GeometryGeometry is all about shapes and their properties.

If you like playing with objects, or like drawing, then geometry is for you!

Geometry can be divided into:

Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper

   

Solid Geometry is about three dimensional objects like cubes, prisms and pyramids.

Plane GeometryPlane geometry is all about shapes like lines, circles and triangles ... shapes that can be drawn on a flat surface called a Plane (it is like on an endless piece of paper).

PlaneA plane is a flat surface with no thickness.

Our world has three dimensions, but there are only two dimensions on a plane.

Examples:

length and height, or x and y

And it goes on forever.

Examples

It is actually hard to give a real example!

When we draw something on a flat piece of paper we are drawing on a plane ...

... except that the paper itself is not a plane, because it has thickness! And it should extend forever, too.

So the very top of a perfect piece of paperthat goes on forever is the right idea!

Also, the top of a table, the floor and a whiteboard are all like a plane.

Imagine

Imagine you lived in a two-dimensional world. You could travel around, visit friends, but nothing in your world would have height.

You could measure distances and angles.

You could travel fast or slow. You could go forward, backwards or sideways. You could move in straight lines, circles, or anything so long as you never go up or down.

What would life be like living on a plane?

Regular 2-D Shapes - Polygons

Move the mouse over the shapes to discover their properties.

Triangle Square

Pentagon Hexagon

Heptagon Octagon

Nonagon Decagon

Hendecagon DodecagonThese shapes are known as regular polygons. A polygon is a many sided

shape with straight sides.

To be a regular polygon all the sides and angles must be the same.

PerimeterPerimeter is the distance around a two-dimensional shape.

Example 1: the perimeter of this rectangle is 7+3+7+3 = 20

Example 2: the perimeter of this pentagon is 3+3+3+3+3 = 5×3 = 15

 

The perimeter of a circle is called the circumference:

Triangles

A triangle has three sides and three angles

The three angles always add to 180°

Equilateral, Isosceles and Scalene

There are three special names given to triangles that tell how many sides (or angles) are equal.

There can be 3, 2 or no equal sides/angles:

Equilateral Triangle

Three equal sides Three equal angles, always 60°

Isosceles Triangle

Two equal sides Two equal angles

Scalene Triangle

No equal sides No equal angles

What Type of Angle?

Triangles can also have names that tell you what type of angle is inside:

Acute Triangle

All angles are less than 90°

Right Triangle

Has a right angle (90°)

Obtuse Triangle

Has an angle more than 90°

Combining the NamesSometimes a triangle will have two names, for example:

Right Isosceles Triangle

Has a right angle (90°), and also two equal angles

Can you guess what the equal angles are?

Area

The area is half of the base times height.

"b" is the distance along the base

"h" is the height (measured at right angles to the base)

Area = ½bh

The formula works for all triangles.

Another way of writing the formula is bh/2

Example: What is the area of this triangle?

Height = h = 12

Base = b = 20

Area = bh/2 = 20 × 12 / 2 = 120

Just make sure that the "h" is measured at right angles to the "b".

Why is the Area "Half of bh"?

Imagine you "doubled" the triangle (flip it around one of the upper edges) to make a square-like shape (it would be a "parallelogram" actually), THEN the whole area would be bh (that would be for both triangles, so just one is ½bh), like this:

You can also see that if you sliced the new triangle and placed the sliced part on the other side you get a simple rectangle, whose area is bh.

Right Angled TrianglesA right angled triangle is (you guessed it), a triangle which has a right angle (90°) in it.

The little square in the corner tells us that it is a right angled triangle (I wrote 90°, but you don't need to!)

Two Types

There are two types of right angled triangle:

An isosceles right angled triangle A scalene right angled triangle

Isosceles right angled triangle

One right angle Two other equal angles always of 45° Two equal sides

Scalene right angled triangle

One right angle Two other unequal angles No equal sides

 

The 3,4,5 Triangle

The "3,4,5 Triangle" has a right angle:

(It is a scalene right angled triangle)

A very useful triangle to draw if you need a right angle!

Quadrilaterals

Quadrilateral just means "four sides" (quad means four, lateral means side).

Any four-sided shape is a Quadrilateral.

