Area & Volume
Area Formulas
Applying area formulas for triangles, parallelograms, trapezoids, and circles.
Where Area Formulas Actually Come From
You already know a triangle's area is half its base times height. Before reading on, think about a trapezoid — a four-sided figure with exactly one pair of parallel sides. Can you find a way to decompose a trapezoid into two triangles, and use that to predict its area formula before seeing it?
Draw a diagonal across a trapezoid with parallel sides b₁ and b₂ and height h — it splits into two triangles, one with base b₁ and the other with base b₂, both sharing the same height h (the trapezoid's height, since both parallel sides are that same perpendicular distance apart). Adding those two triangle areas: (1/2)(b₁)(h) + (1/2)(b₂)(h), which factors into (1/2)(b₁ + b₂)(h) — the trapezoid's area is exactly half the sum of its parallel sides, times its height.
A circle's area formula, πr², can also be reasoned toward rather than just stated. Imagine slicing a circle into many thin, equal pie-shaped wedges and rearranging them, alternating direction, into a shape that looks increasingly like a rectangle as the wedges get thinner and more numerous. That approximate rectangle's height is the circle's radius r, and its base is half the circle's circumference (since the wedges' curved edges, laid end to end, make up half of πd going one way and half going the other): (1/2)(2πr) = πr. Multiplying height by base gives r × πr = πr² — the more wedges used, the closer this rearranged shape gets to a genuine rectangle, and the more exact this reasoning becomes.
Worked Example — Finding a Trapezoid's Area
Worked Example — Finding the Area of a Composite Figure
Geometry Canvas
Construct
Objects
- 1.
P1: a free point, draggable on the plane
- 2.
P2: a free point, draggable on the plane
- 3.
P3: a free point, draggable on the plane
- 4.
poly1: the polygon through P1, P2, P3
Measurements
- poly1area = 15perimeter = 17.66
Tip
Common Mistakes
Using a trapezoid's slanted side length as its height, instead of the actual perpendicular distance between the two parallel sides.
A trapezoid's height, like a triangle's, is always the perpendicular distance between the base and its opposite side — never the length of the slanted leg connecting them.
Adding a composite figure's semicircular piece as a full circle's area instead of half of it.
Check exactly how much of a circle a composite figure actually includes — a semicircular piece attached to a shape is half a circle's area, using half of πr², not the full formula.
Key Takeaways
- A trapezoid's area formula, (1/2)(b₁ + b₂)h, comes from splitting it into two triangles sharing the same height.
- A circle's area formula, πr², can be reasoned toward by rearranging thin pie-shaped wedges into an approximate rectangle.
- Composite figures are handled by decomposing them into simpler shapes with known formulas and adding (or subtracting) their areas.
Summary
Understanding where area formulas come from, not just memorizing them, extends naturally to three dimensions. The next lesson finds the surface area of solid figures using the same net-based reasoning from earlier grades.
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