Polygons
57 flashcards covering Polygons for the ACT Math section.
Polygons are two-dimensional shapes formed by straight lines that connect to enclose a space, with no curves or gaps. Common examples include triangles, squares, and hexagons. Essentially, any shape with three or more sides qualifies as a polygon. Understanding polygons is key in geometry because they help you grasp concepts like angles, sides, and area, which build a foundation for solving real-world problems and more complex math.
On the ACT Math section, polygons show up in questions testing area, perimeter, angles, and properties of shapes, often in multiple-choice formats. Common traps include mistaking similar polygons for congruent ones or overlooking units in calculations. Focus on mastering formulas for regular and irregular polygons, as well as identifying interior and exterior angles. Practice visualizing shapes to avoid errors in word problems or coordinate geometry.
A helpful tip: Always sketch the polygon if it's not provided.
Terms (57)
- 01
Polygon
A closed plane figure with straight sides that are line segments and does not intersect itself.
- 02
Regular polygon
A polygon that has all sides of equal length and all interior angles equal.
- 03
Irregular polygon
A polygon that does not have all sides equal or all angles equal.
- 04
Convex polygon
A polygon where all interior angles are less than 180 degrees and no sides bend inwards.
- 05
Concave polygon
A polygon that has at least one interior angle greater than 180 degrees, creating an inward bend.
- 06
Triangle
A polygon with three sides and three angles.
- 07
Equilateral triangle
A triangle with all three sides of equal length and all angles measuring 60 degrees.
- 08
Isosceles triangle
A triangle with at least two sides of equal length, resulting in two equal angles opposite those sides.
- 09
Scalene triangle
A triangle with all three sides of different lengths and all three angles of different measures.
- 10
Right triangle
A triangle with one angle measuring exactly 90 degrees.
- 11
Acute triangle
A triangle where all three interior angles are less than 90 degrees.
- 12
Obtuse triangle
A triangle with one interior angle greater than 90 degrees but less than 180 degrees.
- 13
Quadrilateral
A polygon with four sides and four angles.
- 14
Square
A quadrilateral with all four sides equal and all four angles measuring 90 degrees.
- 15
Rectangle
A quadrilateral with opposite sides equal and all four angles measuring 90 degrees.
- 16
Parallelogram
A quadrilateral with opposite sides parallel and equal in length.
- 17
Rhombus
A quadrilateral with all four sides equal in length.
- 18
Trapezoid
A quadrilateral with at least one pair of parallel sides.
- 19
Pentagon
A polygon with five sides and five angles.
- 20
Hexagon
A polygon with six sides and six angles.
- 21
Heptagon
A polygon with seven sides and seven angles.
- 22
Octagon
A polygon with eight sides and eight angles.
- 23
Interior angle
An angle inside a polygon formed by two adjacent sides.
- 24
Exterior angle
An angle formed by one side of a polygon and the extension of an adjacent side.
- 25
Sum of interior angles
The total of all interior angles in a polygon, calculated as (n-2) times 180 degrees, where n is the number of sides.
- 26
Perimeter of a polygon
The total length of all sides of a polygon added together.
- 27
Area of a triangle
The space inside a triangle, calculated as one-half base times height.
- 28
Area of a rectangle
The space inside a rectangle, calculated as length times width.
- 29
Area of a square
The space inside a square, calculated as side length squared.
- 30
Area of a parallelogram
The space inside a parallelogram, calculated as base times height.
- 31
Area of a trapezoid
The space inside a trapezoid, calculated as the average of the two parallel sides times the height.
- 32
Area of a regular polygon
The space inside a regular polygon, calculated as (perimeter times apothem) divided by 2.
- 33
Apothem
The distance from the center of a regular polygon to the midpoint of one of its sides, used in area calculations.
- 34
Diagonal of a polygon
A line segment connecting two non-adjacent vertices of a polygon.
- 35
Properties of parallelograms
In a parallelogram, opposite sides are equal and parallel, opposite angles are equal, and diagonals bisect each other.
- 36
Properties of rectangles
In a rectangle, all angles are 90 degrees, opposite sides are equal, and diagonals are equal in length.
- 37
Pythagorean theorem
In a right triangle, the square of the hypotenuse's length equals the sum of the squares of the other two sides.
- 38
Triangle inequality theorem
In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
- 39
Similar polygons
Polygons that have the same shape but not necessarily the same size, with corresponding angles equal and corresponding sides proportional.
- 40
Congruent polygons
Polygons that are identical in shape and size, with all corresponding sides and angles equal.
- 41
Midpoint formula
For a line segment on a coordinate plane, the midpoint is found by averaging the x-coordinates and y-coordinates of the endpoints.
- 42
Distance formula
The length of a line segment between two points on a coordinate plane, calculated as the square root of the sum of the squared differences in x and y coordinates.
- 43
Common mistake with area and perimeter
Confusing area, which is measured in square units, with perimeter, which is measured in linear units, often leading to incorrect units in problems.
- 44
Example of triangle area
To find the area of a triangle with base 5 and height 6, multiply base by height and divide by 2, resulting in 15 square units.
For a triangle with base 5 units and height 6 units, area is (5 6) / 2 = 15 square units.
- 45
Example of quadrilateral perimeter
To find the perimeter of a quadrilateral with sides 3, 4, 5, and 6, add all sides together, resulting in 18 units.
For a quadrilateral with sides 3, 4, 5, and 6 units, perimeter is 3 + 4 + 5 + 6 = 18 units.
- 46
Strategy for polygon area
Divide a complex polygon into triangles or other simple shapes to calculate the total area by summing the areas of the parts.
- 47
Interior angle of regular polygon
For a regular polygon with n sides, each interior angle is [(n-2) 180] / n degrees.
- 48
Sum of exterior angles
The sum of the exterior angles of any polygon is always 360 degrees.
- 49
Heron's formula
A method to find the area of a triangle when all three sides are known, using the square root of [s(s-a)(s-b)(s-c)], where s is the semi-perimeter.
- 50
Angle bisector in a triangle
A line that divides an angle of a triangle into two equal angles.
- 51
Median of a triangle
A line segment from a vertex of a triangle to the midpoint of the opposite side.
- 52
Altitude of a triangle
A perpendicular line segment from a vertex of a triangle to the line containing the opposite side.
- 53
Centroid of a triangle
The point where the three medians of a triangle intersect, also the center of mass.
- 54
Inscribed polygon
A polygon drawn inside a circle such that all its vertices lie on the circle.
- 55
Circumscribed polygon
A polygon around which a circle is drawn such that the circle touches all sides of the polygon.
- 56
Slope for parallel sides
In polygons on a coordinate plane, sides are parallel if they have the same slope.
- 57
Slope for perpendicular sides
In polygons on a coordinate plane, sides are perpendicular if the product of their slopes is -1.