But the sides have to be straight, and it has to be 2-dimensional.

Properties Four sides (or edges) Four vertices (or corners).

The interior angles add up to 360 degrees:

Try drawing a quadrilateral, and measure the angles. They should add to 360°

Types of Quadrilaterals

There are special types of quadrilateral:

Some types are also included in the definition of other types! For example a square, rhombus and rectangle are also parallelograms. See below for more details.

Let us look at each type in turn:

The Rectangle

means "right angle"

and show equal sides

A rectangle is a four-sided shape where every angle is a right angle (90°).

Also opposite sides are parallel and of equal length.

The Rhombus

A rhombus is a four-sided shape where all sides have equal length.

Also opposite sides are parallel and opposite angles are equal.

Another interesting thing is that the diagonals (dashed lines in second figure) of a rhombus bisect each other at right angles.

The Square

means "right angle"

show equal sides

A square has equal sides and every angle is a right angle (90°)

Also opposite sides are parallel.

A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

The Parallelogram

Opposite sides are parallel and equal in length, and opposite angles are equal (angles "a" are the same, and angles "b" are the same)

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

Example:

A parallelogram with all sides equal and angles "a" and "b" as right angles is a square.

 

The Trapezoid (UK: Trapezium)

Trapezoid Isosceles Trapezoid

A trapezoid (called a trapezium in the UK) has one pair of opposite sides parallel.

It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown.

A trapezoid is not a parallelogram because only one pair of sides is parallel.

Language Note: In the US a "trapezium" is a quadrilateral with NO parallel sides!

The Kite

Hey, it looks like a kite. It has two pairs of sides. Each pair is made up of adjacent sides that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.

 

... and that's it for the special quadrilaterals.

Irregular Quadrilaterals

The only regular quadrilateral is a square. So all other quadrilaterals are irregular.

 

The "Family Tree" Chart

Quadrilateral definitions are inclusive.

Example: a square is also a rectangle.

So we include a square in the definition of a rectangle.

(We don't say "A rectangle has all 90° angles, except if it is a square")

This may seem odd because in daily life we think of a square as not being a rectangle ... but in mathematics it is.

Using the chart below you can answer such questions as:

Is a Square a type of Rectangle? (Yes) Is a Rectangle a type of Kite? (No)

Complex Quadrilaterals

Oh Yes! when two sides cross over, you call it a "Complex" or "Self-Intersecting" quadrilateral like these:

They still have 4 sides, but two sides cross over.

Polygon

A quadrilateral is a polygon. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on.

Play with Them

Now that you know the different types, you can play with the Interactive Quadrilaterals.

Other Names

A quadrilateral can sometimes be called:

a Quadrangle ("four angles"), so it sounds like "triangle" a Tetragon ("four and polygon"), so it sounds like "pentagon", "hexagon", etc.

CircleA circle is easy to make:

Draw a curve that is "radius" away from a central point.

And so:

All points are the same distancefrom the center.

 

Also, the circle is a plane shape (two dimensional).

Definition

In fact the definition of a circle is "the set of all points on a plane that are a fixed distance from a center".

Radius and Diameter

The Radius is the distance from the center to the edge.

The Diameter starts at one side of the circle, goes through the center and ends on the other side.

So the Diameter is twice the Radius:

Diameter = 2 × Radius

 

Circumference

The Circumference is the distance around the edge of the circle.

It is exactly Pi (the symbol is π) times the Diameter, so:

Circumference = π × Diameter

And so these are also true:

Circumference = 2 × π × Radius

Circumference/Diameter = π

Area

The area of a circle is π times the Radius squared, which is written:

A = π × r2

Or, in terms of the Diameter:

A = (π/4) × D2

It is easy to remember if you think of the area of the square that the circle would fit inside.

Names

Because people have studied circles for thousands of years special names have come about.

Nobody wants to say "that line that starts at one side of the circle, goes through the center and ends on the other side" when a word like "Diameter" would do.

So here are the most common special names:

Lines

A line that goes from one point to another on the circle's circumference is called a Chord.

If that line passes through the center it is called a Diameter.

If a line "just touches" the circle as it passes it is called a Tangent.

