Polygons - ACT Math

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

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

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

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

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

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

4. What is the sum of the interior angles of a pentagon?

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

5. What is the number of diagonals in an $n$-gon?

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

6. What is the number of sides $n$ if a polygon has $44$ diagonals?

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

7. What is each interior angle of a regular octagon?

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

8. What is the term for a polygon with four sides and four angles?

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

9. What is the formula for the area of a regular hexagon with side length $s$?

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

10. What is the definition of a regular polygon?

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

11. What is the measure of each interior angle of a regular $n$-gon?

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

12. What is each interior angle of a regular hexagon?

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

13. What is the sum of interior angles of an octagon?

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

14. What is the apothem $a$ of a regular polygon with area $150$ and perimeter $50$?

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

15. What is the formula for the perimeter of a regular polygon with side length $s$?

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

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

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

17. What is the number of sides of a regular polygon with each interior angle $165$?

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

18. What is the sum of interior angles of a pentagon?

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

19. How many diagonals does a decagon have?

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

20. What is the sum of interior angles of a dodecagon ($n=12$)?

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

21. Calculate the area of a rectangle with length 10 and width 4.

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

22. What is the definition of a concave polygon?

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

23. What is the formula for the area of a rectangle?

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

24. What is the number of sides of a regular polygon with each interior angle $150$?

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

25. What is the formula for the area of a rhombus?

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

26. What is the area of an equilateral triangle with side length 6?

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

27. What is the sum of the exterior angles of any polygon?

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

28. What is the formula for the area of a parallelogram?

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

29. What is each exterior angle of a regular nonagon ($n=9$)?

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

30. State the formula for the measure of each interior angle of a regular polygon.

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

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

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

32. What is the sum of the interior angles of a hexagon?

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

33. What is the relationship between an interior angle and its adjacent exterior angle?

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

34. What is the measure of each exterior angle in a regular decagon?

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

35. What is the formula for the area of a kite?

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

36. What is the number of sides of a regular polygon with each exterior angle $24$?

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

37. What is the formula for the area of a triangle?

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

38. What is the number of diagonals in a decagon ($n=10$)?

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

39. What is the measure of each exterior angle of a regular $n$-gon?

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

40. What is the area of a regular polygon in terms of perimeter $P$ and apothem $a$?

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

41. What is the number of diagonals in a polygon with $12$ sides?

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

42. What is the interior angle if the exterior angle of a convex polygon is $55$?

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

43. What is the sum of interior angles of a polygon with $15$ sides?

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

44. What is the definition of a diagonal of a polygon?

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

45. Identify the type of polygon with all sides and angles equal.

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

46. What is the number of sides $n$ if the sum of interior angles is $2520$?

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

47. Identify the polygon with all sides and angles congruent.

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

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

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

49. What is the measure of an exterior angle if the interior angle is $112$?

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

50. What is the sum of interior angles of an $n$-gon?

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

51. What is the number of diagonals in a hexagon?

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

52. State the formula for the area of a trapezoid.

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

53. What is a polygon with three sides called?

    Triangle. The simplest polygon with three vertices and sides.

54. What is the number of sides of a regular polygon with each exterior angle $30$?

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

55. What is each exterior angle of a regular polygon with $n=18$ sides?

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

56. What is the number of sides $n$ if the sum of interior angles is $1260$?

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

57. What is the perimeter $P$ of a regular polygon with area $96$ and apothem $8$?

    $24$. From $96 = \frac{1}{2} \cdot 8 \cdot P$, solving gives $P = 24$.

58. What is the area of a regular polygon with apothem $10$ and perimeter $36$?

    $180$. Using $\frac{1}{2} \cdot 10 \cdot 36 = \frac{360}{2} = 180$.

59. What is the formula for the sum of interior angles of an $n$-sided polygon?

    Sum = $(n - 2) \times 180^\circ$. Each interior angle reduces the total by $180^\circ$ from a full circle.

60. What is the measure of each exterior angle in a regular hexagon?

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

61. What is the number of sides $n$ if a polygon has $65$ diagonals?

    $13$. From $\frac{n(n-3)}{2} = 65$, solving gives $n = 13$.

62. Calculate the area of a rhombus with diagonals 10 and 8.

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

63. What is the definition of a convex polygon?

    All interior angles are less than $180$. No interior angle exceeds a straight angle.

64. What type of polygon is a quadrilateral with opposite sides parallel?

    Parallelogram. Opposite sides are parallel and equal in length.

65. What is the sum of exterior angles (one at each vertex) of any convex polygon?

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

66. What is the measure of each interior angle of a regular octagon?

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

67. What is the number of diagonals in a heptagon ($n=7$)?

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

68. What is a four-sided polygon called?

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

69. What is the measure of each exterior angle of a regular pentagon?

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

70. Find the area of a parallelogram with base 8 and height 5.

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

71. What is each interior angle of a regular polygon with $n=20$ sides?

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

72. Find the area of a trapezoid with bases 8, 5, and height 4.

    26. Apply $\frac{1}{2}(8+5) \times 4 = \frac{1}{2} \times 13 \times 4 = 26$.

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