10.4 Polygons, Perimeter, and Circumference
3 min read•june 18, 2024
Polygons are shapes with straight sides and angles. They come in various forms, from triangles to decagons and beyond. Understanding their properties helps us measure and classify them accurately.
Key aspects of polygons include , interior and exterior angles, and classification based on sides and angles. Circles, though not polygons, share some similar concepts like and calculations.
Polygons
Perimeter and circumference calculations
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- Perimeter of a
- Represents the total distance around the by adding the lengths of all sides
- Calculated using the formula , where is the length of each (square, )
- Circumference of a
- Measures the distance around the circle, similar to the perimeter of a polygon
- Determined using the formula , where is the , or , where is the (coin, pizza)
- Measuring lengths
- Involves using measuring tools like rulers or tape measures to find side lengths or radius/diameter
- Requires converting all measurements to the same unit (inches, centimeters) before calculating perimeter or circumference
Interior and exterior angle measures
- Interior angles
- Formed inside the polygon where two adjacent sides meet
- Sum of all interior angles calculated using , where is the number of sides (: , square: )
- Each in a measures (regular pentagon: )
- Exterior angles
- Created by extending a side of the polygon outward
- Always sum to for any polygon, regardless of the number of sides
- Measure of each in a regular polygon is (regular : )
- Relationship between interior and exterior angles
- Adjacent interior and exterior angles form a straight line, making them supplementary (sum to )
Classification of polygons
- Number of sides
- Polygons classified based on the number of sides they have (triangle: 3, : 8)
- Names for polygons with 3 to 10 sides: triangle, , pentagon, hexagon, , octagon, ,
- Angles
- Acute polygons have all angles measuring less than (acute triangle)
- Obtuse polygons have at least one greater than (obtuse triangle)
- Right polygons have at least one angle equal to (right triangle, rectangle)
- Sides
- Equilateral polygons have all sides equal in length (equilateral triangle, square)
- Equiangular polygons have all angles equal in measure (square, regular pentagon)
- Regular polygons are both equilateral and equiangular (square, regular hexagon)
- Convexity
- Convex polygons have all interior angles less than (square, regular pentagon)
- Concave polygons have at least one interior angle greater than (star shape, arrow shape)
- Symmetry
- occurs when a polygon can be divided into two congruent halves by a line (rectangle, )
- happens when a polygon appears unchanged after rotating it about its center by a certain angle (square: , , , equilateral triangle: , )
Basic elements of polygons
- Sides: The line segments that form the boundary of the polygon
- Vertices: The points where two sides of a polygon meet (singular: )
- Angles: Formed by two adjacent sides of the polygon
- Area: The space enclosed within the polygon's boundaries
Circles
- A round shape with all points on its edge equidistant from the center
- Circumference is the distance around the circle (perimeter equivalent)
- Area is calculated using the formula , where is the radius
Key Terms to Review (34)
Acute polygon: An acute polygon is a type of polygon in which all interior angles are less than 90 degrees. This characteristic means that each angle contributes to the overall 'pointed' appearance of the shape, distinguishing it from other polygons with right or obtuse angles. The acute nature of the angles has implications for both the polygon's perimeter and its geometric properties, influencing how measurements and calculations are approached.
Angle: An angle is formed when two rays share a common endpoint, known as the vertex. Angles are measured in degrees and can be classified based on their measures, such as acute, right, obtuse, and straight. Understanding angles is crucial when examining the properties of shapes, especially polygons, and when calculating measurements like perimeter and circumference.
Area: Area is a measure of the amount of space contained within a two-dimensional shape, typically expressed in square units. Understanding area is crucial for calculating how much surface space a shape covers, which connects directly to determining quantities like land size, material needs, or even visual representations. It serves as a foundation for more complex mathematical concepts and is essential in practical applications such as architecture, engineering, and landscaping.
Circle: A circle is a round, two-dimensional shape where every point on its boundary is equidistant from a fixed central point, known as the center. The distance from the center to any point on the circle is called the radius, while the distance across the circle through its center is the diameter, which is twice the radius. This concept is fundamental to understanding polygons, perimeter, circumference, and area calculations.
