What Is a Trapezoid and How to Find the Area of a Trapezoid ?
Before getting to the question on «how to find the area of a trapezoid », first let’s figure out what a trapezoid and an area of a trapezoid are. Trapezoid is a figure with four sides, so it is a quadrilateral, because quadrilaterals are figures with four sides. However, two sides of a trapezoid are parallel and the other two sides are not. Its parallel sides are called the base of a trapezoid, and the other two are called the legs or the lateral sides. So, a trapezoid is a quadrilateral figure with one pair of parallel sides. The distance between the two bases of a trapezoid is height.
Etymology of a Trapezoid
The expression «trapezoid» has been in application in the English language since 1570, from Greek «trapezion» and from Late Latin trapezium, which word for word means a «small table». On the book «Elements» by Euclid, Marinus Proclus at first described «trapezion» when writing the commentary.
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Special Cases
- A right trapezoid.
It has two adjacent right angles. These kinds of trapezoids are applied in the trapezoidal rule for determination areas under a curve. - An acute trapezoid.
It has two adjacent acute angles on the longer base edge. It is also an isosceles trapezoid, if its sides are of the same length, and the angles in the base are of the same measure. It has reflex symmetry. - An obtuse trapezoid.
It has one obtuse and one acute angle on each base. An obtuse trapezoid that has two pairs of sides parallel is a parallelogram, which has central two-fold rotational symmetry.
There are some disputes about whether parallelograms with two pairs of parallel sides should be considered as trapezoids. Some people define a trapezoid as a quadrilateral with just one couple of parallel sides (the exclusive determination), which excludes parallelograms. Others, however, define a trapezoid as a quadrilateral with at least one pair of sides that are parallel (the inclusive determination), making quadrilaterals a special kind of trapezoid. The latter determination is consistent with its applications in higher mathematics. The former definition would make concepts like trapezoidal approximation to a definite integral ill-defined.
According to the inclusive determination, all parallelograms (rectangles, rhombuses, and squares) are trapezoids. Rectangles have reflex symmetry on mid-edges. Rhombuses have reflex symmetry on vertices. Squares have reflex symmetry on both vertices and mid-verges.
A Saccheri quadrilateral is very alike to a trapezoid in the hyperbolic plane with two adjoining right angles, however in the Euclidean plane it is a rectangle. In the hyperbolic plane a Lambert quadrilateral has three right angles.
A tangential trapezoid is a trapezoid with an incircle.
Characterizations
If quadrilateral is convex, the following properties are reciprocal, and each assumes that the quadrilateral is a trapezoid:
- It has two adjoining angles that are supplementary, i.e. they add up to 180 degrees.
- The angel between a diagonal and a side is equal to the angle between the same diagonal and the opposite side.
- The diagonals divide each other in mutually the same ratio.
- The diagonals split the quadrilateral into four triangles, of which one opposite pair is similar.
- The diagonals split the quadrilateral into four triangles, of which one opposite pair has equal areas.
- The product of the areas of the two triangles created by one diagonal equals the product of the two triangles created by the other diagonal.
- The areas T and S of some two opposite triangles of the four triangles created by one diagonal satisfy the equation: √K = √S + √T, where K is the quadrilateral’s area..
- The midpoints of the two opposite sides and the crossing of the two diagonals are collinear.
- sin A sin C = sin B sin D.
- The two adjacent angles’ cosines sum to 0, as the cosines of the other two angles.
- The two adjacent angles’ cotangents sum to 0, as the cotangents of the other two adjacent angles.
- One bimedian cuts the quadrilateral in two quadrilaterals with equal areas.
- Twice the length of the bimedian that connects the midpoints of the two opposite sides equals the total of the lengths of the other sides.
How to Find the Area of a Trapezoid ?
If when writing a reflective essay, you need to measure a trapezoid’s area, here is the easy explanation on how to do it:
- In order to understand how to find the area of a trapezoid, measure the length of the two bases. Let’s call them sides a and b. For example, side a is 8 cm long and side b is 13 cm long.
- Sum the length of the two bases, i.e., add 8 cm and 13 cm. 8 + 13 = 21 cm.
- Measure the height of the trapezoid. The height of a trapezoid is a perpendicular to the bases. In our example, the height equals 7 cm.
- Multiply the sum of the lengths of the two bases by the height: 21 x 7 = 147 cm2.
- Divide the result by two. Divide 147 cm2 by two to get the result of the trapezoid’s area. 147/2 = 73.5 cm2.
All these actions form the formula that measures the trapezoid’s area, which is the following: [(a + b) x h]/2. Now you know how to find the area of a trapezoid using the main formula.
