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Geometry
AREA

Area represents the amount of space enclosed by the boundary of a figure and is expressed in square units. For example, square meters.  In practical terms, area calculations are helpful in determining the size of a piece of land, or the quantity of materials needed for construction. If you’re not super solid on the properties of different polygons and circles, you should review before you move on to this topic.  When you’re ready, this downloadable geometry study guide is here for you and consultations are always a click away.

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Polygon Interior Angle Sum Theorem

Breaking figures down into simpler shapes, such as triangles or rectangles, can be helpful in determining area. Remember that “regular” means a polygon has sides that are of equal length and all angles are of equal measure.  Additionally, remember that the sum of the measures of all angles in a triangle is 180 degrees.  Knowing the number of triangles that a polygon can be split into and knowing the number of degrees in a triangle allows you to determine the sum of the measures of interior angles of any regular polygon.

Determining the Area of Parallelograms

A parallelogram is a four-sided figure with two sets of parallel sides. There are different ways to find the areas of specific types of parallelograms, but the most straight forward method is the one used for squares and rectangles.  A square is an equilateral and equiangular parallelogram.  All its sides are equal in length, and all its angles are right angles (90 degrees).  A rectangle is an equiangular parallelogram (each angle is equal to 90 degrees), but not all sides are equal. It has two sets of sides that are equal in length. To find the area of these figures, simply multiply the length of the base by the length of the height. The base of a parallelogram is any one of its parallel sides.  The height is the perpendicular distance from the base to the opposite parallel side. The formula for the area of a parallelogram is derived from the fact that a parallelogram can be divided into two congruent triangles. The area of each triangle is half the product of the base and height. Multiplying this by 2 gives the total area of the parallelogram.

 

Calculating the Area of a Rhombus

A rhombus is a special type of parallelogram in which all four sides are of equal length, but not all angles are equal.  It is equilateral and has two sets of angles that are equal in measure.  Another way to talk about the angles is to say that opposite angles of a rhombus are equal, but sets of opposite angles are not congruent to each other. Interestingly, the diagonals of a rhombus are perpendicular, splitting the figure into 4 congruent right triangles.  The area of a rhombus is ½ the product of its diagonals.

 

 

How to Calculate the Area of a Triangle

The base of a triangle is any one of its sides. The height is the perpendicular distance from the base to the opposite vertex. The height can be inside the triangle or outside the triangle, depending on its shape.  To find the area of a triangle multiply the length of the base by the length of the height, and then divide the result by 2. This formula is derived from the fact that a triangle can be viewed as half of a parallelogram. The base represents its length, and the height represents its width. Multiplying the base by the height gives the area of the entire parallelogram, and dividing by 2 accounts for the fact that a triangle is half of a rectangle.

 

Area of a Trapezoid

A trapezoid is an irregular polygon with four sides. It is not a parallelogram because it has only one pair of parallel sides. These parallel sides are called the "bases" of the trapezoid, and the non-parallel sides are referred to as the "legs" or "trapezoid legs."  The height is the perpendicular distance between the bases. To find the area of a trapezoid, add the lengths of the two bases, multiply the sum by the height, and then divide the result by 2.

How to Find the Area of a Kite

A kite, in geometry, refers to a quadrilateral where two adjacent sides are of equal length, and the other two adjacent sides are also equal but may have a different length from the first pair.  Even though a kite is not a parallelogram like a rhombus is, its diagonals are also perpendicular so it can be dissected into four right triangles. The area of a kite is ½ the product of its diagonals.

Apothem and The Area of a Regular Polygon Theorem

An apothem is a line segment from the center of a regular polygon to the midpoint of one of its sides. For regular polygons, the apothem is constant, as all sides and angles are congruent. To determine the area of a regular polygon multiply the product of the apothem and the perimeter of the polygon by ½ This essentially breaks down the regular polygon into congruent isosceles triangles, where each triangle shares a common vertex at the center of the polygon and the base is one of the sides of the polygon.

Finding the Area of a Circle

The radius of a circle is the distance from its center to any point on its edge. Squaring the radius and then multiplying by the mathematical constant π (pi, approximately equal to 3.14159) give the area of the circle.

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