ACT Math Flashcards: Polygons

What this deck covers

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

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: Identify the shape: A quadrilateral with only one pair of parallel sides.

Answer: Trapezoid. Has exactly one pair of parallel sides called bases.

Flashcard 2: State the number of diagonals in an $n$-sided polygon.

Answer: Diagonals = $\frac{n(n-3)}{2}$. Each vertex connects to $(n-3)$ non-adjacent vertices, divided by 2.

Flashcard 3: What is the measure of each interior angle in a regular pentagon?

Answer: 108°. Apply $\frac{(5-2) \times 180°}{5} = \frac{540°}{5} = 108°$.

Flashcard 4: What is each exterior angle of a regular nonagon ($n=9$)?

Answer: $40$. Using $\frac{360}{9} = 40°$ for each exterior angle.

Flashcard 5: What is the sum of the interior angles of a pentagon?

Answer: 540°. Apply $(5-2) \times 180° = 3 \times 180° = 540°$.

Flashcard 6: What is the number of diagonals in an $n$-gon?

Answer: $\frac{n(n-3)}{2}$. Each vertex connects to $(n-3)$ non-adjacent vertices.

Flashcard 7: What is the sum of interior angles of a pentagon?

Answer: $540$. Using $180(5-2) = 180(3) = 540°$.

Flashcard 8: What is the measure of each exterior angle in a regular hexagon?

Answer: 60°. Apply $\frac{360°}{6} = 60°$ for each exterior angle.

Flashcard 9: What is the number of sides $n$ if a polygon has $44$ diagonals?

Answer: $11$. From $\frac{n(n-3)}{2} = 44$, solving gives $n = 11$.

Flashcard 10: What is each interior angle of a regular octagon?

Answer: $135^\circ$. Using $\frac{180(8-2)}{8} = \frac{1080}{8} = 135^\circ$

Flashcard 11: What is the measure of each interior angle of a regular octagon?

Answer: 135°. Apply $\frac{(8-2) \times 180°}{8} = \frac{1080°}{8} = 135°$.

Flashcard 12: What is the sum of exterior angles (one at each vertex) of any convex polygon?

Answer: $360$. Always $360°$ regardless of polygon shape or number of sides.

Flashcard 13: What is the number of diagonals in a polygon with $12$ sides?

Answer: $54$. Using $\frac{12(12-3)}{2} = \frac{12 \cdot 9}{2} = 54$.

Flashcard 14: What is the term for a polygon with four sides and four angles?

Answer: Quadrilateral. A polygon with exactly four sides and four vertices.

Flashcard 15: What is the formula for the sum of interior angles of an $n$-sided polygon?

Answer: Sum = $(n - 2) \times 180^\text{o}$. Each interior angle reduces the total by $180°$ from a full circle.

Flashcard 16: Identify the type of polygon with all sides and angles equal.

Answer: Regular polygon. All sides are congruent and all angles are congruent.

Flashcard 17: What is the formula for the area of a regular hexagon with side length $s$?

Answer: Area = $\frac{3\text{√}3}{2} s^2$. Derived from dividing a hexagon into six equilateral triangles.

Flashcard 18: What is the definition of a regular polygon?

Answer: All sides and all interior angles are congruent. Equilateral (equal sides) and equiangular (equal angles).

Flashcard 19: What is the number of sides $n$ if the sum of interior angles is $2520$?

Answer: $16$. From $180(n-2) = 2520$, solving gives $n = 16$.

Flashcard 20: What is the number of diagonals in an $n$-gon?

Answer: $\frac{n(n-3)}{2}$. Each vertex connects to $(n-3)$ non-adjacent vertices.

Flashcard 21: What is the measure of each interior angle of a regular $n$-gon?

Answer: $\frac{180(n-2)}{n}$. Sum of all interior angles divided by the number of angles.

Flashcard 22: What is each interior angle of a regular hexagon?

Answer: $120$. Using $\frac{180(6-2)}{6} = \frac{720}{6} = 120^\circ$

Flashcard 23: What is the sum of interior angles of an octagon?

Answer: $1080^\circ$. Using $180(8-2) = 180(6) = 1080^\circ$.

Flashcard 24: What is the apothem $a$ of a regular polygon with area $150$ and perimeter $50$?

Answer: $6$. From $150 = \frac{1}{2} \cdot a \cdot 50$, solving gives $a = 6$.

