Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Area"

m
Line 3: Line 3:
 
It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elementary means.  One can find the area of even more complex regions via the use of [[calculus]].
 
It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elementary means.  One can find the area of even more complex regions via the use of [[calculus]].
  
−
[[Rectangle]]s are the most basic figures whose area we can study.  It makes sense that the area of a rectangle with length '''l''' and width '''w''' is simply <math> l\cdot w</math>.
+
[[Rectangle]]s are the most basic figures whose area we can study.  It makes sense that the area of a rectangle with length <math>l</math> and width <math>w</math> is simply <math> l\cdot w</math>.
  
−
Once we know the area of a rectangle, we can easily find the area of a [[triangle]] by just noting that if our triangle has base '''b''' and height '''h''', then the rectangle with length '''b''' and width '''h''' has exactly twice as much area as the original triangle.  Thus, the area of a triangle is
+
Once we know the area of a rectangle, we can easily find the area of a [[triangle]] by just noting that if our triangle has base <math>b</math> and height <math>h</math>, then the rectangle with length <math>b</math> and width <math>h</math> has exactly twice as much area as the original triangle.  Thus, the area of a triangle is
  
 
<center><math>A=\frac 12 bh.</math></center>
 
<center><math>A=\frac 12 bh.</math></center>
Line 14: Line 14:
 
== Notation ==
 
== Notation ==
  
−
The letters ''A'' and ''K'' are frequently used to stand for area.  When there are multiple regions under consideration, subscripts are often employed, like <math> A_1, K_2,\ldots</math> or <math> A_{ABC}, K_{BCD},\ldots</math>.  For example, <math> K_{ABCDEF}</math> would mean the area of hexagon ABCDEF.
+
The letters <math>A</math> and <math>K</math> are frequently used to stand for area.  When there are multiple regions under consideration, subscripts are often employed: <math> A_1, K_2,\ldots</math> might be used to denote the areas of particular regions, or <math> A_{ABC}, K_{BCD},\ldots</math>.  For example, <math> K_{ABCDEF}</math> would mean the area of [[hexagon]] <math>ABCDEF</math>.
−
An alternative notation is to use square brackets around the region to denote its area, e.g. <math> [ABC]</math> for the area of triangle <math>\triangle ABC</math>.  
 
  
 +
An alternative notation is to use square brackets around the name of the region to denote its area, e.g. <math> [ABC]</math> for the area of triangle <math>\triangle ABC</math>.
  
−
== Area of Regular Geometric Figures ==
 
  
−
The area of any regular geometric figure can be found as follows:
+
== Area of Regular Polygons ==
  
 +
The area of any [[regular polygon]] can be found as follows:
  
−
Inscribe the figure, with n sides of length s, in a circle and draw a line from two adjacent vertices to the [[circumcenter]].  This creates a triangle that is <math>\frac{1}{n},</math> of the total area (consider the regular octagon below as an example).
+
 
 +
Inscribe the figure, with <math>n</math> sides of length <math>s</math>, in a circle and draw a line from two adjacent vertices to the [[circumcenter]].  This creates a triangle that is <math>\frac{1}{n},</math> of the total area (consider the regular [[octagon]] below as an example).
  
 
<center>[[Image:Regularoctagon.PNG]]</center>
 
<center>[[Image:Regularoctagon.PNG]]</center>
Line 30: Line 31:
  
  
−
Drawing the [[altitude]] in creates two right triangles, with an angle of <math>\frac{180}{n}^{\circ}</math> at the top vertex.  If the polygon has side length ''s'', the height of the triangle can be found using [[trigonometry]] to be of length <math> \displaystyle \frac s2 \cot \frac{180}{n}^{\circ}</math>.
+
Drawing the [[altitude]] creates two [[right triangle]]s, with an [[angle]] of <math>\frac{180}{n}^{\circ}</math> at the top vertex.  If the polygon has side length <math>s</math>, the height of the triangle can be found using [[trigonometry]] to be of length <math> \displaystyle \frac s2 \cot \frac{180}{n}^{\circ}</math>.
  
 
The area of each triangle is <math>\frac12</math> the base times the height, so the area of each triangle is <math>\displaystyle \frac{s^2}{4} \cot\frac{180}{n}^{\circ}</math> and the area of the entire polygon is  
 
The area of each triangle is <math>\frac12</math> the base times the height, so the area of each triangle is <math>\displaystyle \frac{s^2}{4} \cot\frac{180}{n}^{\circ}</math> and the area of the entire polygon is  
Line 40: Line 41:
 
There are many ways to find the area of a [[triangle]].  In all of these formulae, <math>{K}</math> will be used to indicate area.
 
