Area & Perimeter

Perimeter is the distance all the way around — add up the sides. Area is the space inside — measured in squares.

Formulas worth knowing cold:

• Rectangle: A = length × width
• Triangle: A = ½ × base × height
• Parallelogram: A = base × height
• Trapezoid: A = ½(b₁ + b₂) × height

The height is always perpendicular to the base, not the slanted side. That is the most common error in triangle and parallelogram problems.

Units give it away: perimeter in cm, area in cm².

Area is additive over non-overlapping regions, and every polygon formula follows from the rectangle by decomposition. A parallelogram cut along an altitude and reassembled becomes a rectangle, giving A = bh. A triangle is half a parallelogram, giving A = ½bh. A trapezoid is two triangles, or equivalently the average of the parallel sides times the height, A = ½(b₁ + b₂)h.

The perpendicularity requirement on h is essential: the slant side overstates the vertical extent, so using it inflates the area. When only a slant length and an angle are known, the height must be recovered with trigonometry.

Dimensionally, perimeter is a length (degree 1) and area a product of two lengths (degree 2). This is why scaling every length by k multiplies perimeter by k but area by k² — the same scaling result met with similar figures, and the reason area and perimeter do not determine each other. Many different rectangles share a perimeter of 20 while their areas range from near 0 up to 25.