Area of Rectangles, Triangles, & Parallelograms
These three foundational area formulas build on each other: a rectangle's area leads to a parallelogram's, which leads to a triangle's.
Introduction
Past Knowledge
Multiplication. Perpendicular height. Parallelogram properties (9.2.1).
Today's Goal
Derive and apply area formulas for rectangles, parallelograms, and triangles.
Future Success
Trapezoid/kite area (11.1.2), composite figures (11.1.4), surface area (12.2).
Key Concepts
The Three Formulas
| Shape | Formula | Key Variable |
|---|---|---|
| Rectangle | l = length, w = width | |
| Parallelogram | h = perpendicular height | |
| Triangle | h = perpendicular height |
Formula Derivations
Parallelogram = Rectangle (Rearranged)
Cut the left triangle and move it to the right side → it becomes a rectangle with area .
Triangle = ½ Parallelogram
A diagonal splits a parallelogram into two congruent triangles, so each triangle's area is .
Worked Examples
Rectangle Area
Rectangle: l = 12 cm, w = 7 cm.
A = 84 cm²
Parallelogram with Slant Side
Base = 15, slant side = 10, height = 8. Find area.
Use the height, not the slant side:
A = 120 sq units (slant side is irrelevant for area)
Triangle — Find the Height
Triangle: area = 60, base = 15. Find h.
→ →
h = 8
Common Pitfalls
Using the Slant Side as Height
The height must be perpendicular to the base. The slant side of a parallelogram or triangle is NOT the height.
Forgetting ½ for Triangles
Triangles get ½ because they're half a parallelogram. Parallelograms do NOT have the ½.
Real-Life Applications
Carpet & Flooring
Calculating square footage for a rectangular room uses A = lw. Irregularly shaped rooms get broken into rectangles and triangles.
Roofing & Construction
Gable roofs are triangular — the area formula tells builders how many shingles to order.
Practice Quiz
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