**Introduction to Quadrilateral Perimeters and Areas**

*Interior area* is an expression of the size of the region enclosed by a polygon, and that lies in the same plane as all the vertices of the polygon. It is expressed in square units (or units squared). The *perimeter* of a polygon is the sum of the lengths of its sides. Perimeter can also be defined as the distance once around a polygon, starting at some point on one of its sides and proceeding clockwise or counterclockwise along the sides until that point is encountered again. Perimeter is expressed in linear units (or, if you prefer, “plain old units”).

**Perimeter Of Parallelogram**

Suppose we have a parallelogram defined by points *P, Q, R* , and *S* , with sides of lengths *d* and *e* as shown in Fig. 3-15. The two angles labeled *x* have equal measure. Let *d* be the base length and let *h* be the height. Then the perimeter, *B* , of the parallelogram is given by the following formula:

*B* = 2 *d* + 2 *e*

**Fig. 3-15** . Perimeter and area of parallelogram. A parallelogram is a rhombus if and only if *d* = *e* .

**Interior Area Of Parallelogram**

Suppose we have a parallelogram as defined above and in Fig. 3-15. The interior area, *A* , is the product of the base length and the height:

*A* = *dh*

**Perimeter and Area of a Rhombus**

**Perimeter Of Rhombus**

Suppose we have a rhombus defined by points *P, Q, R* , and *S* , and having sides all of which have the same length. The rhombus is a special case of the parallelogram (Fig. 3-15) in which *d* = *e* . Let the lengths of all four sides be denoted *d* . The perimeter, *B* , of the rhombus is given by the following formula:

*B* = 4 *d*

**Fig. 3-15** . Perimeter and area of parallelogram. A parallelogram is a rhombus if and only if *d* = *e* .

**Interior Area Of Rhombus**

Suppose we have a rhombus as defined above and in Fig. 3-15, where *d* = *e* . Let the lengths of all four sides be denoted *d* . The interior area, *A* , of the rhombus is the product of the length of any side and the height:

*A* = *dh*

**Perimeter and Area of a Rectangle**

**Perimeter Of Rectangle**

Suppose we have a rectangle defined by points *P, Q, R* , and *S* , and having sides of lengths *d* and *e* as shown in Fig. 3-16. Let *d* be the base length, and let *e* be the height. The perimeter, *B* , of the rectangle is given by the following formula:

*B* = 2 *d* + 2 *e*

**Fig. 3-16** . Perimeter and area of rectangle. The figure is a square if and only if *d* = *e* .

**Interior Area Of Rectangle**

Suppose we have a rectangle as defined above and in Fig. 3-16. The interior area, *A* , is given by:

*A* = *de*

**Perimeter and Area of a Square**

**Perimeter Of Square**

Suppose we have a square defined by points *P, Q, R* , and *S* , and having sides all of which have the same length. The square is a special case of the rectangle (Fig. 3-16) in which *d* = *e* . Let the lengths of all four sides be denoted *d* . The perimeter, *B* , of the square is given by the following formula:

*B* = 4 *d*

**Fig. 3-16** . Perimeter and area of rectangle. The figure is a square if and only if *d* = *e* .

**Interior Area Of Square**

Suppose we have a square as defined above and in Fig. 3-16, where *d* = *e* . Let the lengths of all four sides be denoted *d* . The interior area, *A* , is equal to the square of the length of any side:

*A* = *d* ^{2}

**Perimeter and Area of a Trapezoid**

**Perimeter Of Trapezoid**

Suppose we have a trapezoid defined by points *P, Q, R* , and *S* , and having sides of lengths *d, e, f* , and *g* as shown in Fig. 3-17. Let *d* be the base length, let *h* be the height, let *x* be the angle between the sides having length *d* and *e* , and let *y* be the angle between the sides having length *d* and *g* . Suppose the sides having lengths *d* and *f* (line segments *RS* and *PQ* ) are parallel. Then the perimeter, *B* , of the trapezoid is:

*B* = *d* + *e* + *f* + *g*

**Fig. 3-17** . Perimeter and area of trapezoid. Dimensions and angles are discussed in the text.

**Interior Area Of Trapezoid**

Suppose we have a trapezoid as defined above and in Fig. 3-17. The interior area, *A* , is equal to the average (or *arithmetic mean* ) of the lengths of the base and the top, multiplied by the height. The formula for calculating *A* is as follows:

*A* = [( *d* + *f* )/2] *h*

= ( *dh* + *fh* )/2

Suppose *m* represents the length of the median of the trapezoid, that is, a line segment parallel to the base and the top, and midway between them. Then the interior area is equal to the product of the length of the median and the height:

*A* = *mh*

Practice problems for these concepts can be found at: Quadrilaterals Practice Test.