Area of Trapezoid
The area of a trapezoid is calculated using the formula A = (1/2)h(b1 + b2), where A represents the area, h is the height, and b1 and b2 are the lengths of the two parallel sides. This formula allows for the quick and accurate determination of the trapezoid's area, making it a valuable tool in geometry and real-world applications.
6 Key excerpts on "Area of Trapezoid"
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...An isosceles trapezoid has two parallel sides (FG and EH in the figure above), and the other two sides (EF = HG) are equal. The base is usually chosen to be the longest parallel side, and the base angles (the angles on either side of it) are equal, although it is also true that the other two angles are equal. In addition, for an isosceles trapezoid, the diagonals (here they would be EG and FH) are of equal length. The perimeter of any trapezoid is just the sum of the sides (AB + BC + CD + DA); for the isosceles trapezoid, it is (EF + FG + GH + HE). So for a trapezoid with four sides of lengths, say, a, b, c, and d, the perimeter is The area of any quadrilateral is based on the simple formula of base × height (also called altitude), but for a trapezoid we have to consider the average of the two bases as the base in this calculation. Otherwise, the area would be two different numbers, depending on which of the parallel sides is considered to be the base. The height is defined as perpendicular to the base, the same as for triangles. Since the bases are parallel to each other, the height is perpendicular to each base and its size doesn’t vary. How do we determine the average of the two bases? Averages are covered in Chapter 8, but basically the average of two quantities is their sum divided by 2. So the formula to use for a trapezoid is where b 1 and b 2 are the lengths of the two parallel sides (bases). The area of a trapezoid is given by This formula is the same as the one provided on the GED ® test formula sheet: since multiplication is commutative (xy is the same as yx; see Chapter 2). Parallelogram A parallelogram is a quadrilateral (four-sided figure) with parallel sides, as the name implies. In this case, it is a step up from the trapezoid because the other two sides are also parallel...
eBook - ePubThe Problem with Math Is English
A Language-Focused Approach to Helping All Students Develop a Deeper Understanding of Mathematics
- Concepcion Molina(Author)
- 2012(Publication Date)
- Jossey-Bass(Publisher)
...The base of the parallelogram was renamed as (b 1 + b 2). Through substitution, the area of the original parallelogram becomes A = h • (b 1 + b 2). Because each of the trapezoids had an area half that of the parallelogram, the area of either trapezoid could be expressed as half of h • (b 1 + b 2), which written mathematically would be A = • [ h • (b 1 + b 2)]. Compare that reorganized version to the standard formula A = • h • (b 1 + b 2). As with the first activity, the only difference between the formulas is in how the values are organized, which corresponds to what happened in this second activity. In the first activity, the height of the parallelogram formed was half the height of the original trapezoid and the base of the parallelogram was the sum of the two bases of the original trapezoid. In the second activity, the area of the trapezoid formed was half the area of the original parallelogram. In each case, how the standard area formula is reorganized and read parallels the process and the new figure that resulted. Combined, the two activities reveal the power of different interpretations of mathematical symbolism and the new perspectives that can result. Trapezoid Perspective III This third perspective relies on making connections to conceptual knowledge of an average (or the mean) and is not as dependent on the manipulation of geometric figures as the two prior activities. However, diagrams are necessary in order to visualize the bridging of numeric and geometric perspectives. As with the previous two activities, this task is best done with students in small groups. Give each group a drawing of trapezoid WXYZ (Box 9.23) on card stock. Box 9.23 Start the activity by investigating what happens when we average two numbers. As an example, compute the average of 8 and 12, which is 10. In the original context, the numbers are 8 and 12 and in the other the numbers would be 10 and 10. Each of the two addends in the second context is the mean...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...formula where d 1 and d 2 are the diagonals. ABCD is a rhombus. AC = 4 and BD = 7. The area of the rhombus is. • The diagonals of a rhombus bisect the angles of the rhombus. • If the diagonals of a parallelogram are perpendicular, the parallelogram is a rhombus. • If a quadrilateral has four equal sides, then it is a rhombus. • A parallelogram is a rhombus if either diagonal of the parallelogram bisects the angles of the vertices it joins. SQUARES A square is a rhombus with a right angle. • A square is an equilateral quadrilateral. • A square has all the properties of rhombi and rectangles. • In a square, the measure of either diagonal can be calculated by multiplying the length of any side by the square root of 2. • The area of a square is given by the formula A = s 2, where s is the side of the square. • Since all sides of a square are equal, it does not matter which side is used. Example: The area of the square shown below is: A = s 2 A = 6 2 A = 36 The area of a square can also be found by taking the product of the length of the diagonal. squared. This comes from a combination of the facts that the area of a rhombus is d 1 d 2 and that d 1 = d 2 for a square. Example: The area of the square shown below is: TRAPEZOIDS A trapezoid is a quadrilateral with two and only two par allel sides. The parallel sides of a trapezoid are called the bases. The median of a trapezoid is the line joining the midpoints of the nonparallel sides. The perpendicular segment connecting any point in the line containing one base of the trapezoid to the line containing the other base is the altitude of the trapezoid. A pair of angles including only one of the parallel sides is called a pair of base angles. • The median of a trapezoid is parallel to the bases and equal to one-half their sum. • The area of a trapezoid equals one-half the altitude times the sum of the bases, or • An isosceles trapezoid is a trapezoid whose non-parallel sides are equal...
