Mensuration flashcards

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What are the formulae for the area of a rectangle, triangle, parallelogram and trapezium?
Rectangle \(lw\); triangle \(\frac12bh\); parallelogram \(bh\); trapezium \(\frac12(a+b)h\), where \(h\) is the perpendicular height.
Find the area of a trapezium with parallel sides 6 cm and 10 cm and height 4 cm.
\(\frac12(6+10)\times4=32\) cm².
State the formulae for the circumference and area of a circle.
Circumference \(C=2\pi r=\pi d\); area \(A=\pi r^2\).
Find the area of a circle of diameter 10 cm, to 3 significant figures.
\(r=5\), so \(A=\pi\times25=78.5\) cm².
How do you find the length of an arc and the area of a sector with angle \(\theta\)?
Arc length \(=\frac{\theta}{360}\times2\pi r\); sector area \(=\frac{\theta}{360}\times\pi r^2\).
A sector has radius 9 cm and angle \(80^\circ\). Find its arc length to 3 s.f.
\(\frac{80}{360}\times2\pi\times9=12.6\) cm.
How do you find the perimeter of a sector?
Arc length plus two radii: \(P=\frac{\theta}{360}\times2\pi r+2r\). Do not forget the straight edges.
State the volume formula for any prism and for a cylinder.
Prism: \(V=\) cross-sectional area \(\times\) length. Cylinder: \(V=\pi r^2h\).
A cylinder has radius 3 cm and height 10 cm. Find its volume to 3 s.f.
\(V=\pi\times9\times10=283\) cm³.
State the curved surface area and total surface area of a cylinder.
Curved surface \(=2\pi rh\). Total \(=2\pi rh+2\pi r^2\) (curved surface plus two circular ends).
(Extended) State the volume and surface area formulae for a sphere.
\(V=\frac43\pi r^3\); surface area \(=4\pi r^2\).
(Extended) Find the volume of a sphere of radius 6 cm, to 3 s.f.
\(\frac43\pi\times216=905\) cm³.
(Extended) State the volume and curved surface area of a cone.
\(V=\frac13\pi r^2h\); curved surface area \(=\pi rl\), where \(l\) is the slant height. The base adds \(\pi r^2\).
(Extended) A cone has radius 5 cm and perpendicular height 12 cm. Find its slant height and volume.
\(l=\sqrt{5^2+12^2}=13\) cm. \(V=\frac13\pi\times25\times12=100\pi\approx314\) cm³.
(Extended) State the volume of a pyramid.
\(V=\frac13\times\) base area \(\times\) perpendicular height.
Convert 1 m² to cm² and 1 m³ to cm³. Why is it not 100 and 100?
\(1\text{ m}^2=10\,000\text{ cm}^2\) and \(1\text{ m}^3=1\,000\,000\text{ cm}^3\). Areas scale by the length factor squared (\(100^2\)), volumes by it cubed (\(100^3\)).
Convert 2.5 litres to cm³ and 3000 cm³ to litres.
\(1\text{ litre}=1000\text{ cm}^3\), so 2.5 litres = 2500 cm³, and 3000 cm³ = 3 litres.
How do you find the area of a compound shape?
Split it into rectangles, triangles and circle parts (or subtract a cut-out from a larger shape), find each area, then add or subtract.
What is the difference between perimeter, area and volume, and their units?
Perimeter is a length around a shape (cm, m). Area is the surface covered (cm², m²). Volume is the space inside a solid (cm³, m³).
Find the perimeter of a semicircle of diameter 10 cm, to 3 s.f.
Curved part \(=\frac12\pi\times10=15.7\) cm, plus the diameter 10 cm. Perimeter \(=25.7\) cm.
A cuboid measures 4 cm by 5 cm by 6 cm. Find its volume and total surface area.
Volume \(=120\) cm³. Surface area \(=2(20+24+30)=148\) cm².
How do you find the radius of a circle given its area \(A\)?
Rearrange \(A=\pi r^2\): \(r=\sqrt{\frac{A}{\pi}}\). E.g. \(A=50\) cm² gives \(r=3.99\) cm.
How do you find the height of a cylinder with volume 500 cm³ and radius 4 cm?
\(h=\frac{V}{\pi r^2}=\frac{500}{16\pi}=9.95\) cm.
How are speed, distance and time related, and what unit conversion is often needed?
\(\text{speed}=\frac{\text{distance}}{\text{time}}\). Convert units to match, e.g. \(1\text{ m/s}=3.6\text{ km/h}\).
A car travels 150 km in 2 h 30 min. Find the average speed.
Time \(=2.5\) h, so speed \(=\frac{150}{2.5}=60\) km/h.
Define density and how to find mass from it.
Density \(=\frac{\text{mass}}{\text{volume}}\), so mass \(=\) density \(\times\) volume. E.g. 20 cm³ at 2.7 g/cm³ has mass 54 g.
(Extended) A solid sphere and a cylinder have the same radius \(r\). The cylinder's height is \(2r\). Find the ratio of their volumes.
Sphere \(\frac43\pi r^3\); cylinder \(\pi r^2\times2r=2\pi r^3\). Ratio \(=\frac43:2=2:3\).
(Extended) What is a hemisphere's volume and total surface area?
\(V=\frac23\pi r^3\). Total surface area \(=2\pi r^2+\pi r^2=3\pi r^2\) (curved part plus flat circle).

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