And a part of the circumference is called an Arc.

Slices

There are two main "slices" of a circle

The "pizza" slice is called a Sector.

And the slice made by a chord is called a Segment.

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

Quarter of a circle is called a Quadrant.

Half a circle is called a Semicircle.

Inside and Outside

A circle has an inside and an outside (of course!). But it also has an "on", because you could be right on the circle.

Example: "A" is outside the circle, "B" is inside the circle and "C" is on the circle.

Pi

Pi (the symbol is the Greek letter π) is:

The ratio of the Circumference to the Diameter

of a Circle.

In other words, if you measure the circumference, and then divide by the diameter of the circle you get the number π

It is approximately equal to:

3.14159265358979323846…

The digits go on and on with no pattern. In fact, pi has been calculated to over one million decimal places and still there is no pattern.

Approximation

A quick and easy approximation to Pi is 22/7

22/7 = 3.1428571...

But as you can see, 22/7 is not exactly right. In fact Pi is not equal to the ratio of any two numbers, which makes it an irrational number.

To 100 Decimal Places

Here is Pi with the first 100 decimal places:

3.14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679...

Circle Sector and Segment

Slices

There are two main "slices" of a circle:

The "pizza" slice is called a Sector.

And the slice made by a chord is called a Segment.

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

Quarter of a circle is called a Quadrant.

Half a circle is called a Semicircle.

 

Area of a Sector

You can work out the Area of a Sector by comparing its angle to the angle of a full circle.

Note: I am using radians for the angles.

This is the reasoning:

A circle has an angle of 2π and an Area of: πr2

So a Sector with an angle of θ (instead of 2π) must have an area of: (θ/2π) × πr2

Which can be simplified to: (θ/2) × r2

Area of Sector = ½ × θ × r2

= ½ × (θ × π/180) × r2   (if θ is in degrees)

 

Arc Length of Sector or Segment

By the same reasoning, the arc length (of a Sector or Segment) is:

Arc Length "L" = θ × r

= (θ × π/180) × r   (if θ is in degrees)

 

Area of Segment

The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here).

There is a lengthy derivation, but the result is a slight modification of the Sector formula:

Area of Segment = ½ × (θ - sin θ) × r2

= ½ × ( (θ × π/180) - sin θ) × r2   (if θ is in degrees)

AreaThe size of a surface!

Area is the amount of space inside the boundary of a flat object (such as a square or circle).

Example:

These shapes all have the same area of 9:

Area of Plane Shapes

There are special formulas for certain shapes:

Example: What is the area of this rectangle?

The formula is:

Area = w × hw =widthh = height

The breadth is 5, and the height is 3, so we know w = 5 and h = 3. So:

Area = 5 × 3 = 15

Read Area of Plane Shapes for more information.

Finding Area by Counting Squares

You can count the number of squares to find an area.

This rectangle has an area of 15

If each square was 1 cm on a side, then the area would be 15 cm2 (15 square cm)

 

The squares may not match the shape exactly, so you will need to "approximate" an answer.

One way is:

more than half a square counts as 1

less than half a square counts as 0

Like this:

This pentagon has an area of approximately 17

Or just use your eyes and count a whole square when the areas seem to add up, like with this circle, where the area marked "4" seems equal to about 1 whole square (also for "8"):

This circle has an area of approximately 14

Area of Plane Shapes 

TriangleArea = ½b × h

b = baseh = vertical height

Square

Area = a2

a = length of side

RectangleArea = w × h

w = widthh = height

ParallelogramArea = w × h

w = widthh = height

Trapezoid (US)Trapezium (UK)

Area = ½(a+b) × hh = vertical height

CircleArea = πr2

Circumference=2πrr = radius

EllipseArea = πab

SectorArea = ½r2θ

r = radiusθ = angle in radians

Here is an example:

Example: What is the area of this rectangle?

The formula is:

Area = w × hw = widthh = height

We know w = 5 and h = 3, so:

Area = 5 × 3 = 15

Pythagoras' TheoremYears ago, a man named Pythagoras found an amazing fact about triangles:

 

If the triangle had a right angle (90°) ...

