area

(noun)

The interior surface of a circle, given by $A = \pi r^2$.

Related Terms

Examples of area in the following topics:

  • Applications of Geometric Series

    • Archimedes' Theorem states that the total area under the parabola is $\displaystyle{\frac{4}{3}}$ of the area of the blue triangle.
    • Assuming that the blue triangle has area $1$, the total area is an infinite series:
    • The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on.
    • Taking the first triangle as a unit of area, the total area of the snowflake is:
    • The first term of this series represents the area of the first triangle, the second term the total area of the three triangles added in the second iteration, the third term the total area of the twelve triangles added in the third iteration, and so forth.

  • Nonlinear Systems of Inequalities

    • Graphing both inequalities reveals one region of overlap: the area where the parabola dips below the line.
    • This area is the solution to the system.
    • Note that the area above $y=x^2$ that is also below $ y=x+2$ is closed between those two points.
    • Whereas a solution for a linear system of equations will contain an infinite, unbounded area (lines can only pass one another a maximum of once), in many instances, a solution for a nonlinear system of equations will consist of a finite, bounded area.
    • This need not be the case with all nonlinear inequalities, but reversing the direction of both inequalities in the previous example would lead to an infinite solution area.

  • Graphs of Linear Inequalities

    • Graphing linear inequalities involves graphing the original line, and then shading in the area connected to the inequality.
    • To find solutions for the group of inequalities, observe where the area of all of the inequalities overlap.
    • The overlapping shaded area is the final solution to the system of linear inequalities because it is comprised of all possible solutions to $y<-\frac{1}{2}x+1$ (the dotted red line and red area below the line) and $y\geq x-2$ (the solid green line and the green area above the line).  

  • Solving Systems of Linear Inequalities

    • Shade, or indicate with hash marks, the area that corresponds to the inequality.
    • For instance, if it is < or $\leq$, shade in the area below the line.
    • If it is > or $\geq$ shade in the area above the line.
    • Again, draw all the inequalities and shade in the area that each inequality covers.
    • Three lines are graphed, with the area that satisfies all three inequalities shaded.

  • Circles as Conic Sections

    • You probably know how to find the area and the circumference of a circle, given its radius.
    • As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference, and whose height equals the circle's radius, which comes to π multiplied by the radius squared:

  • Introduction to Circles

    • Notice that the radius is the only defining parameter for the size of any particular circle, and so it is the only variable that the area depends on.

  • Applications of Circles and Ellipses

    • One example is the orbits of planets, but you should be able to find the area of a circle or an ellipse, or the circumference of a circle, based on information given to you in a problem.
    • The area that is watered by the sprinkler can be labeled $A_{sprinkler}$, and is:
    • Once we know that the area that is watered is completely inside the garden, the percentage of the garden that is watered can be found by dividing the area watered by the total area of the garden, and then multiplying by $100\%$:

  • Right Triangles and the Pythagorean Theorem

    • The sum of the areas of the two squares on the legs ($a$ and $b$) is equal to the area of the square on the hypotenuse ($c$).  

  • Applications of the Parabola

    • They are frequently used in areas such as engineering and physics, and often appear in nature.
    • This is because air resistance is a force acting on the object, and is proportional to the object's area, density, and speed squared.

  • Summing the First n Terms in a Geometric Sequence

    • Each of the purple squares is obtained by multiplying the area of the next larger square by $\displaystyle{\frac{1}{4}}$.
    • The area of the first square is $\displaystyle{\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}}$, and the area of the second square is $\displaystyle{\frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16}}$.