND & APPLY
For a quadrilateral with parallel bases b₁ and b₂ separated by perpendicular height h, area is (b₁ + b₂)h/2. This trapezoid formula can be understood by combining two copies into a parallelogram whose base is b₁ + b₂. Definitions of “trapezoid” vary on whether a second parallel pair is allowed; here the formula requires the named pair of parallel bases and works in either convention.
A rhombus has perpendicular diagonals that bisect one another. Its area can therefore be found as d₁d₂/2 by adding four right-triangle areas. A general quadrilateral does not have perpendicular diagonals, so the formula cannot be applied without the relevant condition. A rectangle’s diagonals are equal but usually not perpendicular.
ORIGINAL WORKED EXAMPLE
A trapezoid has parallel bases 80 cm and 1.4 m, and perpendicular height 0.6 m. Find its area in square meters.
Answer 0.66 m².
ORIGINAL WORKED EXAMPLE
A rhombus has diagonals of lengths 10 cm and 24 cm. Find its area and perimeter.
Answer Area 120 cm²; perimeter 52 cm.
08 / UNDERSTAND & APPLY
An L-shaped region can be treated as a larger rectangle with a rectangular corner removed, or as two nonoverlapping smaller rectangles. Both decompositions must cover the same region exactly once. For perimeter, walk around the remaining boundary and count every exposed edge, including new edges at a notch.
For example, removing a 3 by 2 corner rectangle from a 10 by 7 rectangle reduces area from 70 to 64 square units. The perimeter remains 34 units: the removed exterior lengths 3 and 2 are replaced by interior-facing exposed lengths 3 and 2. Removing a hole entirely inside the rectangle is different: it adds an inner boundary if total boundary length is requested.
For fencing, paving, or paint problems, read what is being covered. A gate may remove part of a fence length; a border has area; a path around a shape may require subtraction of two regions. Do not infer which boundary is included without the wording.
09 / UNDERSTAND & APPLY
Two adjacent side lengths determine the area of a rectangle, but not of a general parallelogram. The parallelogram can shear, changing its perpendicular height while its side lengths remain fixed. Four equal sides determine a rhombus, but still leave its angles and area free.
A QC comparison can also be fixed without a unique area. A nonsquare rectangle of perimeter 20 has area strictly below 25 because the square maximizes area. The word “nonsquare” matters: if a square is allowed, equality is possible, so the comparison against 25 would not always be strictly less.
ORIGINAL WORKED EXAMPLE
QC. A rectangle has perimeter 24. Quantity A: its area. Quantity B: 30.
Answer D: the relationship cannot be determined.
10 / UNDERSTAND & APPLY
For QC choose A, B, C or D using the key at the start. Other questions state the exact selection requirement.
ORIGINAL PRACTICE · 1
900
A heptagon has seven sides; (7 − 2)180 = 900°.
ORIGINAL PRACTICE · 2
C. 12
Consistent exterior turning angles total 360°, so n = 360/30 = 12.
ORIGINAL PRACTICE · 3
A and C
A square satisfies the rectangle definition, and equal sides define a rhombus. A 2 by 5 rectangle disproves B; that same nonsquare rectangle is a parallelogram with nonperpendicular diagonals, disproving D.
ORIGINAL PRACTICE · 4
107
Consecutive parallelogram angles are supplementary: 180 − 73 = 107°.
ORIGINAL PRACTICE · 5
C
Each side is one quarter of the common perimeter, so side lengths and their squares are equal.
11 / UNDERSTAND & APPLY
Draw a small annotated sketch when it helps, but keep every calculation tied to a stated dimension.
ORIGINAL PRACTICE · 1
105
Area = base×height = 15×7 = 105. The 9 cm side is not perpendicular height.
ORIGINAL PRACTICE · 2
78
Area = (8 + 18)6/2 = 78 square units.
ORIGINAL PRACTICE · 3
C. 36 cm
The side is the positive square root of 81, namely 9 cm; perimeter is 4×9 = 36 cm.
ORIGINAL PRACTICE · 4
D
With an 8-unit base, perpendicular heights 2 and 4 are both possible for a 5-unit sloping side. The corresponding areas are 16 and 32, on opposite sides of 30.
ORIGINAL PRACTICE · 5
96
Subtract the nonoverlapping removed area: 12×9 − 4×3 = 108 − 12 = 96.
ORIGINAL PRACTICE · 6
A and B
The side sum is 14. Sides 1 and 13 give area 13; sides 7 and 7 give area 49. The fixed-sum maximum is 49, so 50 and 56 are impossible.
KEEP THE BIG IDEAS
READ WITH CONTEXT
The teaching notes, passages, worked explanations and practice are original. Official references identify the assessment framework and skill scope; they do not endorse this lesson or license official question reproduction.
GRE analytics distinguish current test structure from a small independently classified public sample selection. Writing instruction has no observed prompt-frequency denominator. These are not historical administered-exam frequencies.