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Geometry: Area of Polygons




In this lesson, we will learn

  • the formulas for the area of following polygons
    • square
    • rectangle
    • parallelogram
    • triangle
    • rhombus
    • kite
    • trapezoid
  • the formula for the area of any regular polygon

Related Topics: More Geometry Lessons

Area of Polygons

The area of a polygon measures the size of the region enclosed by the polygon. It is measured in units squared.

See also area of circles.

Area of a Square

The area of a square is equal to the length of one side squared.

square

Area of a square = s2

Area of a Rectangle

The area of a rectangle is equal to the product of the length of its base and the length of its height. Sometimes, the height is called the "altitude". We can also call the longer side the "length" and the shorter side the "width"

rectangle

Area of a rectangle = lw

Worksheet 1, Worksheet 2, Worksheet 3 to calculate the area and perimeter of rectangles.

Area of a Parallelogram

parallelogram

To get the area of a parallelogram, we first draw a perpendicular line segment from one corner of the parallelogram to the opposite side. This is the height ( h) of the parallelogram. The area of a parallelogram is equal to the product of its length and height. (Remember to always use the perpendicular height)

Area of a parallelogram = lh

You may notice that lh is also the area of a rectangle with dimensions l and h. The diagram below will explain why. If we cut out the triangle ABC and add it to the other side (triangle DEF), you will have a rectangle with dimensions l and h that has the same area as the original parallelogram.

parallelogram

Worksheet to calculate the area of parallelograms


 



Area of a Triangle

triangle

To get the area of a triangle, we first choose one of the sides to be the base (b). Then we draw a perpendicular line segment from a vertex of the triangle to the base. This is the height ( h) of the triangle. The area of a triangle is equal to half the product of the base and the height.

area of triangle

There are also other formulas for the area of a triangle. More examples, formulas and videos for areas of triangles

Area of a Rhombus

We can obtain the area of a rhombus, given the lengths of its diagonals.

If the lengths of the diagonals are a and b, then area of the rhombus is equal to half the product of the diagonals.

Area of rhombus =



If you are given the length of one side (s) and the perpendicular height (h) from one side to the vertex then the area of the rhombus is equal to the product of the side and height.

The area of the rhombus is given by the formula:

Area of rhombus = sh

This formula for the area of a rhombus is similar to the area formula for a parallelogram.

Area of a Kite

The area of a kite uses the same formula as the area of a rhombus. The area of a kite is equal to half the product of the diagonals.

Area of kite =area of rhombus


Area of a Trapezoid

trapezoid
a is parallel to b

To get the area of a trapezoid, we sum the length of the parallel sides and multiply that with half of the height. Remember that the height needs to be perpendicular to the parallel sides.

trapezoid area formula


Worksheet to calculate area of polygons



 



The following video gives formulas and examples for how to find the area of squares, rectangles, triangles, parallelograms, and trapezoids. The video also explains the difference between base and height.



The following video shows how to derive the area formula of a kite and a rhombus based on the rectangle formula; how to calculate the area of a rectangle using diagonal lengths.


Area of any Regular Polygon

A regular polygon is a polygon where all the sides are the same length and all the angles are equal.

The apothem of a regular polygon is a line segment from the center of the polygon to the midpoint of one of its sides. The area of any regular polygon is equal to half of the product of the perimeter and the apothem.

Area of regular polygon =  where p is the perimeter and is the apothem


The following video shows how to use a formula to find the area of any regular polygon.



 



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