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10.6 Area

3 min read•june 18, 2024

Geometric shapes are all around us, from the rooms we live in to the objects we use daily. Understanding how to calculate their areas is crucial for many real-world applications, like home improvement or landscaping.

calculations vary depending on the shape. Triangles, rectangles, and circles each have unique formulas. For more complex shapes, we can break them down into simpler components. This skill helps us tackle real-world problems involving composite shapes.

Area of Geometric Shapes

Area of common geometric shapes

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Top images from around the web for Area of common geometric shapes

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  • area calculated using A=12×[base](https://www.fiveableKeyTerm:base)×[height](https://www.fiveableKeyTerm:Height)A = \frac{1}{2} \times [base](https://www.fiveableKeyTerm:base) \times [height](https://www.fiveableKeyTerm:Height) (, , )
  • areas:
    • area found by multiplying and A=length×widthA = length \times width
    • area calculated by squaring the length A=side2A = side^2
    • area determined using A=base×heightA = base \times height where height is the perpendicular distance between parallel sides
    • area calculated with A=12(base1+base2)×heightA = \frac{1}{2}(base_1 + base_2) \times height where base1base_1 and base2base_2 are the parallel sides and height is the perpendicular distance between them
  • area found using A=12×[perimeter](https://www.fiveableKeyTerm:Perimeter)×[apothem](https://www.fiveableKeyTerm:Apothem)A = \frac{1}{2} \times [perimeter](https://www.fiveableKeyTerm:Perimeter) \times [apothem](https://www.fiveableKeyTerm:Apothem) where apothem is the perpendicular distance from the center to any side of the polygon (, )
  • area calculated with A=πr2A = \pi r^2 where rr is the and π≈3.14159\pi \approx 3.14159

Unit conversions for area

  • Conversion factors for common units of length:
    • 1 equals 12 inches
    • 1 equals 3 feet
    • 1 equals 5,280 feet
    • 1 equals 100 centimeters
    • 1 equals 1,000 meters
  • To convert between units, multiply the given value by the appropriate
    • Convert 5 feet to inches: 5 feet×12 inches1 foot=60 inches5 \text{ feet} \times \frac{12 \text{ inches}}{1 \text{ foot}} = 60 \text{ inches}
  • Remember to square the conversion factor when dealing with area units
    • Convert 9 square feet to square inches: 9 ft2×(12 in1 ft)2=1,296 in29 \text{ ft}^2 \times (\frac{12 \text{ in}}{1 \text{ ft}})^2 = 1,296 \text{ in}^2
  • Common area unit conversions:
    • 1 equals 144 square inches
    • 1 equals 9 square feet
    • 1 equals 43,560 square feet
    • 1 equals 10,000 square centimeters
    • 1 equals 10,000 square meters
  • is important when converting between different units of

Composite shape area calculation

  • Identify the basic shapes that make up the (rectangles, circles, triangles)
  • Calculate the area of each basic shape using the appropriate
  • Add or subtract the areas of the basic shapes to find the total area of the composite shape
    1. Addition used when the basic shapes do not overlap ( room)
    2. Subtraction used when one shape is removed from another (circular hole in a square plate)
  • Find the area of a rectangular pool with a circular :
    1. Calculate the area of the rectangular pool using A=length×widthA = length \times width
    2. Calculate the area of the circular jacuzzi using A=πr2A = \pi r^2
    3. Subtract the area of the jacuzzi from the area of the pool to find the total area
  • Determine the area of a square room with a triangular closet:
    1. Calculate the area of the square room using A=side2A = side^2
    2. Calculate the area of the triangular closet using A=12×base×heightA = \frac{1}{2} \times base \times height
    3. Add the areas of the room and closet to find the total area
  • refers to the number of measurements needed to describe a shape (2D shapes have length and width)
  • is the branch of mathematics that deals with shapes, sizes, and positions of figures
  • is the total area of all faces of a three-dimensional object
  • is the amount of space occupied by a three-dimensional object, often calculated using area as a base

