Glossary

11.2 Areas of regular polygons and composite figures

3 min read•july 22, 2024

Regular polygons and composite figures are key players in geometry. They show up everywhere, from simple shapes to complex designs. Knowing how to calculate their areas is super useful in real-life situations.

Area formulas for regular polygons use apothems and perimeters. For irregular shapes, we break them down into simpler parts. These methods help us tackle all sorts of polygon problems, from basic to advanced.

Area Formulas for Regular Polygons

Apothem and perimeter for polygon area

Top images from around the web for Apothem and perimeter for polygon area

  • Polygons - Karnataka Open Educational Resources View original

    Is this image relevant?

  • Polygons - Karnataka Open Educational Resources View original

    Is this image relevant?

1 of 2

Top images from around the web for Apothem and perimeter for polygon area

  • Polygons - Karnataka Open Educational Resources View original

    Is this image relevant?

  • Polygons - Karnataka Open Educational Resources View original

    Is this image relevant?

1 of 2
  • Calculate the using the formula A=12apA = \frac{1}{2}ap
    • aa represents the , the perpendicular distance from the center of the polygon to any side (radius of the inscribed circle)
    • pp represents the of the polygon, the total distance around the polygon
  • Find the perimeter of a with nn sides of length ss using the formula p=nsp = ns
    • For example, a with side length 5 cm has a perimeter of p=6×5=30p = 6 \times 5 = 30 cm
  • Combine the area and perimeter formulas to express the area as A=12ansA = \frac{1}{2}ans
    • This formula is useful when given the number of sides and side length of a regular polygon

Trigonometric ratios in regular polygons

  • Calculate the central angle θ\theta in a regular polygon with nn sides using the formula θ=360∘n\theta = \frac{360^\circ}{n}
    • For instance, in a regular pentagon (n=5n = 5), the central angle is θ=360∘5=72∘\theta = \frac{360^\circ}{5} = 72^\circ
  • Determine the apothem aa using the trigonometric formula a=s2tan⁡(θ2)a = \frac{s}{2\tan(\frac{\theta}{2})}
    • ss represents the side length of the polygon
    • θ\theta represents the central angle
    • This formula is helpful when the side length and number of sides are known
  • Calculate the side length ss using the trigonometric formula s=2atan⁡(θ2)s = 2a\tan(\frac{\theta}{2})
    • This formula is useful when the apothem and number of sides are given

Area of Composite and Irregular Figures

Composite figures and total area

  • Identify the basic shapes that compose the composite figure (squares, rectangles, triangles, circles)
  • Calculate the area of each individual shape using their respective area formulas
    • Rectangle: A=lwA = lw (length ×\times width)
    • Triangle: A=12bhA = \frac{1}{2}bh (base ×\times height)
    • Circle: A=πr2A = \pi r^2 (radius squared ×π\times \pi)
  • Add the areas of all the individual shapes to find the total area of the composite figure
    • For example, a composite figure consisting of a rectangle (6 cm ×\times 4 cm) and a triangle (base 4 cm, height 3 cm) has a total area of 24+6=3024 + 6 = 30 cm²

Irregular polygons and area decomposition

  • Divide the irregular polygon into triangles and quadrilaterals by drawing diagonals from one vertex to another
    • Choose diagonals that create familiar shapes like right triangles or rectangles
  • Calculate the area of each triangle using the formula A=12bhA = \frac{1}{2}bh
    • bb represents the base of the triangle
    • hh represents the height of the triangle
  • Calculate the area of each quadrilateral using their respective area formulas
    • Rectangle: A=lwA = lw
    • Parallelogram: A=bhA = bh
    • Trapezoid: A=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)h (average of parallel sides ×\times height)
  • Add the areas of all the triangles and quadrilaterals to find the total area of the irregular polygon
    • For instance, an irregular pentagon divided into two triangles (areas 12 cm² and 15 cm²) and a trapezoid (area 28 cm²) has a total area of 12+15+28=5512 + 15 + 28 = 55 cm²

Key Terms to Review (19)

