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Geometry – Learning Resource

Good lessons last a lifetime.

Geometry!

Geometry is something you will have been familiar with even before you knew the word itself. It finds itself everywhere around us, in the shape of the screen you’re looking at to the angle with which you see it. In this free learning resource, we’ll plunge into the world of shapes, angles, and lines, covering the basics of Geometry.

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“Geometry is nothing at all, if not a branch of art”

Basic Shapes

Firstly we’ll look at the basic shapes. These are the shapes that you have seen many times (Square, Rectangle…) and maybe some you haven’t seen (Kite, Trapezium).

We can split these shapes up into three basic categories, quadrilaterals (shapes with four sides), parallelograms (quadrilaterals with opposite sides parallel) and non-quadrilaterals. Remember, a parallelogram is a type of quadrilateral.

Square (Parallelogram)

A parallelogram with 4 equal sides and 4 equal right angles.

Rectangle (Parallelogram)

A parallelogram with 2 pairs of equal sides and 4 equal right angles.

Rhombus (Parallelogram)

A parallelogram with all sides equal.

Trapezium (Quadrilateral)

A quadrilateral with one pair of sides parallel.

Kite (Quadrilateral)

A quadrilateral with two pairs of equal sides adjacent to each other.

Triangle (Non-Quadrilateral)

A shape with 3 sides and 3 angles.

Circle (Non-Quadrilateral)

A round figure with no corners or edges.

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Area and Perimeter

Two important elements of basic shapes are the Area and Perimeter

Area

The amount of space enclosed within a flat shape.

Perimeter

The distance around the outside of a flat shape.

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Surface Area and Volume

So far we’ve only been working in two dimensions. However, we live in a 3-dimensional world, and so we have 3 dimensional shapes, including the cube, rectangular prism, triangular prism, sphere, cone, pyramid, and cylinder. To understand these shapes, we’ll look at 2 key concepts, Volume and Surface Area.

SA

Surface Area

The total area on the outside of a 3D shape.

V

Volume

The amount of space enclosed within a 3D shape.

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Struggling with some calculations! Check out our Algebra Resource for some help in solving equations.

Angles

There are 6 main types of angles which you’ll cover in school.

Acute

An angle that is between 0 and 90 degrees.

Right

An angle that is exactly 90 degrees.

Obtuse

An angle that is between 90 and 180 degrees.

Straight

An angle that is exactly 180 degrees.

Reflex

An angle that is between 180 and 360 degrees.

Revolution

An angle that is exactly 360 degrees.

Types of Triangles

Triangles are a fundamental aspect of geometry, and so in this section, we’ll dissect the 4 different types of triangle.

Equilateral

A triangle with all equal sides and equal angles.

Isosceles

A triangle with a pair of equal angles adjacent to a pair of equal sides.

Right-Angled

A triangle containing a right angle. The side directly opposite the right angle is called the hypotenuse and will always be the longest side of the triangle.

Scalene

A triangle with 3 different sides and 3 different angles.

Some More Terminology

Two ideas which are important in understanding geometry are Complementary Angles and Supplementary Angles.

Complementary Angles

Angles which add up to 90 degrees. Two angles in a Right-Angled Triangle which are not the right angle are complementary.

Supplementary Angles

Angles which add up to 180 degrees. Two angles which form a straight line are supplementary. 


“Parallel lines have so much in common, it’s a shame they’ll never meet”

Parallel and Perpendicular Lines

Some of you may already be familiar with the concepts of parallel and perpendicular lines, but if this is new territory for you, here’s a quick explanation.

Parallel Lines

Two or more lines which will never meet. In textbooks, this is shown as two lines with arrows on them.

Perpendicular Lines

Two lines which intersect at right angles. In textbooks, this is shown as a box as the angle between the two lines.

Fun Fact! Parallel lines have the same gradient!

Click below to check out the graphing module where we talk about gradient. 

Graphing Resource

Some Special Angles

In this section, we’ll look at three types of relationships between angles on parallel lines. But first, let’s add a transversal. What’s a transversal? Simply put, it’s any line that intersects a pair of parallel lines.

Now let’s look at these special angles, they are Alternate angles, Corresponding angles, and Co-interior angles.

Congruent and Similar Triangles

The next step of our journey into the world of Geometry is congruence and similarity. Note that these terms refer to a comparison between TWO triangles, to say a single triangle is congruent or similar is nonsensical.

Congruent Triangles

Two triangles are congruent if they have exactly the same size sides and angles. However, congruent triangles can be flipped and/or rotated.

Similar Triangles

Two triangles are similar if one is a scaled version of the other, that is, the angles are the same, but the sides may be scaled differently. Similar triangles can also be flipped and/or rotated.

The Big One … Pythagoras

Pythagoras’ Theorem

Next we’ll move onto the famous Pythagoras’ Theorem. To understand this theorem, we need to know the two shorter sides of a right-angled triangle as well as the hypotenuse. Remember, Pythagoras’ Theorem ONLY works on right-angled triangles.

It is the glory of geometry, that from so few principles, it is able to accomplish so muchIsaac Newton.

Basics of Trigonometry

The last stop on our journey into Geometry is the basic trigonometric ratios, Sine, Cosine, and Tangent, which are commonly abbreviated to Sin, Cos, and Tan.

Like Pythagoras’ Theorem, we’ll only look at Sin, Cos, and Tan on right-angled triangles. Now let’s let an angle (not the right angle) be x.

Sin

Sin(x) is the ratio of the side opposite to the angle (x) to the hypotenuse.

Cos

Cos(x) is the ratio of the side adjacent to the angle (x) to the hypotenuse.

Tan

Tan(x) is the ratio of the side opposite to the angle (x) to the side adjacent to the angle (x).

SOHCAHTOA

This is a useful abbreviation for remembering our trig ratios, Sin(Opposite/Hypotenus), Cos(Adjacent/Hypotenuse), Tan(Opposite/Adjacent).

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