Flashcard 25: What is the formula for the perimeter of a regular polygon with side length $s$?

Answer: Perimeter = $n \times s$. Multiply the number of sides by the common side length.

Flashcard 26: What is the number of triangles formed by drawing diagonals from one vertex of a convex $n$-gon?

Answer: $n-2$. Diagonals from one vertex create $(n-2)$ triangles.

Flashcard 27: What is the number of sides $n$ if the sum of interior angles is $1260$?

Answer: $9$. From $180(n-2) = 1260$, solving gives $n = 9$.

Flashcard 28: What is the number of sides of a regular polygon with each interior angle $165$?

Answer: $24$. Exterior angle is $15°$, so $\frac{360°}{15°} = 24$ sides.

Flashcard 29: What is the sum of interior angles of a pentagon?

Answer: $540$. Using $180(5-2) = 180(3) = 540°$.

Flashcard 30: How many diagonals does a decagon have?

Answer:

1. Apply $\frac{10(10-3)}{2} = \frac{10 \times 7}{2} = 35$.

Flashcard 31: What is the measure of each exterior angle of a regular pentagon?

Answer: 72°. Apply $\frac{360°}{5} = 72°$ for each exterior angle.

Flashcard 32: What is each interior angle of a regular hexagon?

Answer: $120$. Using $\frac{180(6-2)}{6} = \frac{720}{6} = 120^\circ$

Flashcard 33: What is the sum of interior angles of a dodecagon ($n=12$)?

Answer: $1800$. Using $180(12-2) = 180(10) = 1800°$.

Flashcard 34: What is a four-sided polygon called?

Answer: Quadrilateral. The prefix 'quad-' means four in Latin.

Flashcard 35: What is the sum of interior angles of a dodecagon ($n=12$)?

Answer: $1800$. Using $180(12-2) = 180(10) = 1800°$.

Flashcard 36: Calculate the area of a rectangle with length 10 and width 4.

Answer:

1. Apply length × width = $10 \times 4 = 40$.

Flashcard 37: What is the formula for the area of a kite?

Answer: Area = $\frac{d_1 \times d_2}{2}$. Uses the product of diagonals divided by 2.

Flashcard 38: What is the area of a regular polygon in terms of perimeter $P$ and apothem $a$?

Answer: $\frac{1}{2}aP$. Area equals half the product of apothem and perimeter.

Flashcard 39: State the number of diagonals in an $n$-sided polygon.

Answer: Diagonals = $\frac{n(n-3)}{2}$. Each vertex connects to $(n-3)$ non-adjacent vertices, divided by 2.

Flashcard 40: State the formula for the measure of each interior angle of a regular polygon.

Answer: Angle = $\frac{(n-2) \times 180^\text{o}}{n}$. Divides the total interior angle sum by the number of sides.

Flashcard 41: What is the measure of each interior angle in a regular pentagon?

Answer: 108°. Apply $\frac{(5-2) \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ$.

Flashcard 42: What is the term for a polygon with four sides and four angles?

Answer: Quadrilateral. A polygon with exactly four sides and four vertices.

Flashcard 43: What is the definition of a concave polygon?

Answer: At least one interior angle is greater than $180$. Contains at least one reflex interior angle.

Flashcard 44: What is each exterior angle of a regular polygon with $n=18$ sides?

Answer: $20$. Using $\frac{360°}{18} = 20°$ for each exterior angle.

Flashcard 45: What is the formula for the area of a rectangle?

Answer: Area = length × width. A rectangle is a special parallelogram with right angles.

Flashcard 46: What is the number of sides of a regular polygon with each interior angle $150$?

Answer: $12$. Exterior angle is $30°$, so $\frac{360°}{30°} = 12$ sides.

Flashcard 47: What is the formula for the area of a rhombus?

Answer: Area = $\frac{d_1 \times d_2}{2}$. Uses the product of diagonals divided by 2.

Flashcard 48: What is the area of an equilateral triangle with side length 6?

Answer: $9\text{√}3$. Apply $\frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}$.

Flashcard 49: State the formula for the area of a trapezoid.

Answer: Area = $\frac{1}{2} (b_1 + b_2) \times h$. Average of parallel bases times the perpendicular height.

Flashcard 50: What is the sum of the exterior angles of any polygon?

Answer: 360°. This sum is constant for all polygons regardless of side count.