Circumference: Circumference is the distance around the edge of a circle, often referred to as the perimeter of a circle. This measurement is crucial for understanding the properties of circles, as it connects directly to radius and diameter, allowing for the calculation of space within circular shapes. Knowing how to determine circumference plays an important role in areas like geometry and real-life applications such as construction and design.
Concave polygon: A concave polygon is a type of polygon where at least one interior angle measures greater than 180 degrees, causing it to have at least one 'caved in' section. This characteristic differentiates it from convex polygons, where all interior angles are less than 180 degrees. The presence of this inward dent influences various properties, including perimeter calculations and how the shape behaves under certain geometric transformations.
Convex polygon: A convex polygon is a polygon in which all interior angles are less than 180 degrees and any line segment drawn between two points within the polygon lies entirely inside it. This characteristic ensures that no vertices point inwards, making convex polygons a fundamental shape in geometry, particularly in the study of polygons, perimeter, and tessellations.
Decagon: A decagon is a ten-sided polygon that can be regular, with all sides and angles equal, or irregular, with sides and angles that can vary. This shape is important in geometry as it expands our understanding of polygons and their properties, particularly in calculating perimeter and area. Additionally, the decagon serves as a foundational shape in various applications, including architecture and design, illustrating its practical significance beyond theoretical mathematics.
Diameter: Diameter is the length of a straight line that passes through the center of a circle, connecting two points on its circumference. It is the largest distance across the circle and is always twice the radius, which is the distance from the center to any point on the circle. Understanding diameter is crucial for calculating other geometric properties, such as area and circumference, as it directly influences these measurements.
Equiangular polygon: An equiangular polygon is a polygon in which all interior angles are equal in measure. This characteristic ensures that the angles maintain a consistent relationship, impacting the overall shape and properties of the polygon, including its perimeter and how it can be inscribed in a circle, which relates to its circumference.
Equilateral polygon: An equilateral polygon is a type of polygon where all sides have equal length. This property ensures that each side is identical, which often contributes to the overall symmetry and regularity of the shape. These polygons can also be categorized as regular polygons if their angles are equal as well, linking them to concepts of perimeter and area calculations in geometry.
Exterior angle: An exterior angle is formed when a side of a polygon is extended outside of the shape, creating an angle between that extended line and the adjacent side. This concept plays a crucial role in understanding the properties of polygons, particularly in relation to their interior angles and overall geometry. The relationship between exterior angles and their corresponding interior angles is fundamental in calculating the total angle measures in various polygons.
Heptagon: A heptagon is a seven-sided polygon, characterized by its seven edges and seven vertices. This shape can be regular, where all sides and angles are equal, or irregular, where the lengths of the sides and angles vary. The heptagon is an important figure in geometry as it helps illustrate the properties of polygons and contributes to calculations involving perimeter and area.
Hexagon: A hexagon is a polygon with six sides and six angles. This shape can be regular, where all sides and angles are equal, or irregular, where they vary in length and degree. Hexagons are significant in various mathematical concepts, including perimeter calculations, tessellation patterns, and area measurements, making them a versatile and interesting shape in geometry.
Interior angle: An interior angle is the angle formed between two sides of a polygon that meet at a vertex, lying within the boundaries of that polygon. These angles play a crucial role in understanding the properties and classifications of polygons, including their perimeter and area calculations. Additionally, interior angles are essential when analyzing how shapes fit together in tessellations, where the angles determine the arrangement and overall design of the pattern.
Isosceles triangle: An isosceles triangle is a type of triangle that has at least two sides of equal length. This unique property leads to specific characteristics, such as two equal angles opposite the equal sides, which are known as the base angles. Understanding the properties of isosceles triangles is essential when studying the relationships within triangles and how they fit into the larger category of polygons.
Line symmetry: Line symmetry, also known as reflective symmetry, occurs when a figure can be divided into two identical halves that mirror each other across a line. This concept is important in understanding the properties of shapes and polygons, as it helps in determining how they can be folded or divided evenly. Recognizing line symmetry not only aids in visualizing geometric figures but also enhances the understanding of their perimeters and relationships between sides.
Nonagon: A nonagon is a polygon with nine sides and nine angles. It can be classified as either regular, where all sides and angles are equal, or irregular, where the sides and angles vary in length and measure. The nonagon plays a significant role in understanding the properties of polygons and how they relate to concepts like perimeter and area.
