Measurement

Perimeter and area are independent. For example a $1\times 5$ rectangle and a $2\times 4$ rectangle both have perimeter 12, but their areas are 5 and 8. (A square gives the largest area for a fixed perimeter.)

Perimeter $=2\times(8+5)=2\times 13=26$ cm. (40 cm² would be the area, not the perimeter.)

Area is length × width. Doubling both multiplies the area by $2\times 2 = 4$. (The perimeter would only double, since it is a length.)

Add all the side lengths: $7+4+3+5+6=25$ cm. For an irregular polygon you simply add every side, there is no shortcut formula.

Add every side around the outside: $10+5+4+3+6+8=36$ cm. Do not skip the two short inset edges, they are part of the boundary too.

$30=2\times(8+w)$, so $\dfrac{30}{2}=8+w$, giving $15=8+w$ and $w=7$ cm.

Fencing goes around the outside, which is the perimeter: $2\times(12+8)=2\times 20=40$ m. (96 m² would be the area.)

Area units convert by the square of the length factor. Since 1 m = 100 cm, $1\text{ m}^2 = 100\times 100 = 10\,000$ cm². So $\dfrac{50\,000}{10\,000}=5$ m². (Using 100 instead of 10,000 is the most common mistake.)

Area $=8\times 5 = 40$ cm². (26 cm would be the perimeter.)

Area of a triangle $=\dfrac{1}{2}\times 10\times 6 = 30$ cm². (Forgetting the half gives 60, the most common error.)

Area of a parallelogram $=12\times 4 = 48$ cm². (The half factor is only for triangles, not parallelograms.)

Area of a trapezium $=\dfrac{1}{2}\times(10+6)\times 4 = \dfrac{1}{2}\times 16\times 4 = 32$ cm². (Forgetting the half gives 64.)

Find the area of the large outer rectangle ($9\times 8 = 72$ cm²) and subtract the cut-out corner ($3\times 4 = 12$ cm²): $72-12=60$ cm². (Adding instead of subtracting gives 84.)

Area = length × width, so $48 = 8\times w$ and $w=\dfrac{48}{8}=6$ cm.

Floor area $=5\times 4 = 20$ m². Then $20\times 35 = 700$, so the carpet costs $700. (Using the perimeter is the most common slip; carpet covers area, not perimeter.)

Volume of a cube $=4\times 4\times 4 = 64$ cm³. (16 is one face; 96 is the surface area.)

Volume of a rectangular prism $=5\times 3\times 2 = 30$ cm³. (10 would be just adding the dimensions; 62 is the surface area.)

A prism has 6 faces in 3 matching pairs: $2\times(6\times 4 + 6\times 2 + 4\times 2) = 2\times(24+12+8) = 2\times 44 = 88$ cm². (48 cm³ would be the volume.)

$120 = 5\times 4\times h = 20h$, so $h=\dfrac{120}{20}=6$ cm.

Volume $=60\times 30\times 40 = 72\,000$ cm³. Convert using 1 L = 1000 cm³: $\dfrac{72\,000}{1000}=72$ L. (Forgetting to convert leaves 72,000 L, far too large for a fish tank.)

Find the L-shaped cross-section area first: $6\times 5 - 4\times 3 = 30-12 = 18$ cm². Then multiply by the depth: $18\times 4 = 72$ cm³. (18 cm² is the 2D area; multiply by depth to make it a volume.)

Outer area (pool plus path): adding 1 m on each side gives $(8+2)\times(4+2)=10\times 6 = 60$ m². Subtract the pool area $8\times 4 = 32$ m². Path $=60-32=28$ m².

Work in the same units. Floor area $=6\times 4 = 24$ m², which is $240\,000$ cm². Tile area $=25\times 25 = 625$ cm². Number of tiles $=\dfrac{240\,000}{625}=384$.

Original volume $=3^3 = 27$ cm³, new volume $=6^3 = 216$ cm³. Factor $=\dfrac{216}{27}=8$. Doubling every length multiplies the volume by $2^3 = 8$ (area would be multiplied by $2^2=4$).

Full volume $=10\times 4\times 1.5 = 60$ m³ $= 60\,000$ L. The pool is half full, so the top-up needed is $\dfrac{60\,000}{2}=30\,000$ L. Time $=\dfrac{30\,000}{200}=150$ minutes. (300 minutes is the time to fill from empty.)