Flashcard 51: What is the formula for the area of a parallelogram?

Answer: Area = base × height. Height is perpendicular distance between parallel sides.

Flashcard 52: What is each exterior angle of a regular nonagon ($n=9$)?

Answer: $40$. Using $\frac{360}{9} = 40°$ for each exterior angle.

Flashcard 53: State the formula for the measure of each interior angle of a regular polygon.

Answer: Angle = $\frac{(n-2) \times 180^\text{o}}{n}$. Divides the total interior angle sum by the number of sides.

Flashcard 54: What is the number of triangles formed from one vertex in a nonagon ($n=9$)?

Answer: $7$. Using $n-2 = 9-2 = 7$ triangles from one vertex.

Flashcard 55: What is the sum of the interior angles of a hexagon?

Answer: 720°. Apply $(6-2) \times 180° = 4 \times 180° = 720°$.

Flashcard 56: What is the relationship between an interior angle and its adjacent exterior angle?

Answer: $\text{interior} + \text{exterior} = 180$. They form a linear pair on a straight line.

Flashcard 57: What is the measure of each exterior angle in a regular decagon?

Answer: 36°. Apply $\frac{360°}{10} = 36°$ for each exterior angle.

Flashcard 58: What is the number of diagonals in a decagon ($n=10$)?

Answer: $35$. Using $\frac{10(10-3)}{2} = \frac{10 \cdot 7}{2} = 35$.

Flashcard 59: What is the definition of a diagonal of a polygon?

Answer: A segment joining two nonadjacent vertices. Connects vertices that don't share a side.

Flashcard 60: What is the formula for the area of a kite?

Answer: Area = $\frac{d_1 \times d_2}{2}$. Uses the product of diagonals divided by 2.

Flashcard 61: What is the number of sides of a regular polygon with each exterior angle $24$?

Answer: $15$. Since $\frac{360°}{24°} = 15$ sides for the polygon.

Flashcard 62: Identify the polygon with all sides and angles congruent.

Answer: Regular polygon. All sides are congruent and all angles are congruent.

Flashcard 63: What is each interior angle of a regular polygon with $n=20$ sides?

Answer: $162$. Using $\frac{180(20-2)}{20} = \frac{3240}{20} = 162°$.

Flashcard 64: What is the formula for the area of a triangle?

Answer: Area = $\frac{1}{2}$ × base × height. Height is perpendicular distance from base to opposite vertex.

Flashcard 65: What is the number of diagonals in a heptagon ($n=7$)?

Answer: $14$. Using $\frac{7(7-3)}{2} = \frac{7 \cdot 4}{2} = 14$.

Flashcard 66: What is the number of diagonals in a decagon ($n=10$)?

Answer: $35$. Using $\frac{10(10-3)}{2} = \frac{10 \cdot 7}{2} = 35$.

Flashcard 67: What is the sum of the interior angles of a hexagon?

Answer: 720°. Apply $(6-2) \times 180° = 4 \times 180° = 720°$.

Flashcard 68: What is the relationship between an interior angle and its adjacent exterior angle?

Answer: $\text{interior} + \text{exterior} = 180$. They form a linear pair on a straight line.

Flashcard 69: Calculate the area of a rectangle with length 10 and width 4.

Answer:

1. Apply length × width = $10 \times 4 = 40$.

Flashcard 70: What is the measure of each exterior angle of a regular $n$-gon?

Answer: $\frac{360}{n}$. Total exterior angle sum divided by number of vertices.

Flashcard 71: What is the area of a regular polygon in terms of perimeter $P$ and apothem $a$?

Answer: $\frac{1}{2}aP$. Area equals half the product of apothem and perimeter.

Flashcard 72: What is the number of diagonals in a polygon with $12$ sides?

Answer: $54$. Using $\frac{12(12-3)}{2} = \frac{12 \cdot 9}{2} = 54$.

Flashcard 73: What is the number of triangles formed from one vertex in a polygon with $n=14$ sides?

Answer: $12$. Using $n-2 = 14-2 = 12$ triangles from one vertex.

Flashcard 74: What is the measure of each exterior angle in a regular decagon?

Answer: 36°. Apply $\frac{360°}{10} = 36°$ for each exterior angle.

Flashcard 75: What is the formula for the area of a regular hexagon with side length $s$?