Key Terms to Review (51)

Acre: An acre is a unit of area that is commonly used in the United States and other countries for measuring large plots of land. It is defined as 43,560 square feet or approximately 4,047 square meters. This measurement is crucial in real estate, agriculture, and land development, as it provides a standardized way to quantify land area.
Apothem: An apothem is a line segment from the center of a regular polygon perpendicular to one of its sides. This segment is essential for calculating the area of regular polygons, as it connects the center to the midpoint of a side, helping to form right triangles that make area calculations easier. Understanding the apothem is crucial when working with the formulas for the area of polygons, as it directly links to both the radius and the side lengths.
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.
Base: In mathematics, a base is a fundamental component that serves as the reference point for various operations, such as exponentiation or measurement in geometry. It indicates the number that is being multiplied by itself in exponentiation, the unit of measure in area and volume calculations, and the foundational side in geometric shapes like triangles. Understanding the concept of base is essential for grasping how numbers and shapes relate to one another in mathematical expressions and equations.
Centimeter: A centimeter is a metric unit of length equal to one hundredth of a meter. It is commonly used in everyday measurements, particularly in fields like science and engineering, where precise measurements are essential. The centimeter is part of the International System of Units (SI) and plays a critical role in calculating areas when dimensions are measured in centimeters.
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.
Composite shape: A composite shape is a figure made up of two or more simple geometric shapes, such as rectangles, circles, triangles, or polygons, combined together to form a new, complex shape. Understanding composite shapes involves recognizing how these simple figures interact and contribute to the overall area, which is essential for calculating the total area of the composite figure accurately.
Conversion Factor: A conversion factor is a numerical multiplier used to convert a measurement from one unit to another. It allows for the comparison or conversion of quantities that are expressed in different units, ensuring that calculations yield accurate and meaningful results. This concept is crucial when dealing with measurements in volume, weight, and area, as it provides the necessary framework for translating values between systems of measurement.
Cubic meter: A cubic meter (m³) is the volume of a cube with edges one meter in length. It is the standard unit of volume in the International System of Units (SI).
Dimension: Dimension refers to the measurable extent of an object in a specific direction, often described by its length, width, and height. In a broader sense, it is a way to quantify and understand the space that points, lines, and planes occupy. Understanding dimensions helps in visualizing geometric figures and calculating properties like area and volume.
Equilateral: An equilateral shape is one where all sides and angles are equal, commonly associated with triangles. In an equilateral triangle, each angle measures 60 degrees, creating a perfect symmetry that is visually striking. This property of equal sides and angles makes equilateral shapes significant in various mathematical calculations, especially in determining area and perimeter.
Foot: A foot is a unit of length in the imperial system, equal to 12 inches or 1/3 of a yard. It is commonly used in various fields such as construction, engineering, and everyday measurements to quantify dimensions. The foot is particularly significant when calculating areas, as it helps translate linear measurements into two-dimensional spaces.
Formula: A formula is a mathematical expression that represents a relationship between different quantities or variables. It provides a concise way to calculate values, such as area, volume, or other measurements, by using specific mathematical operations. Formulas often involve constants, variables, and mathematical symbols, allowing for easy application in problem-solving across various contexts.
Geometry: Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids. It encompasses various concepts such as measuring area, understanding the nature of points and lines, and calculating the dimensions of different shapes. Geometry plays a crucial role in real-world applications, from architecture to engineering, by providing the tools to quantify and analyze space.
Hectare: A hectare is a metric unit of area equivalent to 10,000 square meters, commonly used to measure land, especially in agriculture and forestry. It provides a standardized way to quantify large areas, making it easier to compare and communicate land size across different regions. The hectare is particularly relevant in the context of land management, urban planning, and environmental conservation.
Height: Height refers to the measurement of an object from its base to its topmost point. This dimension is crucial in various geometric contexts, influencing calculations related to area, volume, and surface area. Understanding height is essential for determining the size and capacity of three-dimensional shapes, as well as for analyzing hierarchical structures in data representation.
