Areas formula

(pi = = 3.141592...)

Area Formulas

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements. Examples

square = a ²

rectangle = ab

parallelogram = bh

trapezoid = h/2 (b₁ + b₂)

circle = pi r ²

ellipse = pi r₁ r₂

triangle =

one half times the base length times the height of the triangle

equilateral triangle =

triangle given SAS (two sides and the opposite angle)
= (1/2) a b sin C

triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula)

regular polygon = (1/2) n sin(360°/n) S²
when n = # of sides and S = length from center to a corner

Units

Area is measured in "square" units. The area of a figure is the number of squares required to cover it completely, like tiles on a floor.

Area of a square = side times side. Since each side of a square is the same, it can simply be the length of one side squared.

If a square has one side of 4 inches, the area would be 4 inches times 4 inches, or 16 square inches. (Square inches can also be written in².)

Be sure to use the same units for all measurements. You cannot multiply feet times inches, it doesn't make a square measurement.

The area of a rectangle is the length on the side times the width. If the width is 4 inches and the length is 6 feet, what is the area?

NOT CORRECT .... 4 times 6 = 24

CORRECT.... 4 inches is the same as 1/3 feet. Area is 1/3 feet times 6 feet = 2 square feet. (or 2 sq. ft., or 2 ft²).

Prallelogram

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Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle. And, since a square is a degenerate case of a rectangle, both squares and rectangles are special types of parallelograms.

The polygon diagonals of a parallelogram bisect each other (Casey 1888, p. 2).

The angles of a parallelogram satisfy the identities

			(1)
			(2)

and

(3)

A parallelogram of base and height has area

(4)

The height of a parallelogram is

(5)

and the polygon diagonals and are

			(6)
			(7)
			(8)
			(9)

(Beyer 1987).

The sides , , , and diagonals , of a parallelogram satisfy

(10)

(Casey 1888, p. 22).

The area of the parallelogram with sides formed by the vectors and is

			(11)
			(12)
			(13)

where is the two-dimensional cross product and is the determinant.

As shown by Euclid, if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in area (and conversely), so in the above figure, (Johnson 1929).

The centers of four squares erected either internally or externally on the sides of a parallelograms are the vertices of a square (Yaglom 1962, pp. 96-97; Coxeter and Greitzer 1967, p. 84).