Geometric Proofs & Calculus

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:

Two identical congruent trapezoids combine to form one large parallelogram.

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 2 Trapezoids = (a + b) × h
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 = Area(Triangle 1) + Area(Triangle 2)
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:

Total Area = ah + ½(b - a)h = h [a + ½b - ½a] = h [½a + ½b] = ½(a + b)h

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:

∫ₐᵇ f(x) dx ≈ (Δx / 2) [ f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ) ]

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!

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Trapezoid vs Triangle Area

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