Derivation and Proof of the Trapezoid Area Formula
Why does Area = ½(a + b)h work? Here are four distinct geometric and algebraic derivations: the classic Parallelogram Duplication method, the Two-Triangle Decomposition, the Rectangle-and-Triangles dissection, and the Calculus Trapezoidal Rule.
Method 1: Duplication into a Parallelogram
This is the most intuitive visual proof taught worldwide in middle school and high school geometry:
- Take the original trapezoid with parallel sides a and b and height h.
- Make an exact identical copy of this trapezoid.
- Rotate the copy by 180° and attach its slanted leg to the original trapezoid.
- The combined composite shape forms a single large parallelogram!
- The base of this parallelogram is (a + b), and its perpendicular height remains h.
- Since the area of any parallelogram is Base × Height:
Area of 1 Trapezoid = ½(a + b)h
Method 2: Decomposition into Two Triangles
Any trapezoid can be split along one of its diagonals into two non-overlapping triangles sharing the exact same vertical height h.
Draw a diagonal line from top-left vertex to bottom-right vertex:
- Triangle 1: Has base a (the top parallel side) and perpendicular altitude h.
Area(Triangle 1) = ½ · a · h - Triangle 2: Has base b (the bottom parallel side) and perpendicular altitude h.
Area(Triangle 2) = ½ · b · h
Summing both triangles yields the total area of the trapezoid:
Total Area = (½ · a · h) + (½ · b · h)
Total Area = ½(a + b)h
Method 3: Central Rectangle & Two Right Triangles
Drop two vertical altitudes from the ends of the shorter top base a straight down to the longer bottom base b. This partitions the trapezoid into:
- A central rectangle of width a and height h: Area = a · h
- Two right-angled flank triangles of bases x₁ and x₂, where x₁ + x₂ = b - a
- Combined area of the two flank triangles: ½(x₁)h + ½(x₂)h = ½(x₁ + x₂)h = ½(b - a)h
Now sum the central rectangle and flank triangles:
Method 4: Trapezoidal Rule in Calculus & Physics
In integral calculus and physics (such as kinematics velocity-time graphs), calculating the area under a curve is approximated or evaluated using trapezoids.
The Trapezoidal Rule for Numerical Integration:
To approximate the definite integral ∫ₐᵇ f(x) dx partitioned into n strips of equal sub-interval width Δx = (b - a)/n:
Each strip is modeled as a vertical trapezoid whose parallel bases are vertical function heights yᵢ = f(xᵢ) and yᵢ₊₁ = f(xᵢ₊₁), and whose height is strip width Δx.
Physics: Displacement from a Velocity-Time Graph
When an object accelerates uniformly from initial velocity u to final velocity v over time interval t, the velocity-time graph forms a trapezoid on the time axis:
Displacement (s) = Area under v-t graph = ½ (u + v) · t
This is the fundamental kinematic equation of motion, directly representing the geometric area of a right trapezoid!
বিশেষায়িত ট্র্যাপিজিয়াম ক্যালকুলেটর
যেকোনো ট্র্যাপিজিয়ামের ক্ষেত্রফল, পরিসীমা এবং অজানা বাহু সহজেই গণনা করুন।