Obtuse polygon: An obtuse polygon is a type of polygon that has at least one interior angle greater than 90 degrees. This distinctive feature sets it apart from other polygons, which can either be acute (all angles less than 90 degrees) or right (one angle equal to 90 degrees). The presence of an obtuse angle affects various properties of the polygon, including its perimeter calculations and overall shape.
Octagon: An octagon is a polygon with eight sides and eight angles. This shape has unique properties that relate to the concepts of perimeter and area, making it significant in geometry. Octagons can also play a role in tessellations, where they can fit together with other shapes to cover a plane without gaps or overlaps, showcasing their versatility in various mathematical contexts.
Pentagon: A pentagon is a five-sided polygon that is a fundamental shape in geometry. Each internal angle in a regular pentagon measures 108 degrees, and the total sum of the internal angles of any pentagon is 540 degrees. Pentagons can be regular, where all sides and angles are equal, or irregular, where they differ in length and angle, showcasing the diversity of polygonal shapes.
Perimeter: Perimeter is the total distance around the boundary of a two-dimensional shape. It plays a crucial role in various geometric calculations, linking to the area, volume, and surface area of shapes by providing a foundational measure of length that helps in understanding the dimensions and characteristics of polygons and circles.
Pi: Pi is a mathematical constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. This irrational number is crucial in understanding properties of circles and is widely used in various fields including geometry, physics, and engineering.
Polygon: A polygon is a closed, two-dimensional geometric shape composed of straight line segments. The segments are called edges or sides, and the points where two edges meet are called vertices.
Polygon: A polygon is a closed, two-dimensional shape made up of straight line segments, which are called sides. The sides connect at points called vertices and can form various types of shapes, ranging from simple triangles to complex forms with many sides. Polygons can be classified based on the number of sides they have, and they play a key role in understanding concepts like perimeter, which is the total length of the polygon's sides.
Quadrilateral: A quadrilateral is a four-sided polygon with four angles. The sum of the interior angles of any quadrilateral is 360 degrees.
Quadrilateral: A quadrilateral is a polygon that has four sides and four angles. It is a fundamental geometric shape that serves as the basis for various types of polygons and is characterized by its specific properties, such as the sum of its interior angles totaling 360 degrees. Understanding quadrilaterals is essential for calculating perimeter, area, and recognizing their different classifications.
Radius: The radius is the distance from the center of a circle or sphere to any point on its boundary. It is a crucial concept that helps define the size and shape of various geometric figures, influencing their perimeter, circumference, area, volume, and surface area.
Regular polygon: A regular polygon is a flat shape with straight sides that are all equal in length and angles that are all equal in measure. This geometric property means that regular polygons can be classified based on the number of sides they have, which also influences their perimeter, area, and how they can fit together to create tessellations.
Right Polygon: A right polygon is a type of polygon where at least one of its interior angles measures exactly 90 degrees. This characteristic connects it to specific shapes like rectangles and squares, which are both examples of right polygons and have important properties related to perimeter and area calculations.
Rotational symmetry: Rotational symmetry is a property of a shape that looks the same after being rotated by a certain angle around a central point. This concept is crucial in understanding how shapes behave under rotation, allowing for insights into their structure and design. A shape has rotational symmetry if there exists an angle less than 360 degrees where the shape can be mapped onto itself through rotation.
Side: A side is a straight line that forms part of the boundary of a polygon, connecting two vertices. In the context of shapes, the number of sides directly influences the type and properties of the polygon, such as its perimeter and area. Each side contributes to the overall geometric structure, and understanding how sides relate to other components like angles and diagonals is essential for solving various mathematical problems.
Triangle: A triangle is a polygon with three edges and three vertices, and it is one of the simplest shapes in geometry. The sum of the internal angles of a triangle always equals 180 degrees, which is a fundamental property that connects it to various mathematical concepts like perimeter, area, and graphing relationships.
Vertex: A vertex is a point where two or more curves, lines, or edges meet. In different contexts, it can represent a significant feature such as the peak of a parabola, a corner of a polygon, or a key point in graph theory. Understanding the concept of a vertex helps in analyzing the properties and relationships of various mathematical structures.