LESSON 17
AREA & PERIMETER
Measuring the Outside and Inside of Shapes
Two Ways to Measure a Shape
Perimeter is the total distance around the outside of a shape — the sum of all its side lengths. Area is the amount of flat space inside a shape, measured in square units. These two measurements answer different questions: perimeter tells you how far around, area tells you how much surface.
Both measurements are essential in everyday life. Perimeter is used when fencing a yard, framing a picture, or running a track. Area is used when carpeting a floor, painting a wall, or planting a garden. Knowing which one to use — and how to calculate it — is a fundamental life skill.
Rectangles and Squares
For a rectangle with length l and width w: the perimeter is P = 2l + 2w (add all four sides) and the area is A = l × w (length times width). A square is a special rectangle where all sides are equal, so P = 4s and A = s². These are the most commonly used formulas in geometry.
Triangles
The perimeter of a triangle is the sum of all three sides: P = a + b + c. The area of a triangle is half the base times the height: A = ½ × b × h. The height must be perpendicular (at a right angle) to the base — it is not always a side of the triangle.
Circles
A circle's perimeter is called its circumference. Both formulas use π (pi ≈ 3.14159). The circumference is C = 2πr or C = πd, where r is the radius and d is the diameter. The area is A = πr². The radius is the distance from the center to the edge; the diameter is twice the radius.
Composite Figures
A composite figure is made up of two or more basic shapes. To find the area, break the figure into recognizable shapes, calculate each area separately, then add (or subtract) them. To find the perimeter, trace the outer boundary and add only the sides that form the outside edge.
Key Facts
Pi (π) is irrational — its decimal expansion never ends or repeats. It has been calculated to over 100 trillion digits, but 3.14 or 22/7 is sufficient for most problems.
A shape can have a large perimeter and a small area, or a small perimeter and a large area. A circle encloses the maximum area for a given perimeter — this is called the isoperimetric inequality.
Area is always measured in square units (cm², m², ft²) because it represents a two-dimensional space. Perimeter is measured in linear units (cm, m, ft).
Ancient Egyptians used area calculations to re-survey farmland after the Nile flooded each year. This practical need was one of the earliest drivers of geometry.