GRE Quantitative Flashcards: Polygons Circles

What this deck covers

This deck focuses on Polygons Circles, giving you a quick way to review the definitions, rules, and examples that matter most for GRE Quantitative.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the number of diagonals in a nonagon.

Answer: $27$. For $n=9$, diagonals are rac{9(6)}{2}=27 in a nonagon.

Flashcard 2: Find the area of a sector when $r=10$ and $\theta=90$ degrees.

Answer: $25\pi$. Sector area is rac{90}{360} imes pi imes 10^2 = 25pi.

Flashcard 3: Find each exterior angle of a regular decagon.

Answer: $36$. For $n=10$, each exterior angle is $360/10=36$ degrees in a regular decagon.

Flashcard 4: What is the relationship between a central angle and its intercepted arc measure (degrees)?

Answer: They are equal in degree measure. The central angle subtends the arc directly at the center, matching its measure.

Flashcard 5: State the Pythagorean theorem for a right triangle with legs $a,b$ and hypotenuse $c$.

Answer: $a^2+b^2=c^2$. Relates the sides of a right triangle where the square of the hypotenuse equals the sum of squares of the legs.

Flashcard 6: Find the circle equation with center $(2,-3)$ and radius $5$.

Answer: $(x-2)^2+(y+3)^2=25$. Standard form with center $(2,-3)$ shifts $x-2$ and $y+3$, radius squared as $25$.

Flashcard 7: What is the measure of an inscribed angle that intercepts an arc of measure $x$ degrees?

Answer: $\frac{x}{2}$. Inscribed angles subtend the arc from the circumference, measuring half the arc.

Flashcard 8: What is the measure of an angle formed by a tangent and a chord intercepting arc $x$ degrees?

Answer: $\frac{x}{2}$. Such angles measure half the intercepted arc per the tangent-chord theorem.

Flashcard 9: Identify the relationship between a radius and a tangent at the point of tangency.

Answer: They are perpendicular. The tangent is perpendicular to the radius at the contact point due to the circle's symmetry.

Flashcard 10: Identify the number of diagonals in a convex $n$-gon.

Answer: $\frac{n(n-3)}{2}$. Counts lines connecting non-adjacent vertices, subtracting sides and adjusting for double-counting.

Flashcard 11: State the area of a sector for central angle $\theta$ in degrees and radius $r$.

Answer: $\frac{\theta}{360}\cdot \pi r^2$. Proportion of the full circle area based on the central angle fraction.

Flashcard 12: What is the equation of a circle with center $(h,k)$ and radius $r$?

Answer: $(x-h)^2+(y-k)^2=r^2$. Defines all points at distance $r$ from the center using squared distances.

Flashcard 13: What is each interior angle of a regular $n$-gon?

Answer: $\frac{180(n-2)}{n}$. Obtained by dividing the sum of interior angles by $n$ for equal distribution in a regular polygon.

Flashcard 14: Find each interior angle of a regular hexagon.

Answer: $120$. For $n=6$, each angle is rac{180(4)}{6}=120 degrees in a regular hexagon.

Flashcard 15: What is the area of a circle with radius $r$?

Answer: $\pi r^2$. Represents the integral of infinitesimal areas within the circle's boundary.

Flashcard 16: What is each exterior angle of a regular $n$-gon?

Answer: $\frac{360}{n}$. Calculated by dividing the total sum of exterior angles by $n$ for regularity.

Flashcard 17: State the arc length of a circle for central angle $\theta$ in degrees and radius $r$.

Answer: $\frac{\theta}{360}\cdot 2\pi r$. Proportion of the full circumference based on the central angle fraction.

Flashcard 18: What is the area of a triangle with base $b$ and height $h$?

Answer: $\frac{1}{2}bh$. Computes the area as half the product of base and corresponding height.

Flashcard 19: What is the sum of exterior angles of any convex polygon (one per vertex)?

Answer: $360$. Results from the fact that exterior angles sum to a full rotation around a point.

Flashcard 20: What is the area of a regular polygon with perimeter $P$ and apothem $a$?

Answer: $\frac{1}{2}aP$. Represents the sum of areas of triangles formed by the apothem and each side.

Flashcard 21: State the sum of interior angles of an $n$-gon.

Answer: $180(n-2)$. Derived by dividing the polygon into $n-2$ triangles, each contributing $180$ degrees.

Flashcard 22: What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Applies the Pythagorean theorem in two dimensions to find straight-line distance.

Flashcard 23: What is the circumference of a circle with radius $r$?

Answer: $2\pi r$. Measures the perimeter as twice pi times the radius.

Flashcard 24: Find the arc length when $r=6$ and $\theta=60$ degrees.

Answer: $2\pi$. Arc length is rac{60}{360} imes 2pi imes 6 = 2pi.