Mensuration flashcards

Key points

• The circumference is the distance all the way round the circle.
• Circumference = πd, or 2πr, because the diameter is two radii.
• "In terms of π" means leave π in: a diameter of 10 cm gives exactly 10π cm.

Formula

Example

Key points

• Area = πr²: square the radius, then multiply by π.
• Given the diameter, halve it first to get the radius.
• Area is in square units, such as cm². An answer in terms of π, like 25π, is exact.

Formula

Diameter 10 cm

Key points

• A sector is a fraction of a circle, set by its angle θ: the fraction is $\frac{\theta }{360}$.
• The arc is that fraction of the circumference.
• So a 60° sector's arc is one sixth of the circumference.

Formula

Radius 6 cm, angle 60°

Key points

• Work out the fraction of the circle: angle ÷ 360.
• Multiply that fraction by the whole circle's area, πr².
• A 90° sector is a quarter of the circle; a 180° sector is a semicircle.

Formula

Radius 4 cm, angle 90°

Key points

• The perimeter is the curved arc plus the two straight radii.
• Find the arc length first, then add 2 × the radius.
• Trace it: along the arc, back to the centre, and out again.

Radius 4 cm, angle 90°

Key points

• A prism has the same cross-section all the way through.
• Volume = area of the cross-section × length.
• A cylinder is a prism with a circle for its cross-section: V = πr²h.

Formulae

Example

Key points

• It unrolls into two circles and a rectangle.
• The rectangle wraps once round, so it is 2πr wide and h high: the curved surface is 2πrh.
• A prism's surface area works the same way: add the area of every face.

Formula

r = 3 cm, h = 10 cm

Key points

• A pyramid or cone holds one third of the prism with the same base and height.
• Volume = $\frac{1}{3}$ × base area × perpendicular height.
• A cone's base is a circle, so V = $\frac{1}{3}$ πr²h, with h the vertical height.

Formulae

Example

Key points

• The curved surface is πrl, where l is the slant height.
• Add the circular base, πr², for the total surface area.
• If you only have the vertical height h, use Pythagoras: l² = r² + h².

Formulae

r = 3 cm, vertical height 4 cm

Key points

• Volume = $\frac{4}{3}$ πr³: cubic units, so r is cubed.
• Surface area = 4πr²: square units, so r is squared.
• A hemisphere is half a sphere, so its volume is $\frac{2}{3}$ πr³.

Formulae

Radius 6 cm

Key points

• A square metre is 100 cm by 100 cm, so 1 m² = 100 × 100 = 10 000 cm².
• Areas convert by the length factor squared: 1 cm² = 100 mm².
• Or convert the lengths first, then multiply.

Example

Key points

• A cubic metre is 100 cm by 100 cm by 100 cm, so 1 m³ = 100³ = 1 000 000 cm³.
• Volumes convert by the length factor cubed: 1 cm³ = 1000 mm³.
• 1 litre = 1000 cm³, so 1 m³ holds 1000 litres.

Example

Key points

• Split it into shapes you have formulae for: cuboids, prisms, cylinders, cones, hemispheres.
• Find each piece's volume, then add them. Subtract a piece cut away, such as a hole.
• Keep full accuracy, or π, until the end, then round once.

Silo: cylinder r = 2 m, h = 5 m, dome

Key points

• Only the faces on the outside count: surface area is what you could paint.
• The faces where the two solids join are hidden inside, so leave them out.
• List every outside face with its area, then add.

The same silo: dome on a cylinder