Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — circumference and area of a circle
A circle has radius 7 cm. Find its circumference and its area, taking π=722.
- 1Circumference =2πr=2×722×7The 7s cancel, which is exactly why this value of π was chosen.
- 2=44 cmA length, so the unit is cm.
- 3Area =πr2=722×49Square the radius first, then multiply.
- 4=22×7=154 cm2An area, so the unit is cm squared.
Example 2 — volume and surface area of a cylinder
A cylinder has radius 5 cm and height 12 cm. Find its volume and total surface area, taking π=3.14.
- 1Volume =πr2h=3.14×25×12The area of the circular face, times the height.
- 2=942 cm3Three lengths multiplied, so cm cubed.
- 3Curved surface =2πrh=2×3.14×5×12=376.8 cm2The label unrolled into a rectangle.
- 4Two circular ends: 2πr2=2×3.14×25=157 cm2Top and bottom.
- 5Total surface area =376.8+157=533.8 cm2"Total" means include the ends; "curved" means leave them out.
Example 3 — arc length and sector area
A sector of a circle of radius 9 cm has an angle of 40° at the centre. Find the arc length and the area of the sector, taking π=3.14.
- 1The fraction of the circle is 36040=91Find the fraction once and use it twice.
- 2Whole circumference =2π(9)=18πWork with π symbolically for as long as you can.
- 3Arc =91×18π=2π=6.28 cm
- 4Whole area =π(81)=81π, so sector area =91×81π=9πSame fraction, applied to the area.
- 5=28.26 cm2If the question wanted the sector's perimeter, add the two radii: 6.28+18=24.28 cm.