Maths SolverThe area and perimeter formulas for the five shapes that come up at school are short, well known, and surprisingly easy to apply wrongly. Most of the wrong answers I see do not come from forgetting a formula — they come from feeding the right formula the wrong number, usually because the diagram had a length labelled that wasn’t actually the “height” the formula wanted.
This guide does two things. It states the five formulas alongside the single-line geometric reason each one works, and it spells out what exactly each variable means — which is where most of the trouble is.
The five formulas
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | $A = w h$ | $P = 2(w + h)$ |
| Triangle | $A = \tfrac{1}{2} b h$ | $P = a + b + c$ |
| Parallelogram | $A = b h$ | $P = 2(a + b)$ |
| Trapezium | $A = \tfrac{1}{2}(a + b)\,h$ | $P = a + b + c + d$ |
| Circle | $A = \pi r^2$ | $C = 2\pi r$ |
Where the “height” goes wrong
The triangle formula $A = \tfrac{1}{2} b h$ asks for the perpendicular height — the distance from the chosen base to the opposite vertex, measured at right angles to the base. It is not the length of the slanted side. The same issue appears in the parallelogram and trapezium formulas: $h$ is the perpendicular distance between the two parallel sides, never the slant.
A worked example. A triangle has base $10$ and the slant side from one end of the base measures $8$, making a $30^\circ$ angle with the base. The perpendicular height is not $8$; it is $8 \sin 30^\circ = 4$. So the area is $\tfrac{1}{2} \cdot 10 \cdot 4 = 20$, not $\tfrac{1}{2} \cdot 10 \cdot 8 = 40$. The most common version of this mistake on a test is to use the slant length without checking whether it is perpendicular.
The circle: radius vs diameter
The two circle formulas use the radius. If the question gives you the diameter, halve it before plugging in. If it gives you the circumference and asks for the area, compute the radius first ($r = C / 2\pi$) and then the area. Skipping the conversion step is the single most common circle mistake.
Use $\pi \approx 3.14159$ if a calculator is unavailable; for exam work that asks for an exact answer, leave $\pi$ in the expression. A circle of radius $7$ has area $49\pi$, not $153.94$ — both are correct but only one is the form expected when the question says “in terms of $\pi$.”
Units — the silent error source
Length has units of, say, metres. Area has units of metres-squared. Perimeter is back to metres. Mixing them is a quiet, common mistake on problems that combine perimeter and area in the same calculation.
A specific trap: doubling the side of a square doubles the perimeter but quadruples the area. Tripling triples the perimeter and multiplies the area by nine. In general, scaling lengths by a factor $k$ scales areas by $k^2$. This shows up constantly in scale-drawing and similarity problems.
The mistakes I see most often
1. Slant for height
Already covered above; it bears repeating because it is everywhere.
2. Diameter for radius (or vice versa)
$A = \pi d^2$ is wrong; the formula uses radius. If the question gives diameter, the area is $A = \pi (d/2)^2 = \pi d^2 / 4$.
3. Forgetting to take the square root for a side from area
If a square has area $36$, the side is $6$, not $36$. The same applies when working backwards from area to length on any shape.
4. Computing perimeter when the question wanted area
Read the question twice. Perimeter is a length; area is a length squared. If your final number has units of square metres but the question asked for the boundary length in metres, you answered the wrong question.
Where this comes up later
Volumes of prisms and cylinders are $V = \text{base area} \times h$, which means almost every solid-geometry problem starts with one of the area formulas above. Coordinate geometry uses the Pythagorean theorem to compute distances and these formulas to compute regions bounded by curves. In calculus, the “area under a curve” literally generalises area-of-a-region to non-polygonal shapes — see the definite integral guide.
References
- Wikipedia: Area, Perimeter, Circle.
- Khan Academy: Area and perimeter.