Area & Perimeter

Derive the area formulas — rectangle, parallelogram, triangle, circle — then measure.

The idea

Definition (Perimeter and area).

The perimeter of a flat shape is the distance around it: the sum of the lengths of its sides. Its area is the space it encloses, measured in unit squares: a region has area $12$ when twelve squares of side $1$ cover it exactly.

Neither one determines the other: two shapes with the same perimeter can enclose very different amounts of space.

Counting unit squares gives the first area formula. A rectangle of length $l$ and width $w$ holds them in $w$ rows of $l$, and multiplying counts the rows at once, so $A = lw.$

Every other basic area formula comes from this one by dissection. Cut the shape into pieces and rearrange the pieces into a rectangle — moving a piece neither gains nor loses area — then read the area off the rectangle's two sides. The dissection also shows which lengths enter each formula: a height measured perpendicular to a base, never a slanted side.

Ways to work on it

• Walkthrough. Derive the parallelogram, triangle, and circle area formulas from the rectangle.
• Practice. Tell height from side: area and perimeter of a parallelogram.
• Hardest. Derive a regular polygon's area via its center-triangles, plus an annulus.

Not sure where to start? Take the ten-question placement test.