Mensuration flashcards

Topic 5: Mensuration

Area and volume unit conversions: m² to cm², m³ to cm³

1 m² = 100² = 10 000 cm². 1 m³ = 100³ = 1 000 000 cm³. Square the length factor for area; cube it for volume.

Capacity conversions

1 cm³ = 1 ml. 1 litre = 1000 cm³. 1 m³ = 1000 litres.

Area of a trapezium

½(a + b)h, where a and b are the parallel sides and h is the perpendicular height between them.

Arc length and sector area for angle θ

Arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr². The perimeter of a sector is the arc length + 2r.

A sector has radius 8 cm and angle 135°. Find the arc length and area in terms of π.

Fraction = 135/360 = 3/8. Arc = 3/8 × 16π = 6π cm. Area = 3/8 × 64π = 24π cm².

Volume and curved surface area of a cone

Volume = ⅓πr²h (h = perpendicular height). Curved surface area = πrl (l = slant height).

A cone has base radius 5 cm and slant height 13 cm. Find its height and volume.

h = √(13² − 5²) = √144 = 12 cm. Volume = ⅓ × π × 25 × 12 = 100π cm³.

Volume and surface area of a sphere

Volume = (4/3)πr³. Surface area = 4πr².

Total surface area of a solid hemisphere of radius r

Curved surface 2πr² plus the flat circle πr² = 3πr². For r = 3 cm: 27π cm².

Volume of a prism and of a cylinder

Prism: area of cross-section × length. Cylinder: πr²h. Its curved surface area is 2πrh.

A sphere of radius 3 cm is melted and recast as a cone of base radius 3 cm. Find the cone height.

The volume stays the same. Sphere: (4/3)π × 27 = 36π. Cone: ⅓ × π × 9 × h = 3πh = 36π, so h = 12 cm.

How do you find the volume of a frustum?

A frustum is a cone (or pyramid) with the top cut off parallel to the base. Volume = volume of the whole cone − volume of the small cone removed (the two cones are similar).