In 499 AD, a great mathematician-astronomer from the classical time of Indian mathematics and astronomy used this formula in the Aryabhata. This yields as a special case the well-known formula for the triangle area by considering a triangle as a degenerate trapezoid, where one of the parallel sides has shrunk to a point.
How to Find the Area of a Trapezoid if You Know the Median?
Let’s remember what the median is, before getting to the point on «how to find the area of a trapezoid». It is a line that connects the points of the sides that are non-parallel. The length of the median is the average of the two parallel sides. If you have a given value of a meridian then you can measure the trapezoid area following this formula: Area = mh, where m is the median and h is the height.
How to Find the Area of a Trapezoid Using a Compound Shape?
One more way that helps you to find out how to find the area of a trapezoid is to treat it as some plainer shapes, and then subtract or add their areas to find the result. For example, a trapezoid could be treated to a rectangle of a smaller shape with two right triangles. There are many ways for calculating the area of a triangle. Exactly how you do it depends on what values you are given at the start. As this can be very diverse, there is no an easy way to do it.
Coordinate Geometry that Helps to Find out How to Find the Area of a Trapezoid
When doing a research summary in coordinate geometry, if the coordinates of the four vertices are given, you can calculate different properties of it, including the perimeter and the area.
Finding a Base of a Trapezoid from the Area
In order to find a base of a trapezoid, give the one of the bases, the area, and the height. The main formula that gives an answer on how to find the area of a trapezoid has four variables (two bases, area, and the height). If you know those three variables you can easily calculate the fourth one. So, for example, if you know the height, the area, and one base, you can find the other base. In order to find that, follow this formula: Base length = (2a/h)-b, where a is the area, b is the known base, and h is the height.
Finding the Height of a Trapezoid if You Know the Area
The main formula that explains how to find the area of a trapezoid has four variables (two bases, the area, and the height). If you know those three variables, you can easily measure the fourth one. For example, if you know the two bases and the area, you can find the height, by easily reposition the main formula: Height = (2a)/(b1+b2), where a is the area, and b1, b2 are the bases.
Where Does the Formula of How to Find the Area of a Trapezoid Come from?
All the trapezoids can be divided into two triangles. In this case, if you find the area of each two triangles and then add it up together, you will get an area of a trapezoid. In order to find the area of a triangle, you need to use this formula: A = (bxh)/2. Let’s say that the base of the bottom triangle is 6 cm and its height is 5 cm. So, that gives us A = (6×5)/2 = 15 cm2. Now, you can find the area of the top triangle. For example, its base equals 4 cm, and its height is the length of the line segment drawn from its upper vertex down to its base, so this height is exactly the same as of the other bottom triangle. So, the A = (bxh)/2 and this gives us A = (4×5)/2 = 10 cm2. Now, we know that the area of one triangle is 15 cm2 and the area of the other triangle is 10 cm2. And now, you can add the areas together to get the area of a trapezoid: A = 15 cm2 + 10 cm2 = 25 cm2. This value is the same as the one we got when used the main formula for calculating a trapezoid’s area. Now, you know where the main formula comes from and how to find the area of a trapezoid in different ways.
More about the Terminology of the Term
When writing an essay introduction you can specify more details on the term trapezium. In the U.S., for example, a trapezium is sometimes determined as a quadrilateral without parallel sides with the shape that is usually called an irregular quadrilateral. Once in Britain and elsewhere, the term trapezoid was defined as a quadrilateral without any parallel sides.
Oxford English Dictionary states that the sense of a figure that has no parallel sides is the meaning for which Proclus introduced the term trapezoid. This is kept in the French trapézoïde, German trapezoid, and other languages, although it is not used anymore. In Proclus’s sense a trapezium is a quadrilateral, which has one pair of its opposite sides parallel. This sense was specific in England in 17th and 18th centuries, and the prevalent one in recent use. A trapezium described as any quadrilateral is more general than a parallelogram is the sense of Euclid’s term. The sense of a trapezium as an irregular quadrilateral with no parallel sides was once used in England from 1800 to 1875.
Application in Geometry
In geometry, a trapezoid is used in the crossed ladder puzzle – the problem of finding the distance between the two parallel sides of a right trapezoid, when the values of the diagonal length and the distance from the perpendicular leg to the diagonal intersection are given.
Application in Architecture
In architecture the word is applied to describe symmetrical doors, windows, and buildings built wider at the base, getting refined to the top (mostly in Egyptian style). If the buildings have even sides and sharp anguled corners, their shapes are usually referred to as isosceles trapezoids. This shape was a standard style of doors and windows of the Incas.
Application in Biology
In taxonomy, morphology, and other descriptive disciplines, where the term to describe shapes is necessary, trapezoid commonly is applied in descriptions of certain forms or organs.
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