Adding areas: Adding areas refers to the process of calculating the total area of multiple shapes by summing their individual areas. This concept is crucial when dealing with composite figures, which are made up of two or more simple geometric shapes, as well as when finding the surface area of three-dimensional objects like prisms and cones. Mastering this technique allows for a clear understanding of how different shapes can combine to form larger structures and how to approach real-world problems involving space and measurement.
Apothem: An apothem is a line segment from the center of a regular polygon perpendicular to one of its sides. This segment is crucial in calculating the area of regular polygons and plays a significant role when dealing with composite figures that involve these shapes. Understanding the apothem helps in finding other important characteristics like the perimeter and area, linking it closely to formulas for regular polygons.
Area of a Circle: The area of a circle is calculated using the formula $$A = ext{pi} \times r^2$$, where 'A' represents the area and 'r' is the radius. This concept is essential when working with various geometric shapes, particularly in finding areas of regular polygons and composite figures that incorporate circular sections. Understanding this relationship helps in comparing circular areas to other shapes, allowing for better problem-solving strategies in geometry.
Area of a regular polygon: The area of a regular polygon can be calculated using the formula: $$A = \frac{1}{2} \times P \times a$$, where $A$ is the area, $P$ is the perimeter, and $a$ is the apothem. This formula emphasizes the relationship between the polygon's side lengths and height, allowing for efficient computation of the area for any regular polygon, which is essential when dealing with complex geometric shapes.
Area of a Square: The area of a square is the measure of the space contained within its four equal sides. It is calculated using the formula $$A = s^2$$, where 's' represents the length of one side. This concept connects to various geometric shapes and properties, particularly in understanding regular polygons and composite figures, as squares are fundamental building blocks in geometry.
Area of a trapezoid: The area of a trapezoid is calculated using the formula $$A = \frac{1}{2}(b_1 + b_2) \times h$$, where $$b_1$$ and $$b_2$$ are the lengths of the two parallel sides, and $$h$$ is the height. This formula allows for the determination of the space enclosed within the trapezoidal shape, which is essential for understanding the properties of composite figures that include trapezoids. It also connects to the study of regular polygons, as trapezoids can be components of larger geometric configurations that involve polygons.
Area of a Triangle: The area of a triangle is calculated using the formula $$A = \frac{1}{2} \times b \times h$$, where 'A' represents the area, 'b' is the length of the base, and 'h' is the height measured perpendicular to the base. This formula illustrates how the area is essentially half of the product of the base and height, which connects directly to understanding how triangles fit into larger shapes. It serves as a fundamental concept when calculating areas of various geometric figures, including regular polygons and composite figures that incorporate triangles.
Centimeters squared: Centimeters squared is a unit of area measurement that indicates how many square centimeters are contained within a given shape. This term becomes particularly significant when calculating the area of two-dimensional figures, such as polygons and composite shapes, enabling us to quantify the space they occupy. Understanding this unit allows for better comprehension of geometry and the relationships between different shapes in terms of their sizes.
Composite area: Composite area refers to the total area of a figure that is made up of two or more simple shapes, such as rectangles, triangles, and circles. Understanding composite area allows for calculating the area of irregular figures by breaking them down into manageable parts, which can be easily computed and then combined for a final result. This concept is crucial for solving problems related to regular polygons and complex figures.
Convex polygon: A convex polygon is a simple polygon in which all interior angles are less than 180 degrees, meaning that no line segment between two points on the boundary lies outside the polygon. This characteristic ensures that every line drawn between any two points inside the polygon remains within its boundaries, which plays a crucial role in determining the areas of regular polygons and composite figures. Convex polygons can have various numbers of sides, but they are distinct from concave polygons, where at least one interior angle exceeds 180 degrees.
Decomposing Figures: Decomposing figures is the process of breaking down complex geometric shapes into simpler, more manageable components, such as triangles, rectangles, or other polygons. This method allows for easier calculation of areas and understanding of the properties of composite figures, particularly when dealing with regular polygons and shapes that are not straightforward in form.
Equilateral Triangle: An equilateral triangle is a triangle in which all three sides are of equal length and all three angles are equal, each measuring 60 degrees. This unique property creates a perfect balance in the shape, influencing various geometric concepts such as symmetry, area calculations, and congruence relationships.
Perimeter: Perimeter is the total distance around the outer boundary of a two-dimensional shape. It plays a crucial role in geometry, especially when calculating the dimensions of various figures, such as regular polygons and composite shapes. Understanding how to find the perimeter helps in determining the space that a shape occupies and can also be linked to various real-world applications, such as fencing a yard or framing a picture.
Pythagorean Theorem in Area Calculations: The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides, expressed as $$a^2 + b^2 = c^2$$. This theorem is instrumental in determining the dimensions and area of various geometric shapes, especially when dealing with right triangles and regular polygons. By utilizing the relationships between the sides of these figures, one can calculate areas accurately, whether for single shapes or composite figures that can be broken down into simpler components.
Regular Hexagon: A regular hexagon is a six-sided polygon (hexagon) with all sides and angles equal. This unique shape is notable for its symmetry and can be perfectly inscribed within a circle, making it a fundamental figure in geometry, especially in the study of areas of regular polygons and composite figures.
Regular polygon: A regular polygon is a geometric figure with all sides and angles equal, creating a symmetrical and uniform shape. This property of equal length sides and equal angles is crucial for understanding various mathematical concepts, such as area calculations and symmetry. Regular polygons include shapes like equilateral triangles, squares, and regular pentagons, and they can be used to form more complex figures.
Square Units: Square units are the standard measurement used to quantify area, representing the amount of space within a two-dimensional shape. This unit is essential in determining the size of various geometric figures, such as circles, polygons, and composite shapes, as well as the surface area of three-dimensional objects. The concept of square units allows for a consistent way to communicate and calculate area across different shapes and dimensions.
Subtracting areas: Subtracting areas is a mathematical operation that involves calculating the difference between the areas of two shapes, which is essential when determining the area of a composite figure or when one shape overlaps another. This process is crucial for understanding how to find the remaining space within complex shapes by removing the area of one shape from another. Mastering this concept helps in solving real-world problems where one needs to find available space or understand how shapes interact with each other.
Theorem for Area of a Regular n-gon: The theorem for the area of a regular n-gon states that the area can be calculated using the formula $$A = \frac{1}{2} n s a$$, where $n$ is the number of sides, $s$ is the length of one side, and $a$ is the apothem (the perpendicular distance from the center to a side). This theorem connects geometry with algebra by utilizing both the polygon's side lengths and its apothem to find the total area, providing a structured way to approach area calculations for regular polygons.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide