Perimeter, Area and Volume

Memorise these formulas.

Shape	Perimeter	Area
Rectangle	$2(l + w)$	$lw$
Square (side $s$)	$4s$	$s^2$
Triangle	sum of all sides	$\tfrac{1}{2} bh$
Parallelogram	$2(a + b)$	$bh$
Trapezium	sum of all sides	$\tfrac{1}{2}(a + b)h$
Circle (radius $r$)	$2\pi r$ (circumference)	$\pi r^2$

Worked example. A trapezium has parallel sides of 6 and 10 cm, and a height (perpendicular distance) of 4 cm. Find its area.

• Area = $\tfrac{1}{2}(6 + 10) \times 4 = \tfrac{1}{2} \times 16 \times 4 = 32\,\text{cm}^2$.

Worked example. Find the circumference and area of a circle with radius 5 cm.

• Circumference: $2\pi \times 5 = 10\pi \approx 31.4$ cm.
• Area: $\pi \times 5^2 = 25\pi \approx 78.5\,\text{cm}^2$.

Arc length and sector area (Extended) — for a sector with central angle $\theta°$:

• Arc length: $\ell = \dfrac{\theta}{360} \times 2\pi r$.
• Sector area: $A = \dfrac{\theta}{360} \times \pi r^2$.

(In radians: $\ell = r\theta$ and $A = \tfrac{1}{2} r^2 \theta$.)

Standard volume formulas.

Solid	Volume	Surface Area
Cuboid	$lwh$	$2(lw + lh + wh)$
Cube (side $s$)	$s^3$	$6s^2$
Cylinder (radius $r$, height $h$)	$\pi r^2 h$	$2\pi r^2 + 2\pi rh$
Cone (radius $r$, height $h$)	$\tfrac{1}{3}\pi r^2 h$	$\pi r^2 + \pi r l$ (l = slant)
Sphere (radius $r$)	$\tfrac{4}{3}\pi r^3$	$4\pi r^2$
Square Pyramid (base side $a$, height $h$)	$\tfrac{1}{3}a^2 h$	$a^2 + 2al$
Prism (general)	base area × length	$2 \times \text{base} + \text{lateral}$

Worked example. Volume of a cone with $r = 3$ cm and $h = 8$ cm.

• $V = \tfrac{1}{3}\pi r^2 h = \tfrac{1}{3} \times \pi \times 9 \times 8 = 24\pi \approx 75.4\,\text{cm}^3$.

Worked example. Surface area of a sphere with $r = 5$ cm.

• $SA = 4\pi r^2 = 100\pi \approx 314\,\text{cm}^2$.

Worked example. A cylinder has radius 4 cm and height 10 cm. Find its volume and total surface area.

• Volume: $\pi \times 16 \times 10 = 160\pi \approx 503\,\text{cm}^3$.
• Surface: top + bottom = $2 \times \pi \times 16 = 32\pi$. Side: $2\pi r h = 80\pi$. Total: $112\pi \approx 352\,\text{cm}^2$.

Real-world tips:

• Check units. Areas in $\text{cm}^2$, volumes in $\text{cm}^3$. Mixed units cause errors.
• Convert before computing. If a question gives length in m and width in cm, convert to a common unit first.
• Draw the shape if not given. A sketch labels lengths and prevents confusion.

Worked example (criterion D). A cylindrical water tank has radius 1.5 m and height 3 m. (a) How much water (litres) does it hold? (b) What is the outside surface area in $\text{m}^2$?

(a) Volume = $\pi r^2 h = \pi \times 2.25 \times 3 = 6.75\pi \approx 21.21\,\text{m}^3$.

• Convert to litres: $1\,\text{m}^3 = 1000\,\text{L}$, so $\approx 21\,200$ litres.

(b) Surface area = $2\pi r^2 + 2\pi r h = 2\pi (2.25) + 2\pi (1.5)(3) = 4.5\pi + 9\pi = 13.5\pi \approx 42.4\,\text{m}^2$.

Worked example (criterion D). A sphere of radius 5 cm is melted and recast into a cube. Find the side of the cube.

• Volume of sphere: $\tfrac{4}{3}\pi \times 125 = \tfrac{500\pi}{3} \approx 523.6\,\text{cm}^3$.
• Volume conservation: cube volume = 523.6, so side = $\sqrt[3]{523.6} \approx 8.06$ cm.