Geometry — IMAT Mathematics Formulas and Traps

Angles, Triangles and Polygons

Triangles: angle sum, Pythagoras and special right triangles

Pythagoras and triangle area

a^2 + b^2 = c^2 \qquad A = \tfrac{1}{2}bh

$a, b$ the two legs, which meet at the right angle
$c$ the hypotenuse
$b, h$ in the area formula: a base and the height perpendicular to it

Congruent and similar triangles

Similar triangles

\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c} = k \qquad \frac{\text{Area}'}{\text{Area}} = k^2

$a, b, c$ sides of one triangle
$a', b', c'$ the matching sides of the other
$k$ scale factor

Polygons: interior and exterior angles

Polygon angles

S_{\text{int}} = (n - 2) \times 180^\circ \qquad \text{regular: each exterior} = \frac{360^\circ}{n}

$n$ number of sides

Angle facts for crossing lines and parallel lines

Situation	Fact
Angles on a straight line	Add up to $180^\circ$
Angles around a point	Add up to $360^\circ$
Vertically opposite angles, where two lines cross	Are equal
Corresponding angles on parallel lines	Are equal
Alternate angles on parallel lines	Are equal
Co-interior angles on parallel lines	Add up to $180^\circ$ Co-interior angles are supplementary, not equal.

Common traps

Only parallel lines give equal alternate and corresponding angles

The equal-angle facts need the two lines to be parallel. If a diagram does not say so (arrows on the lines or words in the question), you cannot assume it, even if the lines look parallel.

The hypotenuse is the side opposite the right angle

In

a^2 + b^2 = c^2

,

c

must be the longest side, facing the right angle. Adding the squares of a leg and the hypotenuse gives nonsense; when the hypotenuse is known, subtract:

a^2 = c^2 - b^2

.

Match the sides through the angles, not through their position on the page

In the cut-off triangle, AD matches AB (the whole side), not DB. Use

AD/AB

, not

AD/DB

, for the scale factor; the second ratio gives a wrong answer that is usually among the options.

Exterior angles total 360° for every polygon

The interior sum grows with the number of sides, but the exterior angles of any convex polygon add up to

360^\circ

. For a regular polygon, start from the exterior angle: it is the quickest route to

n

.

Areas, Circles, Arcs and Circle Theorems

Learn this subtopic in the notes

Areas and perimeters of plane figures

Trapezium area

A = \tfrac{1}{2}(a + b)\,h

$a, b$ the two parallel sides
$h$ the perpendicular distance between them

Circle measures: circumference, area, arcs and sectors

Arc and sector

\text{arc} = \frac{\theta}{360} \times 2\pi r \qquad \text{sector} = \frac{\theta}{360} \times \pi r^2

$\theta$ angle at the centre, in degrees
$r$ radius

Circle facts: chords, tangents and inscribed angles

Fact	Statement
Tangent and radius	A tangent is perpendicular to the radius at the point of contact
Two tangents from one point	Have equal lengths
Perpendicular from the centre to a chord	Bisects the chord
Angle at the centre	Is twice the inscribed angle standing on the same arc
Angles in the same segment	Inscribed angles on the same arc are equal
Angle in a semicircle	An inscribed angle standing on a diameter is $90^\circ$
Cyclic quadrilateral	Opposite angles add up to $180^\circ$
Tangent and chord (alternate segment theorem)	The angle between a tangent and a chord equals the inscribed angle on that chord in the other segment They are equal: not complementary, not supplementary.

Common traps

Perimeter counts every side

A rectangle has two lengths and two widths, so its perimeter is

2(l + w)

. Setting

l + w

equal to the perimeter gives a rectangle far too big, and its area is usually among the options.

Check whether you are given the radius or the diameter

A circle of diameter 10 has radius 5 and area

25\pi

, not

100\pi

. Using the diameter in

\pi r^2

makes the area four times too big.

The centre angle is double the edge angle, not half

On the same arc, the angle at the centre is the larger one: twice the inscribed angle. Halving the inscribed angle to get the centre angle reverses the theorem.

