Area, volume, circles and Pythagoras: recap

A parallelogram is a rectangle pushed over. Cut the triangle off one end along the perpendicular height and move it to the other end: it becomes a rectangle with the same base and height. So:

$$A = b \times h$$

where $h$ is the perpendicular height, at a right angle to the base, not the slanted side. Area is in square units: cm², m².

Two copies of a triangle, one turned round, make a parallelogram with the same base and height. So the triangle's area is half of base × height:

$$A = \tfrac{1}{2} \times b \times h$$

The height is perpendicular to the base. If the top leans past the end of the base, extend the base: the height meets it outside the triangle. Moving the top along a line parallel to the base keeps the area.

Area problems have three moves:

• Choose the base and the height at a right angle to it; a slanted side is extra information.
• Convert every length to the same unit first (250 cm = 2.5 m). Area is in square units, and 1 m² = 10 000 cm².
• Split a shape into rectangles and triangles and add the parts, or subtract a missing part from a bigger shape.

A composite shape is made of shapes you know. Cut it into pieces and add their areas, or see a big rectangle and take away the missing pieces: both give the same area, if each piece uses its own lengths.

Missing sides are differences of other sides. The perimeter adds every side, in units of length.

For an irregular shape, put a grid over it: count the full squares, add half the part squares, and multiply by one square's area. A finer grid gives a better estimate.

For the perimeter, join points on the edge with straight pieces and add them: more pieces get closer (always slightly short). State how precise the estimate is.

Volume counts unit cubes (cm³, m³). A prism fills up in layers: each layer covers the base, so it has as many cubes as the base has squares. The number of layers is the height:

$$V = A \times h$$

For a box, $A = l \times w$, so $V = l \times w \times h$. The cubes fill the inside, not just the outside you can see.

A right prism has the same cross-section all the way along. Its volume is that slice's area times the length:

$$V = A \times h$$

For a triangular prism, the cross-section is a triangle: halve first ($\tfrac{1}{2} \times 6 \times 4 = 12$ cm², then $\times 10 = 120$ cm³). Capacity: 1 cm³ holds 1 mL, and 1000 cm³ = 1 L.

Volume is in cubic units; capacity is in mL, L or kL. 1 cm³ holds 1 mL, 1000 cm³ holds 1 L, and 1 m³ holds 1000 L.

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

For any right prism, $V = A \times h$: the area of the end times the length. Find the end's area with its own formula (rectangle, triangle or composite).

To find a missing length, divide: $h = V \div A$. Convert units first, and answer in the units the problem asks for (mL, L or m³).

A circle is all the points the same distance (the radius, $r$) from its centre.

• The diameter ($d$) goes right across through the centre: $d = 2r$, and $r = d \div 2$.
• A chord joins any two points; the diameter is the longest chord.
• The circumference ($C$) is the distance all the way round.

Roll a circle out flat for one turn and its circumference is a bit more than 3 diameters. For every circle, the circumference divided by the diameter is the same number, π:

$$\pi = \frac{C}{d} = 3.14159\ldots$$

π's digits never end and never settle into a repeating pattern; 3.14 and $\tfrac{22}{7}$ are approximations. A quick estimate: the circumference is a bit more than 3 diameters.

Because $C \div d = \pi$, the circumference is

$$C = \pi d = 2\pi r$$

Use the diameter (or double the radius). Going backwards, divide by π: $d = C \div \pi$. An exact answer keeps π: a circle with diameter 10 cm has $C = 10\pi$ cm, about 31.4 cm.

Cut a circle into thin slices and lay them top to tail: the shape is nearly a rectangle πr long (half the circumference) and r high. So $A = \pi r^2$.

Check: the area is between $2r^2$ and $4r^2$. Do not mix it up with the circumference $C = 2\pi r$, a length.

Choose the formula: circumference $C = 2\pi r$ (a length) or area $A = \pi r^2$ (a space). From an area, $r = \sqrt{A \div \pi}$.

Parts of circles: a semicircle is half the area; its perimeter is half the circumference plus the diameter. Say which value of π you used.

In a right-angled triangle, the hypotenuse $c$ is the side opposite the right angle (the longest). Pythagoras' theorem: the squares on the two shorter sides add to the square on the hypotenuse,

$$a^2 + b^2 = c^2$$

To find $c$: square $a$ and $b$, add, then take the square root. With 6 and 8: $36 + 64 = 100$, $c = 10$.

Pythagoras finds any side of a right-angled triangle:

• the hypotenuse: add the squares, $c^2 = a^2 + b^2$;
• a shorter side: subtract, $b^2 = c^2 - a^2$.

Then take the square root (to two decimal places if it is not whole). Check: the hypotenuse must be the longest side.

To test a triangle, call its longest side $c$ and compare $a^2 + b^2$ with $c^2$:

• equal: right-angled;
• $a^2 + b^2 < c^2$: obtuse (the angle opposite $c$ is over 90°);
• $a^2 + b^2 > c^2$: acute.

Whole numbers that fit exactly are Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, and their multiples.

A time zone is an offset from UTC. A city with a bigger offset is ahead; a smaller one is behind. Find the difference, add or take it away, and change the day when you cross midnight.

For a flight, add the flight time to the departure (in the departure zone), then change to the arrival zone. Use 24-hour time to avoid am and pm mistakes.