Length and area

Converting units

Length converts in steps of 10, 100 and 1000: mm $\to$ cm $\to$ m $\to$ km. So $4500$ cm $= 45$ m.

Area units square the length factor: $1$ m$^2 = 100 \times 100 = 10000$ cm$^2$, so $0.37$ m$^2 = 3700$ cm$^2$ and $700$ mm$^2 = 7$ cm$^2$. For volume, $1$ m$^3 = 1000$ L.

Perimeter

Perimeter is the total distance around the outside. On composite (L-shaped or step-shaped) figures, find the missing sides first using the sides opposite them, and check every edge has been counted once. Algebraic shapes work the same way — an answer “in terms of $a$ and $b$” is just the sum of the sides, simplified.

Area formulas

• Rectangle: $A = lw$
• Triangle: $A = \frac{1}{2}bh$
• Parallelogram: $A = bh$
• Trapezium: $A = \frac{1}{2}(a + b)h$

In every formula $h$ is the perpendicular height — the height at right angles to the base, not a slanted side. Exams sometimes ask you to explain a formula: a parallelogram becomes a rectangle when you cut a triangle off one end and slide it to the other, which is why $A = bh$.

Circles and exact answers

Circumference is $C = \pi d = 2\pi r$; area is $A = \pi r^2$. When a question says “exact value” or “in terms of $\pi$”, leave $\pi$ in the answer rather than multiplying it out: a semicircle of diameter 10 has perimeter $5\pi + 10$ (half the circumference plus the straight edge).

A sector is a fraction of a circle: multiply the full circumference or area by $\frac{\text{angle}}{360}$.

Composite figures

Split awkward shapes into rectangles, triangles and semicircles, find each area, then add. For shaded regions, subtract: whole shape minus the unshaded parts. Write down which pieces you are using — the working carries most of the marks, and a labelled split makes arithmetic slips easy to find.