Solids: Surface Area, Volume and Scaling

Learn this subtopic in the notes

How area and volume scale with length

Scale factors

\frac{A_2}{A_1} = k^2 \qquad \frac{V_2}{V_1} = k^3

$k$ ratio of matching lengths, large to small

Volume and surface area of the standard solids

Solid	Volume	Surface area
Prism, cross-section area $A$ , length $L$	$AL$	Two end faces plus the rectangles round the side
Cuboid $a \times b \times c$	$abc$	$2(ab + bc + ca)$
Cylinder, radius $r$ , height $h$	$\pi r^2 h$	$2\pi r^2 + 2\pi r h$ (curved part $2\pi r h$ )
Cone, radius $r$ , height $h$ , slant $l$	$\tfrac{1}{3}\pi r^2 h$	$\pi r^2 + \pi r l$ (curved part $\pi r l$ )
Sphere, radius $r$	$\tfrac{4}{3}\pi r^3$	$4\pi r^2$
Pyramid, base area $B$ , height $h$	$\tfrac{1}{3}Bh$	Base plus the triangular faces

Common traps

A cone's volume uses the vertical height, not the slant height

V = \tfrac{1}{3}\pi r^2 h

needs the height measured straight up. If you are given the slant height, find

h

with Pythagoras first. The slant height belongs only in the curved surface area

\pi r l

.

Sphere: 4πr² for area, (4/3)πr³ for volume

The surface area of a sphere is

4\pi r^2

, with no

\tfrac{1}{3}

. Mixing the two formulas gives a wrong answer with the wrong units: area must come out in squared units, volume in cubed units.

Volume goes with the cube of the length

Doubling every length makes a solid 8 times as heavy, not 2 or 4 times. A percentage change in length must be cubed for volume and squared for area, never just multiplied by 3 or 2.

Distance, Midpoint, Gradient and Lines

Learn this subtopic in the notes

Distance and midpoint between two points

Distance and midpoint

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Gradient and the equation of a straight line

Gradient and line through a point

m = \frac{y_2 - y_1}{x_2 - x_1} \qquad y - y_1 = m(x - x_1)

Parallel and perpendicular lines, and where lines meet

Parallel and perpendicular gradients

\text{parallel: } m_1 = m_2 \qquad \text{perpendicular: } m_1 m_2 = -1

Common traps

The midpoint uses sums, the distance uses differences

Average the coordinates for a midpoint:

(x_1 + x_2)/2

. Half the difference,

(x_2 - x_1)/2

, is how far the midpoint is from each end, not where it is.

Gradient is change in y over change in x

Rise over run:

(y_2 - y_1)/(x_2 - x_1)

. Putting the x-difference on top gives the reciprocal, which IMAT includes as an option. Subtract in the same order on top and bottom.

A perpendicular gradient needs both a flip and a sign change

The perpendicular to a line of gradient

m

has gradient

-1/m

. Taking only the reciprocal (

1/m

) or only the negative (

-m

) gives a line that is not perpendicular; both wrong versions appear among the options.

Equations of Circles and Parabolas

Learn this subtopic in the notes

Equation of a circle and completing the square

Circle with centre (a, b) and radius r

(x - a)^2 + (y - b)^2 = r^2

$(a, b)$ the centre
$r$ the radius

The parabola y = ax² + bx + c: vertex, axis and where a line meets it

Vertex form

y = a(x - p)^2 + q \qquad \text{vertex } (p, q), \quad \text{axis } x = p

Common traps

(x minus a) squared means the centre is at plus a

In

(x + 4)^2 + (y - 5)^2 = 9

the centre is

(-4, 5)

, not

(4, -5)

. And the right-hand side is

r^2

: here the radius is 3, not 9.

Divide through before reading off the centre

The rule centre

= (-D/2, -E/2)

only works when

x^2

and

y^2

have coefficient 1. If the equation starts

2x^2 + 2y^2

, divide every term by 2 first.

Touching means a repeated root

A line touches a parabola when the combined quadratic has exactly one solution, which means a discriminant of zero. Two different roots mean the line cuts through at two points; no real roots mean it misses.

Geometry formulas

Angles, Triangles and Polygons

Triangles: angle sum, Pythagoras and special right triangles

Congruent and similar triangles

Polygons: interior and exterior angles

Angle facts for crossing lines and parallel lines

Common traps

Areas, Circles, Arcs and Circle Theorems

Areas and perimeters of plane figures

Circle measures: circumference, area, arcs and sectors

Circle facts: chords, tangents and inscribed angles

Common traps

Solids: Surface Area, Volume and Scaling

How area and volume scale with length

Volume and surface area of the standard solids

Common traps

Distance, Midpoint, Gradient and Lines

Distance and midpoint between two points

Gradient and the equation of a straight line

Parallel and perpendicular lines, and where lines meet

Common traps

Equations of Circles and Parabolas

Equation of a circle and completing the square

The parabola y = ax² + bx + c: vertex, axis and where a line meets it

Common traps

More IMAT Mathematics formula sheets