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Home » Area of Geometric Shapes » Area of a Trapezoid

Area of a Trapezoid

Last updated: Oct 29, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

It is often possible to find the area of polygons with irregular shapes by dividing them into smaller shapes with easily computed areas, like triangles and rectangles. Sometimes this leads to a simple function for the area of the polygon, as is the case with a trapezoid.

Problem

Find a formula for the area of a trapezoid using its height and bases.

Area of trapezoid

Strategy

We will divide the trapezoid into triangles and rectangles, whose areas we can compute easily. We will rely on the fact that in a trapezoid, the bases are parallel to each other.

The Problem asks us to find the area using the height of the trapezoid - so let's do that, and construct two lines which are perpendicular to the longer base, AE and BF, from the corners of the shorter base.

Since we constructed both lines as perpendicular to the base DC, then the AE and BF are parallel to each other by the Converse Perpendicular Transversal Theorem.

As a result, the quadrilateral ABFE is a rectangle - it has two pairs of parallel sides (making it a parallelogram), and interior angles that measure 90° (∠AEF=m∠BFE= 90°). In a rectangle, like any parallelogram, the opposing sides are equal, so AB=EF=x, and AE=BF=h.

The area of the rectangle is x·h, and the area of the two triangles is a·h/2 and b·h/2, respectively. The area of the trapezoid is the sum of the areas of the two triangles and the rectangle - x·h+a·h/2 + b·h/2. combining like terms we get (h/2)·(2x+a+b), or (h/2)·(x+x+a+b). But x is simply the short base, and x+a+b is the long base, so the simplified formula for the area of a trapezoid is (short base+long base)·height/2.

Solution

(1) AE ⊥ DC //Construction
(2) BF ⊥ DC //Construction
(3) AE || BF //Converse Perpendicular Transversal Theorem.
(4) AB||DC //Given, ABCD is a trapezoid
(5) ABFE is a rectangle //(3), (4) - definition of a parallelogram, (1), (2) definition of rectangle
(6) AB=EF=x //Opposing sides of rectangle are equal
(7) AE=BF=h //Opposing sides of rectangle are equal
(8) AABCD=AABFE+AAED+ABFC //Area of trapezoid is sum of area of its parts
(9) AABFE=x·h
(10) AAED=a·h/2
(11) ABFC=b·h/2
(12) AABCD=x·h+a·h/2 + b·h/2=(h/2)·(2x+a+b)=(h/2)·(x+x+a+b)
(13) AABCD=(short base+long base)·height/2

« Obtuse Triangle - Definition and Properties
Area of a Parallelogram »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
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  • Pentagons and Hexagons
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    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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