Comprehensive Educational Resources for Singapore Students

Elementary Mathematics Notes (Diagrams)

Reproduced from http://teach.sg

Angles, Triangles & Polygons

Names of Angles

Right Angle
90°

$(90°)$

Acute Angle
45°

$(<90°)$

Obtuse Angle
120°

$(90° < θ < 180°)$

Reflex Angle
240°

$(180° < θ < 360°)$

Types of Angles

Angles on a Straight Line

Angles on a straight line sum to $180°$

$a$ $b$ $a + b = 180°$

Angles at a Point

Angles around a point sum to $360°$

$x$ $y$ $z$ $w$

Vertically Opposite Angles

Vertically opposite angles are equal

Parallel Lines

Alternate Angles: Equal when lines are parallel

Corresponding Angles: Equal when lines are parallel

Interior Angles: Sum to $180°$ when lines are parallel

Mensuration

Perimeter & Area

Parallelogram
$a$ $b$ $a+b=180°$

Angles on straight line = $180°$

Trapezium
$a$ $b$ $h$

Area = $\frac{a+b}{2} \times h$

Circle
$r$ $O$

Circumference = $2\pi r$

Area = $\pi r^2$

Surface Area & Volume

Cuboid
$l$ $h$ $b$

Surface Area = $2(lb + lh + bh)$

Volume = $l \times b \times h$

Cylinder
$h$ $r$

Surface Area = $2\pi r^2 + 2\pi r h$

Volume = $\pi r^2 h$

Prism
$l$

Volume = cross-sectional area × $l$

Pyramid

Volume = $\frac{1}{3} \times$ base area × height

Cone
$r$ $l$ $h$

Volume = $\frac{1}{3}\pi r^2 h$

Surface Area = $\pi r^2 + \pi r l$

Sphere
$r$

Volume = $\frac{4}{3}\pi r^3$

Surface Area = $4\pi r^2$

Functions & Graphs

Graphs of Power Functions ($y = ax^n$)

$n = 2$ (Quadratic)

$x$ $y$ $O$ $a > 0: y = x^2$ $a < 0: y = -x^2$

$n = 3$ (Cubic)

$x$ $y$ $O$ $a > 0: y = x^3$ $a < 0: y = -x^3$

$n = -1$ (Reciprocal)

$x$ $y$ $O$ $a > 0: y = \frac{1}{x}$ $a < 0: y = -\frac{1}{x}$

Exponential Function (Graphs)

$y = a^x$ (where $a > 1$)

$x$ $y$ $O$ $1$ $y = 5^x$

Properties of Circles

Angle Properties

Tangent Perpendicular to Radius

$O$

A tangent to a circle is perpendicular to the radius at the point of contact.

Right Angle in Semicircle

$O$

The angle in a semicircle is always a right angle (90°).

Angles in Same Segment

Angles subtended by the same arc in the same segment are equal.

Angle at Centre vs Angle at Circumference

$O$

The angle at the centre is twice the angle at the circumference when subtended by the same arc.

Angles in Opposite Segments

Opposite angles in a cyclic quadrilateral sum to 180°.

Tangents from External Point

O P B A

Tangents drawn from an external point to a circle are equal in length.

Bisectors

Constructing Perpendicular Bisector

1
Draw Line Segment

Draw the line segment you want to bisect.

2
Draw Arcs

From each endpoint, draw arcs with the same radius (greater than half the line length).

3
Join Intersections

Draw a line through the two intersection points. This is the perpendicular bisector.

Constructing Angle Bisector

1
Draw Angle

Start with the angle you want to bisect.

2
Draw Arc

From the vertex, draw an arc that intersects both arms of the angle.

3
Draw Angle Bisector

From the intersection points, draw arcs with equal radius. Join the vertex to the intersection of these arcs.

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