← Reference

Area, volume, circles and Pythagoras: cheat sheet

Area, volume, circles and Pythagoras · The words and methods from the lessons, with an example of each.

Area

perpendicular height

The distance from the base to the opposite side or corner, at a right angle (∟) to the base. A slanted side is longer, and is not the height.

h at 90∘ to bh \text{ at } 90^\circ \text{ to } b

area of a triangle

Two copies make a parallelogram with the same base and height; the triangle is half.

A=12×b×hA = \tfrac{1}{2} \times b \times h
A triangle and its turned copy make a parallelogram
A triangle and its turned copy make a parallelogram

square units

Area counts squares: cm², m². A 1 m square is 100 cm by 100 cm.

1 m2=10 000 cm21 \text{ m}^2 = 10\,000 \text{ cm}^2

composite area

Cut the shape into pieces you know and add their areas, or take the missing pieces away from a big rectangle. Each piece uses its own lengths.

A=8×6−5×3=33 m2A = 8 \times 6 - 5 \times 3 = 33\ \text{m}^2
An L-shape with its side lengths
An L-shape with its side lengths

area of a parallelogram

Cut the triangle off one end and move it: a rectangle with the same base and height.

A=b×hA = b \times h
A parallelogram cut and rearranged into a 6 by 4 rectangle
A parallelogram cut and rearranged into a 6 by 4 rectangle

composite area

Split the shape into rectangles and triangles and add the parts, or subtract a missing part from a bigger shape. Put every length in the same unit first.

24+9=33 m224 + 9 = 33 \text{ m}^2
The board

estimating an irregular area

Count the full squares, add half the part squares, multiply by the area of one square. A finer grid gives a closer estimate.

A≈(full+12×part)×one squareA \approx (\text{full} + \tfrac{1}{2} \times \text{part}) \times \text{one square}

Volume

volume of a prism

Each layer is the base (the cross-section); the height counts the layers. For a box, V=l×w×hV = l \times w \times h.

V=A×hV = A \times h
A 4 by 3 by 5 box in layers of 12 cubes
A 4 by 3 by 5 box in layers of 12 cubes

volume and capacity

A cubic metre is 100 × 100 × 100 = 1 000 000 cm³: convert each of the three lengths.

1 cm3=1 mL,1000 cm3=1 L,1 m3=1000 L1\ \text{cm}^3 = 1\ \text{mL}, \quad 1000\ \text{cm}^3 = 1\ \text{L}, \quad 1\ \text{m}^3 = 1000\ \text{L}

cross-section

The shape that is the same all the way along a prism, like a slice of bread. A triangular prism's cross-section is a triangle: halve first.

V=12×6×4×10=120 cm3V = \tfrac{1}{2} \times 6 \times 4 \times 10 = 120 \text{ cm}^3
A triangular prism, half of a 4 by 3 box, 5 layers high
A triangular prism, half of a 4 by 3 box, 5 layers high

capacity

How much a container holds. A 10 cm cube holds 1 litre.

1 cm3=1 mL1000 cm3=1 L1 \text{ cm}^3 = 1 \text{ mL} \qquad 1000 \text{ cm}^3 = 1 \text{ L}

volume of a prism

The area of the end times the length, for any right prism. Divide to find a missing length.

V=Ahh=V÷AV = A h \qquad h = V \div A
The board

Circles

radius and diameter

The radius goes from the centre to the edge; the diameter right across through the centre. A chord joins any two points; the diameter is the longest.

d=2rr=d÷2d = 2r \qquad r = d \div 2
A circle with its centre, radius, diameter, a chord and the circumference
A circle with its centre, radius, diameter, a chord and the circumference

area of a circle

Slices of a circle make a near-rectangle πr long and r high. The area is between 2r² and 4r². Do not mix it up with C = 2πr.

A=πr2A = \pi r^2
24 slices laid top to tail: nearly a rectangle πr by r
24 slices laid top to tail: nearly a rectangle πr by r

pi

The circumference divided by the diameter: the same for every circle. 3.14 and 22/7 are approximations.

π=Cd=3.14159…\pi = \frac{C}{d} = 3.14159\ldots
A circle rolled out flat: just over 3 diameters
A circle rolled out flat: just over 3 diameters

circumference

Use the diameter, or double the radius. Backwards: d=C÷πd = C \div \pi. Exact answers keep π: 10π10\pi cm.

C=πd=2πrC = \pi d = 2\pi r

radius from area

Divide by π, then take the square root. A semicircle is half the area; its perimeter is half the circumference plus the diameter.

r=A÷πr = \sqrt{A \div \pi}

Pythagoras

Pythagoras' theorem

In a right-angled triangle, the squares on the two shorter sides add to the square on the hypotenuse c, the side opposite the right angle.

a2+b2=c2a^2 + b^2 = c^2
Squares of 9, 16 and 25 on a 3-4-5 triangle
Squares of 9, 16 and 25 on a 3-4-5 triangle

finding a side

Add the squares for the hypotenuse; subtract for a shorter side. Check: the hypotenuse is the longest.

c=a2+b2b=c2−a2c = \sqrt{a^2 + b^2} \qquad b = \sqrt{c^2 - a^2}

right, acute or obtuse

With c the longest side: equal, right-angled; a² + b² smaller, obtuse; bigger, acute. Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 and multiples.

a2+b2  vs  c2a^2 + b^2 \;\text{vs}\; c^2

Time zones

time zones

Find the difference between the UTC offsets, add or take it away, and change the day if you cross midnight. For a flight, add the flight time in the departure zone, then change zones.

bigger offset: aheadsmaller offset: behind\text{bigger offset: ahead} \qquad \text{smaller offset: behind}
9:00 am Monday in Sydney, and the same moment in Perth, London and Tokyo
9:00 am Monday in Sydney, and the same moment in Perth, London and Tokyo