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.
Inch: An inch is a unit of length in the imperial system, commonly used in the United States and the UK. It is defined as exactly 2.54 centimeters, making it a vital measurement in various applications, especially in areas such as construction, design, and manufacturing where precise dimensions are essential. The inch is also frequently used to describe dimensions of objects, especially when discussing area and space.
Isosceles: An isosceles triangle is a type of triangle that has at least two sides of equal length, which also means that the angles opposite these sides are equal. This property leads to several unique characteristics, especially in relation to finding the area, as the height can often be determined easily from its vertex to the base. The isosceles triangle plays a crucial role in various geometric calculations and concepts, particularly when calculating area and understanding symmetry.
Jacuzzi: A Jacuzzi is a brand name synonymous with hot tubs and whirlpool baths, designed to provide relaxation through hydrotherapy. These units often feature jets that create a massaging effect, which can aid in stress relief and improve circulation. The term has become widely used to refer to any similar type of hot tub or spa experience, making it an important aspect of leisure and wellness culture.
Kilometer: A kilometer is a metric unit of measurement equal to 1,000 meters, commonly used to measure long distances. It is a standard unit in the International System of Units (SI) and provides a convenient way to quantify distances in various fields such as geography, transportation, and science. Understanding kilometers is essential for converting measurements and calculating areas, particularly when dealing with large-scale maps or geographical data.
L-shaped: An L-shaped figure refers to a two-dimensional geometric shape that resembles the letter 'L', often created by joining two rectangles or segments at a right angle. This configuration is commonly used in various mathematical contexts, especially when calculating area, as it can be divided into simpler shapes for easier measurement and analysis.
Length: Length is a measure of distance or extent in one dimension, often represented as the measurement of an object from one end to another. In mathematical contexts, length is crucial for understanding more complex concepts such as area and perimeter, as it provides the foundation upon which these measurements are built. It is typically measured in standard units such as meters, feet, or inches and is a fundamental aspect of geometry that facilitates the calculation of various properties of shapes and figures.
Measurement: Measurement is the process of determining the size, quantity, or degree of something using specific units or standards. This concept is fundamental in various fields, allowing for the quantification and comparison of physical properties, such as length, mass, and area, through consistent units. Understanding measurement is crucial as it provides a common framework for analysis and problem-solving across different disciplines.
Meter: A meter is the fundamental unit of length in the International System of Units (SI), defined as the distance light travels in a vacuum in 1/299,792,458 seconds. This precise definition links the meter to universal constants, establishing its importance in scientific measurements. Additionally, it serves as the base unit for other measurements, such as area and volume, creating a cohesive system of measurement that is consistent and easily understandable.
Mile: A mile is a unit of length commonly used in the United States and the United Kingdom, equivalent to 5,280 feet or approximately 1.609 kilometers. It is often utilized to measure distances in both land and sea contexts, making it a critical component for understanding area, especially when calculating the dimensions of larger geographical spaces.
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.
Parallelogram: A parallelogram is a four-sided polygon (quadrilateral) where opposite sides are both equal in length and parallel to each other. This unique property allows for the application of various mathematical concepts, particularly in calculating area, as the area can be determined by multiplying the base by the height. Parallelograms include special types such as rectangles and rhombuses, which further illustrate the diversity of properties within this shape.
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.
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.
Rectangle: A rectangle is a four-sided polygon, or quadrilateral, with opposite sides that are equal in length and four right angles (90 degrees). This shape has unique properties that make it fundamental in geometry, particularly in calculating area and understanding various spatial relationships.
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.
Scale: Scale refers to the relative size or proportion of an object in relation to another, which is crucial for understanding and comparing measurements. It plays a significant role in determining the weight of objects, where different scales can measure weight accurately depending on their design, and in calculating area, where scale influences the representation and comparison of two-dimensional spaces.