- Alfred S. Posamentier, Stephen Krulik(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...In geometry, there are other formulas for the area of triangles that students may already have learned. For a right triangle, the area is one-half the product of its legs, and for a triangle where the lengths of two sides and the measure of the included angle are given, the area is one-half the product of the lengths of the two given sides and the sine of the included angle (i.e.,). However, how can we find the area of a triangle, if we are only given the lengths of its three sides? The answer is to use Heron’s formula, which students can be motivated to discover by initially giving them the following problems to work out. Present the following three problems at the start of the lesson: Find the area of a triangle whose base length is 8 and height to that base is 6. Find the area of a triangle whose side lengths are 3, 4, and 5. Find the area of a triangle whose sides have lengths 13, 14, and 15. For students to find the area of a triangle with base length 8 and height 6, they can easily use the formula, and find the area to be square units. Next, for students to find the area of a triangle whose sides are lengths 3, 4 and 5, they should recognize that this is a right triangle, with legs 3 and 4. Thus the area is given by square units. Now students are faced with the task of finding the area of a triangle whose sides are 13, 14, and 15. This is not a right triangle, so no side can be thought of as an altitude. There must be another way to find the area. Here the students should recognize a void in their knowledge of finding the area of a triangle. One of the first student reactions is that, since 3-4-5 are sides of a right triangle, what about 13-14-15? Sadly, the triangle is not a right triangle...
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...Any other triangle can then be seen as a shearing of a right-angled triangle with the same base length, height and area. This ‘model to think with’ can be applied to any parallelogram or triangle, so that the learner can mentally transform the figure on the page back into a rectangle or right-angled triangle. Then the base and height can be identified easily, and so its area may be found. There is no need for a formula: the process is visual and kinaesthetic, not symbolic. PowerPoint PowerPoint 9-2, Areas of Straight-Sided Shapes, shows how to find the area of a parallelogram or a triangle by first shearing it so that it has a right angle, and then basing the calculations on the area of the resulting rectangle. e) Capacity and Volume We have seen that area can be thought of as ‘an amount of flatness’. It is a strictly two-dimensional concept. Capacity and volume, on the other hand, are three-dimensional. But our world is three-dimensional, so these concepts may actually be easier to understand. Capacity may be the best place to start. A capacity relates to a particular container, and it tells you how much that container can hold. This idea may be established using informal measures, such as the number of small cups-full that are needed to fill a big jug. Volume, on the other hand, is the amount of ‘stuff’ that is needed to make a solid. Activities which involve building up cuboids a layer at a time out of cubic-centimetre cubes to find their volumes are commonly used to introduce a more formal measure of volume. This is a good practical approach, and it may be extended in due course to other solids – but it is important to emphasise that the volume relates to the whole block of cubes, not to an empty container. An empty carton has a capacity – but it is empty, so its only ‘volume’ is the volume of plastic or cardboard of which it is composed...
eBook - ePubProject Surveying
Completely revised 2nd edition - General adjustment and optimization techniques with applications to engineering surveying
- Peter Richardus(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...as 2 A = [ (X n +X n+1) (Y n+1 − Y n) ]. (2.12) It can easily be corroborated that 2 A = [ (X n+1 − X n) (Y n − Y n+1) ], (2.13) counting anticlockwise. These formulae (2.12) and (2.13) are known as the trapezoidal formulae, since each product represents the double area of a trapezoid, the area of the polygon being the sum of these. It may seem more convenient to apply a translation of the origin of the coordinate system to the point A of the triangle ABC in Fig. 2.1. This is only ostensibly so since the machine takes care of the coordinate differences automatically. The numerous graphical methods of calculation of areas will not be treated. Attention is drawn only to the development of electronic planimeters and digitizers, which will show their economy especially in projects where many areas are to be determined. 2.2 Problems 1. Calculate the area enclosed by the polygon given by the coordinates of following points: 2. Calculate the area of the block of land, shown in the Fig. 2.4, obtained by a chain survey. The chainage along the traverse line AB and the lengths of the offsets at right angles to this line, to the boundary points are as given in the figure. Fig. 2.4...
