Do two rectangles with the same area always have the same perimeter?

No. A 4 × 10 rectangle has area 40 and perimeter 28. A 5 × 8 rectangle also has area 40 but perimeter 26. Same area, different perimeters. Area measures the 2D inside (length × width), perimeter measures the 1D outside (the sum of the four sides). They are different quantities and they don't track each other.

If a circle has diameter 14, what is its circumference?

Circumference = π × diameter = 14π. If you use 14 as the radius by mistake, you get 28π — which is what AQA flagged as the most common wrong answer on 1F NOV20 Q26. The radius is half the diameter, and many circle formulas use the radius, which is why mixing up the two is so common.

If I double all the lengths of a cuboid, does the surface area double too?

No. Surface area is a 2D measure. Doubling all lengths quadruples the surface area (×2²) and multiplies the volume by 8 (×2³). The scale factor for area is the square of the linear scale factor, and the scale factor for volume is the cube. This is the same dimension rule that says perimeter scales linearly, area scales quadratically, volume scales cubically.

GCSE Maths Foundation · AQA · Geometry

Area-perimeter conflation: same perimeter doesn't mean same area

Foundation geometry consistently exposes a single underlying confusion: students treat perimeter, area and volume as if they obey the same rules. AQA examiner reports flag the pattern every series — 3F NOV24 Q21 noted students who "had difficulty with the difference between area and perimeter and worked out the area of both rectangles as 40 cm² and answered 1 : 1"; 1F JUN22 Q16 saw students computing the area of a triangle as $\text{base} \times \text{height} = 192$ , dropping the $\tfrac{1}{2}$ ; 1F NOV20 Q26 had students using 21 as a radius when it was the diameter.

The thirty-second fix is to read the formula by its dimensions. Perimeter is a 1-D total (add lengths); area is a 2-D product (multiply two dimensions); volume is 3-D (multiply three). Different dimensions, different rules. If you can name the dimension first, the formula and the scaling rule both fall out of it.

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How to spot it in your own work

You said two rectangles must have the same area because they have the same perimeter (or vice versa).
You computed the area of a triangle as $\text{base} \times \text{height}$ with no $\dfrac{1}{2}$ .
You used the diameter where the formula needed the radius (e.g. $\pi r^2$ with $r = 21$ when 21 was the diameter).
You doubled a cuboid's side lengths and said the surface area also doubled (it quadruples).

An exam question that triggers it

Here is a question very close to AQA 3F NOV24 Q21, which the examiner flagged as triggering exactly this conflation:

Rectangle A measures 8 cm by 5 cm. Rectangle B measures 10 cm by 4 cm.

(a) Find the area of each rectangle. (b) Find the perimeter of each rectangle. (c) Do equal areas mean equal perimeters?

The conflation answer is to compute one quantity, notice that the answers match, and conclude that the other quantity must match too. Both rectangles have area $40 \text{ cm}^2$ ( $8 \times 5 = 40$ , $10 \times 4 = 40$ ). The shortcut answer is "so the perimeters are also equal". They are not. Rectangle A has perimeter $2(8 + 5) = 26$ cm. Rectangle B has perimeter $2(10 + 4) = 28$ cm.

The correct conclusion is that area and perimeter measure different things — one a 2-D product, the other a 1-D total. Two rectangles can match on one and differ on the other.

Why students fall for this

Area, perimeter and volume all live on the same shape, so the brain treats them as "facts about the shape" rather than as measurements in different dimensions. Once they're facts about the shape, they feel like they should covary — bigger looks like bigger. The mental model is one-dimensional even though the maths is multi-dimensional. The same root produces "double the sides, double the area" (1F JUN22 Q19, surface area) — the linear scale factor is applied as if it were also the area scale factor.

The circle formulas amplify the same confusion. The student is given a picture with a single labelled length — and that length might be the radius or the diameter. The formulas $\pi r^2$ and $2\pi r$ both reach for the radius, but $\pi d$ reaches for the diameter. If the student doesn't pause to identify which length is labelled, the wrong number ends up in the formula and the answer is off by a factor of 2 or 4. AQA explicitly flagged this on 1F NOV20 Q26: "most common answer given for part (b) was 21π, coming from an attempt to find the circumference of the larger circle but using the radius instead of the diameter."

