Area, volume and angle facts

Triangle area is base × perpendicular height/2. A parallelogram uses base × perpendicular height; a trapezium uses half the sum of its parallel sides times their perpendicular separation. Circle area is πr² and circumference is 2πr. Match length units before calculating; an area has square units.

A prism or cylinder has volume equal to cross-sectional area times perpendicular length or height. A pyramid or cone has one third of the corresponding base-area-times-height product. A sphere has volume 4πr³/3 and surface area 4πr². Volume uses cubic units, so changing every length by factor k changes volume by k³.

For a cylinder, include two circular ends when calculating total surface area: 2πr² + 2πrh. A cone has curved surface area πrl, where l is its slant height; adding πr² includes its circular base. Slant height and perpendicular height are different lengths, often connected by Pythagoras.

Angles on a straight line total 180° and angles around a point total 360°. A triangle has angle sum 180°; a simple n-sided polygon has interior-angle sum (n − 2)180°. A regular polygon has equal interior angles. Corresponding and alternate angles are equal for parallel lines, while co-interior angles total 180°. Record the parallel-line condition when using these facts.

Worked example

A cone has radius 3 cm and perpendicular height 4 cm. Find its volume and total surface area.

1. Its slant height is √(3² + 4²) = 5 cm.
2. Volume = (1/3)π × 3² × 4 = 12π cm³.
3. Total surface area = π × 3 × 5 + π × 3².

Answer: Volume 12π cm³; total surface area 24π cm²