Math101 · Independent Ontario learning library
Math101Area
Area measures two-dimensional coverage and connects geometric formulas to decomposition, scale, design, construction, and optimization.
Area measures how much two-dimensional surface a region covers, expressed in square units.
Why units are squared
A square centimetre is a $1$ cm by $1$ cm tile. Covering a rectangle with rows and columns of these tiles produces length times width. A $5$ cm by $3$ cm rectangle holds $5\times3=15$ square-centimetre tiles.
Perimeter measures the boundary in linear units; area measures the interior in square units.
Core formulas
| Shape | Area |
|---|---|
| rectangle | $A=lw$ |
| parallelogram | $A=bh$ |
| triangle | $A=\tfrac12bh$ |
| trapezoid | $A=\tfrac12(b_1+b_2)h$ |
| circle | $A=\pi r^2$ |
The height $h$ is perpendicular to the chosen base. A slanted side is not automatically the height.
Why triangle area is half
Two congruent copies of a triangle can form a parallelogram with the same base and height. The triangle therefore covers half the parallelogram:
This reasoning works for acute, right, and obtuse triangles as long as height is perpendicular to the base line.
Worked example
The formula averages the two base lengths, then multiplies by height.
Common mistakes
Using a slanted side as height. Height must meet the base at $90^\circ$.
Forgetting the factor $1/2$. Triangles and trapezoids use half of a related parallelogram expression.
Using diameter in $\pi r^2$. Convert to radius first.
Reporting linear units. Area requires squared units.
Double-counting overlap. Composite pieces should cover the region exactly once.
Quick self-check
- What region is actually being covered?
- Which dimensions are perpendicular?
- Are all lengths in the same unit?
- Should pieces be added or subtracted?
- Does the result use square units?
