Area of Trapezoids, Rhombi, & Kites
Trapezoids use the average of the bases. Rhombi and kites both use half the product of the diagonals.
Introduction
Past Knowledge
Rectangle/triangle area (11.1.1). Trapezoid properties (9.3.1). Kite/rhombus properties (9.2.4, 9.3.3).
Today's Goal
Derive and apply the area formulas for trapezoids, rhombi, and kites.
Future Success
Regular polygon area (11.1.3), composite figures (11.1.4).
Key Concepts
Trapezoid
“Half the sum of the bases times the height” — or: average base × height.
Rhombus & Kite
Works because perpendicular diagonals divide the shape into 4 right triangles whose areas total half the rectangle formed by the diagonals.
Formula Derivations
Trapezoid: Two Copies Make a Parallelogram
Rotate a copy of the trapezoid 180° and attach it to the original. The result is a parallelogram with base and height .
Parallelogram area = . The trapezoid is half:
∎ Two trapezoids = one parallelogram → halve it.
Worked Examples
Trapezoid Area
Bases 8 and 14, height 6.
A = 66
Rhombus from Diagonals
Rhombus with diagonals 10 and 24.
A = 120
Find Missing Diagonal
Kite area = 54, one diagonal = 9. Find the other.
→ →
d₂ = 12
Common Pitfalls
Using Leg Instead of Height
For trapezoids, the height is the perpendicular distance between bases, not the slant leg.
Rhombus: Side ≠ Diagonal
The diagonal formula uses the full diagonals, not half-diagonals! If given half-diagonals, double them first.
Real-Life Applications
Retaining Walls
Cross-sections of retaining walls are trapezoidal — wider at the base for stability. The area formula helps calculate the volume of concrete needed.
Kite Design
The fabric area for a kite-shaped kite is exactly ½d₁d₂ — knowing the stick lengths gives the material needed.
Practice Quiz
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