Area of a Trapezium and a Polygon
191. The Rectangle's Rhythm · how length and breadth dictate area and perimeter
Area always equals l × b, and perimeter always equals 2(l + b).
Rectangle formulas: Area = l × b (length × breadth) and Perimeter = 2(l + b). Both update live as you change either dimension.
What this lesson covers
Try to break it
Drag L to change the length and B to change the breadth. Area = l × b, perimeter = 2(l + b), diagonal = √(l² + b²) — all three readouts update together. Make the rectangle long and thin: the area drops sharply but the perimeter can stay similar. Try to break any of the three formulas; impossible.
How you build it
Construct a rectangle.
- Draw a horizontal line segment of length l.
- At one endpoint, construct a perpendicular of length b.
- Through the top endpoint, draw a line parallel to the base.
- Close the rectangle by drawing a line parallel to the perpendicular side.
The proof, step by step
Prove that the area of a rectangle is l × b and its perimeter is 2(l + b).
- Divide the rectangle into a grid of 1×1 unit squares.
- Count how many squares fit along the length l.
- Count how many squares fit along the breadth b.
- Total squares = l × b. Therefore, Area = l × b.
- Perimeter is the total boundary length: l + b + l + b = 2(l + b).
- The diagonal splits the rectangle into two right triangles. By Pythagoras, d² = l² + b².
Worked example
A rectangular garden measures 15 m in length and 8 m in breadth. What is its area and perimeter?
Area = l × b = 15 × 8 = 120 m². Perimeter = 2(l + b) = 2(15 + 8) = 46 m.
- Area = 120 m², Perimeter = 46 m — correct
- Area = 120 m², Perimeter = 23 m
- Area = 23 m², Perimeter = 120 m
- Area = 150 m², Perimeter = 46 m