... and you made a square on each of the three sides, then ...

... the biggest square had the exact same area as the other two squares put together!

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle the square of the hypotenuse is equal tothe sum of the squares of the other two sides.

So, the square of a (a²) plus the square of b (b²) is equal to the square of c (c²):

a2 + b2 = c2

Sure ... ?

Let's see if it really works using an example. A "3,4,5" triangle has a right angle in it, so the formula should work.

Let's check if the areas are the same:

32 + 42 = 52

Calculating this becomes:

9 + 16 = 25

Yes, it works !

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows us to find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a2 + b2 = c2

Now you can use algebra to find any missing value, as in the following examples:

Example: Solve this triangle.

 

a2 + b2 = c2

52 + 122 = c2

25 + 144 = c2

169 = c2

c2 = 169

c = √169

c = 13

Example: Solve this triangle.

 

a2 + b2 = c2

92 + b2 = 152

81 + b2 = 225

Take 81 from both sides:

b2 = 144

b = √144

b = 12

Example: What is the diagonal distance across a square of size 1?

 

a2 + b2 = c2

12 + 12 = c2

1 + 1 = c2

2 = c2

c2 = 2

c = √2 = 1.4142...

It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ?

(√3)2 + (√5)2 = (√8)2

3 + 5 = 8

Yes, it does!

So this is a right-angled triangle

Pythagorean TriplesThese are simply sets of whole numbers which fit the rule:

a2 + b2 = c2

(this is the equation for the Pythagorean Theorem)

Examples of these are:

3,4,5 Triangle 5,12,13 triangle 9,40,41 Triangle32 + 42 = 52 52 + 122 = 132 92 + 402 = 412

There are an infinite number of triangles like these!

The simplest way to create further pythagorean triples is to simply scale up a set of triples.

Example: scale 3,4,5 by 2 gives 6,8,10 which also fits the formula a2 + b2 = c2

Polygons

A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons. Here are some more:

Pentagon

Pentagram

Hexagon

Symbols in GeometryCommon Symbols Used in Geometry

Symbols save time and space when writing. Here are the most common geometrical symbols:

Symbol Meaning Example In Words

TriangleABC has 3 equal

sidesTriangle ABC has three equal sides

Angle ABC is 45° The angle formed by ABC is 45

degrees.

Perpendicular AB CD The line AB is perpendicular to line

CD

Parallel EF GH The line EF is parallel to line GH

Degrees360° makes a full

circle

Right Angle (90°) is 90° A right angle is 90 degrees

Line Segment "AB" AB The line between A and B

Line "AB" The infinite line that includes A and B

Ray "AB"The line that starts at A, goes through B and continues on

Congruent (same shape and size)

ABC DEF Triangle ABC is congruent to triangle

DEF

Similar (same shape, different size)

DEF MNO Triangle DEF is similar to triangle

MNO

Therefore a=b b=a a equals b, therefore b equals a

Naming AnglesFor angles the central letter is where the angle is. For example when you see " ABC is 45°", then the point "B" is where the angle is.

Short Example So now, when someone writes: In ABC, BAC is

You know they are saying:"In triangle ABC, the angle BAC is a right angle"

CongruentIf one shape can become another using Turns, Flips and/or Slides, then the two shapes are called Congruent:

Rotation Turn!

Reflection Flip!

Translation Slide!

After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

Examples

These shapes are all Congruent:

Rotated Reflected and Moved Reflected and Rotated

Congruent or Similar?

The two shapes need to be the same size to be congruent. (If you need to resize one shape to make it the same as the other, the shapes are called Similar)

If you ... Then the shapes are ...

... only Rotate, Reflect and/or Translate Congruent

... need to ResizeSimilar

Congruent AnglesCongruent Angles have the same angle in degrees. That's all.

These angles are congruent.

They don't have to point in the same direction.

They don't have to be on similar sized lines.

Similar

Two shapes are Similar if the only difference is size (and possibly the need to turn or flip one around).

Resizing is the Key

If one shape can become another using Resizing (also called dilation, contraction, compression, enlargement or even expansion), then the shapes are Similar:

These Shapes are Similar!

There may be Turns, Flips or Slides, Too!