The fix — Dimensions tell you the rule

Perimeter is a 1-D total (add lengths); area is a 2-D product (multiply dimensions); volume is 3-D (multiply three). Different dimensions, different rules. Before you compute anything, name the dimension of the quantity you're asked to find. Once you've named it, the formula and the scaling rule fall out: 1-D quantities add, 2-D quantities multiply two lengths and have units of (length)², 3-D quantities multiply three lengths and have units of (length)³.

The dimension also fixes the scaling rule. Scale every length by a factor of $k$ : perimeter scales by $k$ , area by $k^2$ , volume by $k^3$ . Double the sides of a cuboid: surface area goes up by $2^2 = 4$ , volume by $2^3 = 8$ . The same one rule generalises across rectangles, circles, prisms, hemispheres — anywhere a length-vs-area-vs-volume question appears.

Worked example

Compare the two rectangles from the trigger question: A is $8 \times 5$ , B is $10 \times 4$ .

Name the dimensions. Area is 2-D, units cm². Perimeter is 1-D, units cm. Write the units down before computing — it stops you mixing the two.
Compute areas (2-D product). $\text{Area}_A = 8 \times 5 = 40 \text{ cm}^2$ . $\text{Area}_B = 10 \times 4 = 40 \text{ cm}^2$ . Equal areas.
Compute perimeters (1-D total).
$\text{Perimeter}_A = 2(8 + 5) = 26 \text{ cm}$
$\text{Perimeter}_B = 2(10 + 4) = 28 \text{ cm}$
Different perimeters.
Conclude. Same area, different perimeter. Equal areas do not force equal perimeters, because area and perimeter are different measurements. The units alone make this obvious: cm² ≠ cm. You can't transfer a result from one to the other any more than you can transfer a result from kilograms to seconds.

The same logic settles the cuboid-scaling question. Double a 2 × 3 × 4 cuboid to 4 × 6 × 8: volume goes from 24 to 192, a factor of $8 = 2^3$ , not 2. Surface area goes from 52 to 208, a factor of $4 = 2^2$ , not 2. The exponent on the scale factor matches the dimension of the quantity.

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Common questions

What's the difference between radius and diameter?: The diameter is the full distance across a circle through the centre. The radius is half of that — from the centre to the edge. Circumference uses either one ( $\pi d$ or $2\pi r$ ), but area uses the radius only ( $\pi r^2$ ). On a diagram labelled with one length, always check whether it's the radius or the diameter before plugging it in.
What's the area formula for a triangle?: $\text{Area} = \dfrac{1}{2} \times \text{base} \times \text{height}$ . The half is critical — without it, you've computed the area of the parallelogram (or rectangle) the triangle fits inside, which is twice as big. AQA flagged this on 1F JUN22 Q16 where many students computed base × height = 192 and lost the half.
If a sphere has radius 6, what's the volume of a hemisphere with radius 6?: Sphere volume: $\dfrac{4}{3}\pi r^3 = \dfrac{4}{3}\pi (6)^3 = 288\pi$ . Hemisphere is half of that: $144\pi$ . The trap is using diameter instead of radius — if the diagram shows diameter 12, you must halve it to $r = 6$ before substituting into $r^3$ . AQA flagged this on 3F NOV24 Q26 where students used the diameter as the radius and then divided by 2, ending up off by a large factor.
If I double the side lengths of a shape, do all measurements double?: No. Only the 1-D measurements double. 2-D measurements (area, surface area) go up by $2^2 = 4$ . 3-D measurements (volume) go up by $2^3 = 8$ . The exponent on the scale factor matches the dimension of the quantity. This is why "double the sides of a cuboid, double the surface area" — the 1F JUN22 Q19 wrong answer — fails: it treats surface area as a 1-D quantity when it is 2-D.

Related misconceptions

Decimal place value— Decimal arithmetic in area and volume — a misplaced point shifts the answer by a factor of ten.
Percentage change family— Scaling by percentage feeds geometry: 'increase the area by 20%' uses the multiplier method.
Ratio: additive vs multiplicative scaling— Similar shapes scale ratios of corresponding sides — the multiplicative scaling rule applies.

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