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Math101learn.math101.caArea
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.
Composite figures
Break an irregular region into familiar non-overlapping shapes, find each area, and add. For a cut-out, find the outer area and subtract the missing region.
Different decompositions should produce the same total. Sketch divider lines and label which regions are added or removed before calculating.
Circles and sectors
A circle’s area is $\pi r^2$. If diameter is given, divide by $2$ before squaring. A sector occupying angle $\theta$ degrees has fraction $\theta/360$ of the full area:
Keep $\pi$ for an exact answer unless a decimal is requested.
Scale factor
If all lengths are multiplied by $k$, area is multiplied by $k^2$. Doubling length and width creates four times as many square units. This explains why a map or image enlargement changes area faster than perimeter.
Context and estimation
Real projects may require extra material for cutting, overlap, or waste. First calculate the geometric area; then apply a stated allowance. Estimate dimensions before multiplying so a misplaced decimal is easier to detect.
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?
Related topics
Explore the idea
Geometry measurement
Change one quantity at a time and connect what moves to Area.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
A trapezoid has parallel bases 9 cm and 15 cm and height 7 cm. Find its area in square centimetres.
- A = (1/2)(b₁ + b₂)h
- = (1/2)(9 + 15)(7)
- = 84 cm²
End of lesson
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