There are many ways to find the area of a [[triangle]].  In all of these formulae, <math>{K}</math> will be used to indicate area.
  
−
* <math>K=\frac{bh}{2}</math> where b is a base, and h is the altitude of the triangle to that base.
+
* <math>K=\frac{bh}{2}</math> where <math>b</math> is a base and <math>h</math> is the altitude of the triangle to that base.
−
* [[Heron's formula]]: <math>K=\sqrt{s(s-a)(s-b)(s-c)}</math>, with [[semi-perimeter]] <math>s=\frac{a+b+c}{2}</math>.
+
* [[Heron's formula]]: <math>K=\sqrt{s(s-a)(s-b)(s-c)}</math>, where <math>a, b</math> and <math>c</math> are the lengths of the sides and <math>s</math> is the [[semi-perimeter]] <math>s=\frac{a+b+c}{2}</math>.
−
* <math>\displaystyle K=rs</math>, where r is the radius of the [[Incircle | incircle]], and s is the semi-perimeter.
+
* <math>\displaystyle K=rs</math>, where <math>r</math> is the radius of the [[incircle]] and s is the semi-perimeter.
−
* <math>K=\frac{ab\sin{\theta}}{2}</math> where a and b are adjacent sides of the triangle, and <math>\theta</math> is the angle between them.
+
* <math>K=\frac{ab\sin{\theta}}{2}</math> where <math>a</math> and <math>b</math> are adjacent sides of the triangle and <math>\theta</math> is the measure of the angle between them.
−
* <math>K=\frac{abc}{4R}</math>, where <math>\displaystyle a,b,c </math> are the sides of the triangle and <math> \displaystyle R </math> is the circumradius.
+
* <math>K=\frac{abc}{4R}</math>, where <math>\displaystyle a,b,c </math> are the lengths of the sides of the triangle and <math> \displaystyle R </math> is the circumradius.
  
 
== Area of a Quadrilateral ==
 
== Area of a Quadrilateral ==
  
−
To find the area of most [[quadrilateral | quadrilaterals]], you must divide the quadrilateral up into smaller triangles and find the area of each triangle.  However, some quadrilaterals have special formulas to find their areas.  Again, <math>K</math> is the area.
+
To find the area of most [[quadrilateral]]s, you must divide the quadrilateral up into smaller triangles and find the area of each triangle.  However, some quadrilaterals have special formulas to find their areas.  Again, <math>K</math> is the area.
  
  
  
−
* [[Kite]] - <math>K=\displaystyle\frac{d_1\cdot d_2}{2}</math> where the ds represent the lengths of the diagonals of the kite.
+
* [[Kite]] - <math>K=\displaystyle\frac{d_1\cdot d_2}{2}</math> where the <math>d</math>s represent the lengths of the diagonals of the kite.
  
−
* [[Parallelogram]] - <math>{K=bh}</math>, where b is the base and h is the height to the base.
+
* [[Parallelogram]] - <math>{K=bh}</math>, where <math>b</math> is the base and <math>h</math> is the height to that base.
  
−
* [[Trapezoid]] - <math>K=\displaystyle\frac{b_1+b_2}{2}\cdot h</math>, where the bs are the set of parallel sides and h is the distance between those bases.
+
* [[Trapezoid]] - <math>K=\displaystyle\frac{b_1+b_2}{2}\cdot h</math>, where the <math>b</math>s are the parallel sides and <math>h</math> is the distance between those bases.
  
 
* [[Rhombus]] - a special case of a kite and parallelogram, so either formula may be used here.
 
* [[Rhombus]] - a special case of a kite and parallelogram, so either formula may be used here.
  
−
* [[Rectangle]] - <math>{\displaystyle K=lw}</math>, where l is the length of the rectangle and w is the width.
+
* [[Rectangle]] - <math>{\displaystyle K=lw}</math>, where <math>l</math> is the length of the rectangle and <math>w</math> is the width. (This is a special case of the formula for a parallelogram where the height and a side happen to coincide.)
  
−
* [[Square]] - <math>\displaystyle K=s^2</math>, where s is the length of a side.
+
* [[Square (geometry) | Square]] - <math>\displaystyle K=s^2</math>, where <math>s</math> is the length of a side.
  
