Geometry/Area

Area of Circles

The method for finding the area of a circle is

$Area=\pi r^{2}$

Where $r$ is the radius of the circle; a line drawn from any point on the circle to its center.

Area of triangles

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:

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

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 $90^{\circ }$ angle can be determined.

Second method

Main article: Hero's Formula

, also known as 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 $a,b,c$ :

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

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:

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

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

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

Third method

In a triangle with sides length $a,b,c$ and angles $\alpha ,\beta ,\gamma$ opposite them,

$A={\frac {ab\sin(\gamma )}{2}}={\frac {bc\sin(\alpha )}{2}}={\frac {ac\sin(\beta )}{2}}$

This formula is true because $h=\sin(\gamma )$ in the formula $A={\frac {Bh}{2}}$ . 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 $a\cdot b\cdot c$ 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

Parallelograms are described in their own chapter.

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 a rhombus

Remember in a rombus all sides are equal in length.

$A={\frac {d_{1}\cdot d_{2}}{2}}$

where $d_{1},d_{2}$ 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 $b_{1},b_{2}$ are the lengths of the two parallel bases.

Area of kites

Standard kite quadrilateral

The area of a kite is based on splitting the kite into four pieces by halving it along each diagonal and using these pieces to form a rectangle of equal area.

$A={\frac {ab}{2}}$

Where $a,b$ are the diagonals of the kite.

Alternatively, the kite may be divided into two halves, each of which is a triangle, by the longer of its diagonals, $a$ . The area of each triangle is thus

${\frac {a\cdot {\dfrac {b}{2}}}{2}}$

Where $b$ is the other (shorter) diagonal of the kite.

And the total area of the kite (which is composed of two identical such triangles) is

$2\cdot {\frac {a\cdot {\dfrac {b}{2}}}{2}}=a\cdot {\frac {b}{2}}={\frac {ab}{2}}.$

For more details see w:Kite (geometry)#Properties.

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 with arithmetic.

Areas defining angles

The area of a circular sector is a fraction of the area of the whole circle. When the circle has radius that is the square root of two, the circle has area 2 π, and the radian measure of the sector corresponds to the fraction of the total circular area. In calculus, another type of angle called hyperbolic is related to the exponential function. This type of angle also corresponds to the area of a sector of the hyperbola xy=1. Use of area measure provides a means to unify these angle types. See Unified Angles.

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• Geometry/Chapter 4 Congruence and Similarity
• Geometry/Chapter 5 Triangle: Congruence and Similiarity
• Geometry/Chapter 6 Triangle: Inequality Theorem
• Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
• Geometry/Chapter 8 Perimeters, Areas, Volumes
• Geometry/Chapter 9 Prisms, Pyramids, Spheres
• Geometry/Chapter 10 Polygons
• Geometry/Chapter 11 $\mathbb {R} ,\mathbb {R} ^{2},\mathbb {R} ^{3}$
• Geometry/Chapter 12 Angles: Interior and Exterior
• Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
• Geometry/Chapter 14 Pythagorean Theorem: Proof
• Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
• Geometry/Chapter 16 Constructions
• Geometry/Chapter 17 Coordinate Geometry
• Geometry/Chapter 18 Trigonometry
• Geometry/Chapter 19 Trigonometry: Solving Triangles
• Geometry/Chapter 20 Special Right Triangles
• Geometry/Chapter 21 Chords, Secants, Tangents, Inscribed Angles, Circumscribed Angles
• Geometry/Chapter 22 Rigid Motion
• Geometry/Appendix A Formulae
• Geometry/Appendix B Answers to problems
• Appendix C. Geometry/Postulates & Definitions
• Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry