SAT® · Math: Geometry and Trigonometry
Area and Perimeter of Plane Figures
Perimeter adds up the outside edges of a shape; area measures the space it covers. They depend on a figure's dimensions in different ways, so changing one does not change the other by the same factor.
Perimeter is a sum of lengths — walk around the outside and add up every side. Area is a product, built from pairs of perpendicular lengths: base times height for a rectangle, half of base times height for a triangle. Because one is addition and the other is multiplication, a shape can keep the same perimeter while its area swings wildly. A long skinny rectangle and a near-square rectangle can share a perimeter, but the square-ish one packs in more area, since for a fixed sum, two numbers multiply to the largest product when they are close to equal.
Any polygon, no matter how irregular, breaks into rectangles and triangles. Drop in some horizontal and vertical lines, or connect vertices, and you get pieces whose area formulas you already know. Add the pieces and you have the whole figure's area. This is why rectangles and triangles anchor the whole area-and-perimeter domain: every other plane figure gets reduced back to them.
These area and perimeter problems usually dress up as algebra: you are given a perimeter and a relationship between length and width, or a triangle's sides in a ratio like 3x, 4x, 5x, and you solve for the unknown before plugging into an area formula. That overlap with linear and quadratic equations is deliberate — the geometry sets up the equation, and solving it is where the real work happens. This topic sits right next to circles and volume in the same domain, and the same instinct — break a shape into known pieces, then use algebra to pin down the missing lengths — carries over directly.
Key terms
- Perimeter
- The total distance around the outside edges of a two-dimensional figure.
- Area
- The amount of flat space enclosed by a figure, measured in square units.
- Decomposition
- Splitting a complex polygon into simpler rectangles and triangles to compute its total area.
- Pythagorean triple
- A set of three whole numbers, like 3-4-5, that satisfy a2 + b2 = c2 and form a right triangle.
Watch it explained
- How it worksHow does chopping any polygon into rectangles and triangles give you its area?
- Worked exampleFind the area of a rectangle whose perimeter is 34 and whose length is 4 more than its width.
- Worked exampleFind the perimeter of a triangle with sides 3x, 4x and 5x whose area is 54.
- Common mistakeWhy doesn't doubling a rectangle's perimeter double its area?
Check yourself
End of topic.