IMAT Mathematics · Free notes

Geometry: Shapes, Solids and Coordinates

Geometry has 21 past questions since 2011, and the ministry papers from 2023 on have asked 5 of them. It is the largest Mathematics chapter in the bank, but most items need one well-known fact or formula: a volume, an angle theorem, a gradient, the equation of a circle. The work is in choosing that fact quickly and in the algebra after it: completing the square for a circle, the negative reciprocal for a perpendicular line, squaring a scale factor for an area. Diagrams are often marked not to scale, so trust the numbers you are given, not your eye.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

IMAT home All formulas on one page

Subtopic notes

Formula & revision sheet

11 formulas · 3 reference tables · 16 gotchas across all subtopics — the exam-eve cheat-sheet

Angles, Triangles and Polygons

Formulas (3)

Triangles: angle sum, Pythagoras and special right triangles · Pythagoras and triangle area $a^2 + b^2 = c^2 \qquad A = \tfrac{1}{2}bh$
Congruent and similar triangles · Similar triangles $\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c} = k \qquad \frac{\text{Area}'}{\text{Area}} = k^2$
Polygons: interior and exterior angles · Polygon angles $S_{\text{int}} = (n - 2) \times 180^\circ \qquad \text{regular: each exterior} = \frac{360^\circ}{n}$

Reference tables (1)

Angle facts for crossing lines and parallel lines6 rows

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.

Watch out for (4)

Only parallel lines give equal alternate and corresponding angles
→ Angle facts for crossing lines and parallel lines
The hypotenuse is the side opposite the right angle
→ Triangles: angle sum, Pythagoras and special right triangles
Match the sides through the angles, not through their position on the page
→ Congruent and similar triangles
Exterior angles total 360° for every polygon
→ Polygons: interior and exterior angles

Areas, Circles, Arcs and Circle Theorems

Formulas (2)

Areas and perimeters of plane figures · Trapezium area $A = \tfrac{1}{2}(a + b)\,h$
Circle measures: circumference, area, arcs and sectors · Arc and sector $\text{arc} = \frac{\theta}{360} \times 2\pi r \qquad \text{sector} = \frac{\theta}{360} \times \pi r^2$

Reference tables (1)

Circle facts: chords, tangents and inscribed angles8 rows

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.

Watch out for (3)

Perimeter counts every side
→ Areas and perimeters of plane figures
Check whether you are given the radius or the diameter
→ Circle measures: circumference, area, arcs and sectors
The centre angle is double the edge angle, not half
→ Circle facts: chords, tangents and inscribed angles

Solids: Surface Area, Volume and Scaling

Formulas (1)

How area and volume scale with length · Scale factors $\frac{A_2}{A_1} = k^2 \qquad \frac{V_2}{V_1} = k^3$

Reference tables (1)

Volume and surface area of the standard solids6 rows

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

Watch out for (3)

A cone's volume uses the vertical height, not the slant height
→ Volume and surface area of the standard solids
Sphere: 4πr² for area, (4/3)πr³ for volume
→ Volume and surface area of the standard solids
Volume goes with the cube of the length
→ How area and volume scale with length

Distance, Midpoint, Gradient and Lines

Formulas (3)

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

Watch out for (3)

The midpoint uses sums, the distance uses differences
→ Distance and midpoint between two points
Gradient is change in y over change in x
→ Gradient and the equation of a straight line
A perpendicular gradient needs both a flip and a sign change
→ Parallel and perpendicular lines, and where lines meet

Equations of Circles and Parabolas

Formulas (2)

Equation of a circle and completing the square · Circle with centre (a, b) and radius r $(x - a)^2 + (y - b)^2 = r^2$
The parabola y = ax² + bx + c: vertex, axis and where a line meets it · Vertex form $y = a(x - p)^2 + q \qquad \text{vertex } (p, q), \quad \text{axis } x = p$

Watch out for (3)

(x minus a) squared means the centre is at plus a
→ Equation of a circle and completing the square
Divide through before reading off the centre
→ Equation of a circle and completing the square
Touching means a repeated root
→ The parabola y = ax² + bx + c: vertex, axis and where a line meets it

PYQ weightage by concept

14 concepts · 0 PYQs — where the marks actually sit, so you know what to drill first

Angles, Triangles and Polygons0 PYQs · 0%

Concept	PYQs	Share
Angle facts for crossing lines and parallel linesfoundation	—	—
Triangles: angle sum, Pythagoras and special right trianglesfoundation	—	—
Congruent and similar trianglesfoundation	—	—
Polygons: interior and exterior anglesfoundation	—	—

Areas, Circles, Arcs and Circle Theorems0 PYQs · 0%

Concept	PYQs	Share
Areas and perimeters of plane figuresfoundation	—	—
Circle measures: circumference, area, arcs and sectorsfoundation	—	—
Circle facts: chords, tangents and inscribed anglesfoundation	—	—

Solids: Surface Area, Volume and Scaling0 PYQs · 0%

Concept	PYQs	Share
Volume and surface area of the standard solidsfoundation	—	—
How area and volume scale with lengthfoundation	—	—

Distance, Midpoint, Gradient and Lines0 PYQs · 0%

Concept	PYQs	Share
Distance and midpoint between two pointsfoundation	—	—
Gradient and the equation of a straight linefoundation	—	—
Parallel and perpendicular lines, and where lines meetfoundation	—	—

Equations of Circles and Parabolas0 PYQs · 0%

Concept	PYQs	Share
Equation of a circle and completing the squarefoundation	—	—
The parabola y = ax² + bx + c: vertex, axis and where a line meets itfoundation	—	—