Scalene: A scalene triangle is a type of triangle where all three sides have different lengths and all three angles are also different. This unique property makes scalene triangles distinct from other types of triangles, such as isosceles and equilateral, which have equal sides or angles. Understanding scalene triangles is crucial when exploring concepts like area, as the formula for calculating the area of a scalene triangle often involves the use of Heron's formula or coordinate geometry.
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.
Square: A square is a special type of polygon that has four equal sides and four right angles, making it a specific case of a rectangle and a rhombus. This geometric shape is significant in various contexts, as it serves as a fundamental building block for calculating area and comparing graphical data. Its properties allow for clear calculations and comparisons when dealing with two-dimensional space and data visualization.
Square foot: A square foot is a unit of area measurement equivalent to a square that measures one foot on each side. This term is commonly used in real estate, construction, and interior design to convey the size of a space or the area of a property. Understanding square footage is essential for calculating areas, determining costs, and comparing different spaces effectively.
Square inch: A square inch is a unit of area measurement that represents a square with each side measuring one inch in length. This unit is commonly used in various fields, including engineering, design, and construction, as it helps quantify space and surface area. Understanding square inches is essential when calculating areas of objects, comparing sizes, and determining material requirements for projects.
Square meter: A square meter (m²) is the SI unit of area, defined as the area of a square with sides that are one meter in length. It is commonly used to measure surface areas in various applications such as land, rooms, and other spaces.
Square meter: A square meter is a unit of area measurement that represents a square with sides measuring one meter in length. This metric unit is widely used in various fields, including construction, real estate, and land measurement, to quantify the size of a given space. The square meter is essential for determining the area of irregular shapes by breaking them down into smaller squares or rectangles.
Square units: Square units are a measurement used to quantify area, representing the size of a two-dimensional surface. This term is significant in calculating and expressing the area of various shapes, such as rectangles, squares, and circles. By using square units, we can standardize the measurement of area, making it easier to compare and analyze different surfaces.
Square yard: A square yard is a unit of area measurement that equals the area of a square with sides that each measure one yard. This term is commonly used in various contexts, including construction, landscaping, and interior design, to quantify space. Understanding square yards helps in calculating material quantities, estimating costs, and determining land usage, making it a vital concept in practical applications.
Surface area: Surface area is the total area that the surface of a three-dimensional object occupies. It plays a crucial role in understanding properties like volume, material use, and heat transfer. Knowing how to calculate surface area helps in practical applications, such as determining the amount of paint needed for a wall or the capacity of containers.
Trapezoid: A trapezoid is a four-sided polygon, known as a quadrilateral, that has at least one pair of parallel sides. The parallel sides are referred to as the bases, while the other two sides are called the legs. This unique structure allows for various properties and formulas related to area and perimeter, making trapezoids important in both geometry and real-world applications.
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.
Volume: Volume refers to the amount of three-dimensional space occupied by a substance or object, typically measured in cubic units. Understanding volume is essential for various applications, such as calculating the capacity of containers, determining the amount of material needed for construction, or analyzing physical properties in science and engineering. It connects closely with area, as the base area of a solid and its height determine its overall volume.
Width: Width refers to the measurement of an object's extent from one side to the other, typically representing the shorter dimension in the context of geometric shapes. It plays a vital role in calculating area, as understanding width helps determine how much space a shape occupies on a flat surface. In practical applications, width is essential for designing and organizing spaces, ensuring that objects fit appropriately within designated areas.
Yard: A yard is a unit of length in both the imperial and US customary systems, equivalent to 3 feet or 36 inches. It's commonly used in measuring distances and areas, particularly in construction and landscaping, where dimensions of outdoor spaces are often expressed in yards.
π: π, or pi, is an irrational number that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14159 and is crucial in various mathematical contexts, particularly when dealing with circular shapes and their properties. Its unique characteristics as an irrational number mean that it cannot be expressed as a simple fraction, making it an interesting and essential component in geometry and calculus.
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10.7 Volume and Surface Area