Answer: Area = $\frac{3\text{√}3}{2} s^2$. Derived from dividing a hexagon into six equilateral triangles.

Flashcard 76: What is each interior angle of a regular octagon?

Answer: $135$. Using $\frac{180(8-2)}{8} = \frac{1080}{8} = 135°$.

Flashcard 77: What is the interior angle if the exterior angle of a convex polygon is $55$?

Answer: $125$. Interior and exterior angles are supplementary: $180° - 55° = 125°$.

Flashcard 78: What is the sum of interior angles of a polygon with $15$ sides?

Answer: $2340$. Using $180(15-2) = 180(13) = 2340°$.

Flashcard 79: What is the definition of a diagonal of a polygon?

Answer: A segment joining two nonadjacent vertices. Connects vertices that don't share a side.

Flashcard 80: Identify the type of polygon with all sides and angles equal.

Answer: Regular polygon. All sides are congruent and all angles are congruent.

Flashcard 81: What is the number of sides $n$ if the sum of interior angles is $2520$?

Answer: $16$. From $180(n-2) = 2520$, solving gives $n = 16$.

Flashcard 82: What is the definition of a regular polygon?

Answer: All sides and all interior angles are congruent. Equilateral (equal sides) and equiangular (equal angles).

Flashcard 83: What is the number of triangles formed by drawing diagonals from one vertex of a convex $n$-gon?

Answer: $n-2$. Diagonals from one vertex create $(n-2)$ triangles.

Flashcard 84: Identify the polygon with all sides and angles congruent.

Answer: Regular polygon. All sides are congruent and all angles are congruent.

Flashcard 85: What is the formula for the perimeter of a regular polygon with side length $s$?

Answer: Perimeter = $n \times s$. Multiply the number of sides by the common side length.

Flashcard 86: What is the number of triangles formed from one vertex in a polygon with $n=14$ sides?

Answer: $12$. Using $n-2 = 14-2 = 12$ triangles from one vertex.

Flashcard 87: What is the number of diagonals in a hexagon?

Answer: $9$. Using $\frac{6(6-3)}{2} = \frac{6 \cdot 3}{2} = 9$.

Flashcard 88: What is the measure of each exterior angle of a regular $n$-gon?

Answer: $\frac{360}{n}$. Total exterior angle sum divided by number of vertices.

Flashcard 89: What is the measure of an exterior angle if the interior angle is $112$?

Answer: $68$. Interior and exterior angles are supplementary: $180° - 112° = 68°$.

Flashcard 90: What is the sum of interior angles of an $n$-gon?

Answer: $180(n-2)$. Divides polygon into $(n-2)$ triangles, each contributing $180°$.

Flashcard 91: What is the number of diagonals in a hexagon?

Answer: $9$. Using $\frac{6(6-3)}{2} = \frac{6 \cdot 3}{2} = 9$.

Flashcard 92: State the formula for the area of a trapezoid.

Answer: Area = $\frac{1}{2} (b_1 + b_2) \times h$. Average of parallel bases times the perpendicular height.

Flashcard 93: Calculate the area of a rhombus with diagonals 10 and 8.

Answer:

1. Apply $\frac{10 \times 8}{2} = \frac{80}{2} = 40$.

Flashcard 94: What is a polygon with three sides called?

Answer: Triangle. The simplest polygon with three vertices and sides.

Flashcard 95: What is the number of sides of a regular polygon with each exterior angle $30$?

Answer: $12$. Since $\frac{360°}{30°} = 12$ sides for the polygon.

Flashcard 96: What is each exterior angle of a regular polygon with $n=18$ sides?

Answer: $20$. Using $\frac{360°}{18} = 20°$ for each exterior angle.

Flashcard 97: What is the apothem $a$ of a regular polygon with area $150$ and perimeter $50$?

Answer: $6$. From $150 = \frac{1}{2} \cdot a \cdot 50$, solving gives $a = 6$.

Flashcard 98: What is the area of an equilateral triangle with side length 6?

Answer: $9\text{√}3$. Apply $\frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}$.

Flashcard 99: Find the area of a parallelogram with base 8 and height 5.

Answer:

1. Apply base × height = $8 \times 5 = 40$.

Flashcard 100: What is the number of sides $n$ if the sum of interior angles is $1260$?

Answer: $9$. From $180(n-2) = 1260$, solving gives $n = 9$.