IMAT Mathematics · Geometry

Areas, Circles, Arcs and Circle Theorems

Areas and perimeters of flat shapes, the measures of a circle and its parts, and the angle facts about chords, tangents and inscribed angles.

All Geometry notes

Why this matters

Area questions appear across the years: a rectangle from its perimeter and the ratio of its sides (2025), shapes with circular pieces removed (2011, 2013) and a sector between two circles (2022). In 2024 the ministry paper asked how the angle between a tangent and a chord compares with an inscribed angle.

Concept 1 of 3: Areas and perimeters of plane figures

Area counts unit squares inside a shape; perimeter is the length of its boundary. Every common formula comes from the rectangle: a parallelogram is a rectangle with a slice moved across, a triangle is half a parallelogram, and a trapezium is the average of its two parallel sides times the height. Awkward shapes are made of these pieces, added or cut away.

Definition

Rectangle $l \times w$ : area $lw$ , perimeter $2(l + w)$ . Square of side $s$ : area $s^2$ , perimeter $4s$ .
Parallelogram: base times perpendicular height (not the slanted side).
Triangle: $\tfrac{1}{2} \times$ base $\times$ height.
Trapezium with parallel sides $a$ and $b$ , $h$ apart: $\tfrac{1}{2}(a + b)h$ .
Composite shapes: split into these pieces, then add, or subtract the pieces cut out.

Trapezium area

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

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

Worked example

A lawn is a trapezium with parallel sides of 12 m and 20 m, which are 7 m apart. A square pond of side 3 m is dug in the lawn. What area of grass is left?

Practice this conceptself-check · 4 quick reps

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.

Concept 2 of 3: Circle measures: circumference, area, arcs and sectors

An arc is a fraction of the circumference and a sector is the same fraction of the area. The fraction is the angle at the centre out of 360°. So every arc or sector problem is the whole-circle formula times

\theta/360

Definition

For a circle of radius $r$ (diameter $2r$ ):

Circumference $2\pi r$ ; area $\pi r^2$ .
An arc cut off by a centre angle $\theta^\circ$ has length $\dfrac{\theta}{360} \times 2\pi r$ .
A sector (a slice like a pizza piece) with centre angle $\theta^\circ$ has area $\dfrac{\theta}{360} \times \pi r^2$ .
In radians: arc $r\theta$ , sector $\tfrac{1}{2}r^2\theta$ .
The perimeter of a sector is the arc plus two radii.

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

Worked example

A circle has radius 9 cm. A sector has a centre angle of

80^\circ

. Find its arc length, its area and its perimeter.

Practice this conceptself-check · 4 quick reps

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.

Concept 3 of 3: Circle facts: chords, tangents and inscribed angles

Most circle theorems come from one picture: two radii make an isosceles triangle. From it you get the angle at the centre being twice the angle at the edge, and from that the angle in a semicircle, angles in the same segment and the tangent-chord rule.

Definition

A chord joins two points of the circle; a tangent touches it at one point only.
An inscribed angle has its vertex on the circle, its arms passing through the ends of a chord.
A chord divides the circle into two segments: the major (larger) and the minor (smaller).
A cyclic quadrilateral has all four corners on a circle.

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.

Practice this conceptself-check · 4 quick reps

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.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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

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