Geometry/Area

Area of [[../Triangle/]]s

Three ways of calculating the area inside of a triangle are mentioned here.

First method: If one of the sides of the triangle is chosen as a base, then a height for the triangle and that particular base can be defined. The height is a line segment perpendicular to the base or the line formed by extending the base and the endpoints of the height are the corner point not on the base and a point on the base or line extending the base. Let B = the length of the side chosen as the base. Let
h = the distance between the endpoints of the height segment which is perpendicular to the base. Then the area of the triangle is given by:

$Area=\frac{B\times h}{2}$

This method of calculating the area is good if the value of a base and its corresponding height in the triangle is easily determined. This is particularly true if the triangle is a right triangle, and the lengths of the two sides sharing the 90^o angle can be determined.

Second method: Hero's Formula (or Heron's Formula)

If the lengths of all three sides of a triangle are known, Hero's formula may be used to calculate the area of the triangle. First, the semiperimeter, s, must be calculated by dividing the sum of the lengths of all three sides by 2. For a triangle having side lengths a, b, and c :

$s=\frac{a+b+c}{2}$

Then the triangle's area is given by:

$Area=\sqrt{s(s-a)(s-b)(s-c)}$

If the triangle is needle shaped, that is, one of the sides is very much shorter than the other two then it can be difficult to compute the area because the precision needed is greater than that available in the calculator or computer that is used. In otherwords Heron's formula is numerically unstable. Another formula that is much more stable is:

$Area=\frac{\sqrt{(a+(b+c))\cdot (c-(a-b))\cdot (c+(a-b))\cdot (a+(b-c))}}{4}$

where $a$, $b$, and $c$ have been sorted so that $a\ge b\ge c$.

See also Heron's Formula at MathWorld and How JAVA's Floating-Point Hurts Everyone Everywhere

Third method: In a triangle with sides length a, b, and c and angles A, B, and C opposite them,

$A=\frac{1}{2}ab*sin(C)=\frac{1}{2}bc*sin(A)=\frac{1}{2}ac*sin(B)$

This formula is true because $h=sin(C)$ in the formula $A=\frac{1}{2}Bh$. It is useful because you don't need to find the height from an angle in a separate step, and is also used to prove the law of sines (divide all terms in the above equation by a*b*c and you'll get it directly!)

Area of Rectangles

The area calculation of a rectangle is simple and easy to understand. One of the sides is chosen as the base, with a length b. An adjacent side is then the height, with a length h, because in a rectangle the adjacent sides are perpendicular to the side chosen as the base. The rectangle's area is given by:

$A=b\cdot h$

Sometimes, the baselength may be referred to as the length of the rectangle, l, and the height as the width of the rectangle, w. Then the area formula becomes:

$A=l\cdot w$

Regardless of the labels used for the sides, it is apparent that the two formulas are equivalent.

Of course, the area of a square with sides having length s would be:

$A=s^2$

Area of [[../Parallelograms/]]

The area of a parallelogram can be determined using the equation for the area of a rectangle. The formula is:

$A=b\cdot h$

A is the area of a parallelogram. b is the base. h is the height.

The height is a perpendicular line segment that connects one of the vertices to its opposite side (the base).

Area of [[../Rhombus/]]

Remember in a rombus all sides are equal in length.

$A=\frac{d_1\cdot d_2}{2}$ $d_1$ and $d_2$ represent the diagonals.

Area of [[../Trapezoids/]]

The area of a trapezoid is derived from taking the arithmetic mean of its two parallel sides to form a rectangle of equal area.

$A=\frac{(b_1+b_2)\cdot h}{2}$

Where $b_1$ and $b_2$ are the lengths of the two parallel bases.

Areas of other [[../Quadrilaterals/]]

The areas of other quadrilaterals are slightly more complex to calculate, but can still be found if the quadrilateral is well defined. For example, a quadrilateral can be divided into two triangles, or some combination of triangles and rectangles. The areas of the constituent polygons can be found and added up.