  
 
== See also ==
 
== See also ==
 
* [[Areas]] -- An article about areas of various geometric figures.
 
* [[Areas]] -- An article about areas of various geometric figures.

Revision as of 16:40, 17 October 2006

In mathematics, area refers to the size of the region that a two-dimensional figure occupies.

It is often possible to find the area of a region bounded by parts of circles and line segments through elementary means. One can find the area of even more complex regions via the use of calculus.

Rectangles are the most basic figures whose area we can study. It makes sense that the area of a rectangle with length $l$ and width $w$ is simply $l\cdot w$.

Once we know the area of a rectangle, we can easily find the area of a triangle by just noting that if our triangle has base $b$ and height $h$, then the rectangle with length $b$ and width $h$ has exactly twice as much area as the original triangle. Thus, the area of a triangle is

$A=\frac 12 bh.$

We can now find the area of any polygon by breaking it up into triangles.


Notation

The letters $A$ and $K$ are frequently used to stand for area. When there are multiple regions under consideration, subscripts are often employed: $A_1, K_2,\ldots$ might be used to denote the areas of particular regions, or $A_{ABC}, K_{BCD},\ldots$. For example, $K_{ABCDEF}$ would mean the area of hexagon $ABCDEF$.

An alternative notation is to use square brackets around the name of the region to denote its area, e.g. $[ABC]$ for the area of triangle $\triangle ABC$.


Area of Regular Polygons

The area of any regular polygon can be found as follows:


Inscribe the figure, with $n$ sides of length $s$, in a circle and draw a line from two adjacent vertices to the circumcenter. This creates a triangle that is $\frac{1}{n},$ of the total area (consider the regular octagon below as an example).

Regularoctagon.PNG



Drawing the altitude creates two right triangles, with an angle of $\frac{180}{n}^{\circ}$ at the top vertex. If the polygon has side length $s$, the height of the triangle can be found using trigonometry to be of length $\displaystyle \frac s2 \cot \frac{180}{n}^{\circ}$.

The area of each triangle is $\frac12$ the base times the height, so the area of each triangle is $\displaystyle \frac{s^2}{4} \cot\frac{180}{n}^{\circ}$ and the area of the entire polygon is

$\displaystyle \frac{n\cdot s^2}{4} \cot\frac{180}{n}^{\circ}$.

Area of Triangle

There are many ways to find the area of a triangle. In all of these formulae, ${K}$ will be used to indicate area.

  • $K=\frac{bh}{2}$ where $b$ is a base and $h$ is the altitude of the triangle to that base.
  • Heron's formula: $K=\sqrt{s(s-a)(s-b)(s-c)}$, where $a, b$ and $c$ are the lengths of the sides and $s$ is the semi-perimeter $s=\frac{a+b+c}{2}$.
  • $\displaystyle K=rs$, where $r$ is the radius of the incircle and s is the semi-perimeter.
  • $K=\frac{ab\sin{\theta}}{2}$ where $a$ and $b$ are adjacent sides of the triangle and $\theta$ is the measure of the angle between them.
  • $K=\frac{abc}{4R}$, where $\displaystyle a,b,c$ are the lengths of the sides of the triangle and $\displaystyle R$ is the circumradius.

Area of a Quadrilateral

To find the area of most quadrilaterals, you must divide the quadrilateral up into smaller triangles and find the area of each triangle. However, some quadrilaterals have special formulas to find their areas. Again, $K$ is the area.


  • Kite - $K=\displaystyle\frac{d_1\cdot d_2}{2}$ where the $d$s represent the lengths of the diagonals of the kite.
  • Parallelogram - ${K=bh}$, where $b$ is the base and $h$ is the height to that base.
  • Trapezoid - $K=\displaystyle\frac{b_1+b_2}{2}\cdot h$, where the $b$s are the parallel sides and $h$ is the distance between those bases.
  • Rhombus - a special case of a kite and parallelogram, so either formula may be used here.
  • Rectangle - ${\displaystyle K=lw}$, where $l$ is the length of the rectangle and $w$ is the width. (This is a special case of the formula for a parallelogram where the height and a side happen to coincide.)
  • Square - $\displaystyle K=s^2$, where $s$ is the length of a side.


See also

  • Areas -- An article about areas of various geometric figures.
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Area&oldid=10371"