Flashcards · J560 · Topic 10

Mensuration flashcards

28 cards on Topic 10, subtopics 10.01 to 10.05. Pick a subtopic or study them all.

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Area and volume conversions from 1 m = 100 cm: 1 m² = 100² = 10 000 cm². 1 m³ = 100³ = 1 000 000 cm³. 1 litre = 1000 cm³; 1 m³ = 1000 litres.
Convert 4 h 45 min to hours, and 1.4 h to hours and minutes: 45 ÷ 60 = 0.75, so 4.75 h. 0.4 × 60 = 24, so 1 h 24 min.
Convert 8 m/s to km/h: × 3.6: 8 × 3.6 = 28.8 km/h. (m/s to km/h is × 3.6; km/h to m/s is ÷ 3.6.)
Density and pressure formulae and unit links: Density = mass ÷ volume; pressure = force ÷ area. 1 g/cm³ = 1000 kg/m³. Pressure in N/m² and N/cm² differ by a factor of 10 000.
A map has scale 1 : 2000. What real length is 8 cm on the map?: 8 × 2000 = 16 000 cm = 160 m.
Bearings and back bearings: Measured from north, clockwise, with three figures (such as 047°). Back bearing: add or subtract 180°. A bearing of 305° from X to Y gives 125° from Y to X.

Circumference of a circle: C = πd = 2πr
Arc length for angle θ and radius r: Arc length = (θ/360) × 2πr
Perimeter of a sector and of a semicircle: Sector: arc length + 2r. Semicircle: πr + 2r.
Perimeter of a rectangle with a corner notch cut out: The same as the outer rectangle: the two inner edges of the notch replace the two missing outer lengths.

Area of a triangle, parallelogram and trapezium: Triangle: ½ × base × perpendicular height. Parallelogram: base × perpendicular height. Trapezium: ½ × (a + b) × h, where a and b are the parallel sides.
Sector area, and the area of a sector of radius 5 cm and angle 288°: Sector area = (θ/360) × πr². 288/360 = 4/5, so 4/5 × 25π = 20π ≈ 62.8 cm².
Area of a ring with outer radius 7 cm and inner radius 3 cm: π × 7² − π × 3² = 49π − 9π = 40π ≈ 126 cm². It is not π(7 − 3)².
A trapezium has area 66 cm² and parallel sides 8 cm and 14 cm. Find its height.: 66 = ½ × (8 + 14) × h = 11h, so h = 6 cm.
Area of a triangle from two sides and the included angleHigher: Area = ½ab sin C, where C is the angle between sides a and b. Example: 7 cm and 12 cm with 30°: ½ × 7 × 12 × 0.5 = 21 cm².

Volume and surface area of a prism: Volume = area of cross-section × length. Surface area = 2 × cross-section + perimeter of cross-section × length.
Cylinder of radius r and height h: Volume = πr²h. Curved surface = 2πrh; total surface area = 2πr² + 2πrh.
Cone: volume, curved surface and slant height: Volume = ⅓πr²h (perpendicular height). Curved surface = πrl (slant height). l = √(r² + h²). Example: r = 6, h = 8 gives l = 10 and curved area 60π cm².
Sphere and solid hemisphere: Sphere: V = (4/3)πr³, surface area 4πr². Solid hemisphere: V = (2/3)πr³, total surface area 3πr² (curved part 2πr²).
Volume of a pyramid: ⅓ × base area × perpendicular height. Example: 6 cm by 5 cm base, height 9 cm: ⅓ × 30 × 9 = 90 cm³.
Volume units: 1 cm³ in ml, and 1000 cm³: 1 cm³ = 1 ml; 1000 cm³ = 1 litre. Convert all lengths to one unit before multiplying.

Pythagoras: finding the hypotenuse compared with a shorter side: a² + b² = c², c the hypotenuse (opposite the right angle). Hypotenuse: add the squares, then √ (5 and 12 give 13). Shorter side: subtract the squares, then √ (10 and 6 give 8).
Right-angled trigonometry ratios: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. Unknown side on top: multiply; on the bottom: divide. For an angle use sin⁻¹, cos⁻¹ or tan⁻¹.
Exact values of sin, cos and tan for 0°, 30°, 45°, 60°, 90°: sin: 0, 1/2, √2/2, √3/2, 1 cos: 1, √3/2, √2/2, 1/2, 0 tan (0° to 60°): 0, √3/3, 1, √3
Angles of elevation and depression: Both are measured from the horizontal: elevation looking up, depression looking down.
Space diagonal of an 8 × 6 × 5 cuboidHigher: Base diagonal: √(64 + 36) = 10. Space diagonal: √(10² + 5²) = √125 ≈ 11.2. Keep exact values until the end.
Sine ruleHigher: a/sin A = b/sin B = c/sin C (or sin A/a = … for angles). Use it when you know a side and its opposite angle. For an angle, check whether 180° minus the answer also fits.
Cosine ruleHigher: a² = b² + c² − 2bc cos A, for two sides and the included angle. cos A = (b² + c² − a²)/(2bc), for an angle from three sides. A negative cosine means an obtuse angle.