Sometimes it can be hard to see if two shapes are Similar, because you may need to turn, flip or slide one shape as well as resizing it.

Rotation Turn!

Reflection Flip!

Translation Slide!

Examples

These shapes are all Similar:

Resized Resized and Reflected Resized and Rotated

Why is it Useful?

When two shapes are similar, then:

corresponding angles are equal, and

the lines are in proportion.

This can make life a lot easier when solving geometry puzzles, as in this example:

Example: What is the missing length here?

Notice that the red triangle has the same angles as the main triangle ...

... they both have one right angle, and a shared angle in the left corner

In fact you could flip over the red triangle, rotate it a little, then resize it and it would fit exactly on top of the main triangle. So they are similar triangles.

So the line lengths will be in proportion, and we can calculate:

? = 80 × (130/127) = 81.9

(No fancy calculations, just common sense!)

Congruent or Similar?

If you don't need to resize to make the shapes the same, they are Congruent.

So, if the shapes become the same:

When you ... Then the shapes are ...

... only Rotate, Reflect and/or Translate Congruent

... need to ResizeSimilar

AnglesAn angle measures the amount of turn

Names of Angles

As the Angle Increases, the Name ChangesType of Angle

Description

Acute Angle

an angle that is less than 90°

Right Anglean angle that is 90° exactly

Obtuse Angle

an angle that is greater than 90° but less than 180°

Straight Angle

an angle that is 180° exactly

Reflex Angle

an angle that is greater than 180°

Try It Yourself!

View Larger

Be Careful What You Measure

This is an Obtuse Angle. And this is a Reflex Angle.

But the lines are the same ... so when naming the angles make sure that you know which angle is being asked for!

Parts of an Angle

The corner point of an angle is called the vertex

And the two straight sides are called arms

The angle is the amount of turn between each arm.

Labelling Angles

There are two main ways to label angles:

1. by giving the angle a name, usually a lower-case letter like a or b, or sometimes a Greek letter like α (alpha) or θ (theta)

2. or by the three letters on the shape that define the angle, with the middle letter being where the angle actually is (its vertex).

Example angle "a" is "BAC", and angle "θ" is "BCD"

Degrees (Angles)

We can measure Angles in Degrees.

There are 360 degrees in one Full Rotation (one complete circle around).

(Angles can also be measured in Radians)

(Note: "Degrees" can also mean Temperature, but here we are talking about Angles)

The Degree Symbol: °

We use a little circle ° following the number to mean degrees.

For example 90° means 90 degrees

One Degree

This is how large 1 Degree is

 

The Full Circle

A Full Circle is 360°

Half a circle is 180°(called a Straight Angle)

Quarter of a circle is 90°(called a Right Angle)

Why 360 degrees? Probably because old calendars (such as the Persian Calendar) used 360 days for a year - when they watched the stars they saw them revolve around the North Star one degree per day.

Measuring DegreesWe often measure degrees using a protractor:

The normal protractor measures 0° to 180°

You can also get full-circle protractors.

But they are not as commonly used because they are a bit big and don't do anything special.

Acute AnglesAn acute angle is one which is less than 90°

This is an acute angle

All the angles below are acute angles:

Remember to look carefully at which angle you are being asked to name. It is the small angle which is less than 90° which is the acute angle.

Right AnglesA right angle is an internal angle which is equal to 90°

This is a right angle

Note the special symbol like a box in the angle. If you see this, it is a right angle. The 90° is rarely written in. If you see the box in the corner, you are being told it is a right angle.

 

All the angles below are right angles:

A right angle can be in any orientation or rotation as long as the internal angle is 90°

Obtuse AnglesAn obtuse angle is one which is more than 90° but less than 180°

This is an obtuse angle !

 

All the angles below are obtuse angles:

Remember to look carefully at which angle you are being asked to name. It is the smallest angle which is between the lines. The obtuse angle is more than 90° and less than 180°.

I have actually used the same angles as on the Reflex Angles page. The reflex angle is the other side of the lines. If you look at both pages and add the reflex and the obtuse angle for the same shapes you will always come to 360°

Straight AngleA straight angle is 180 degrees

This is a straight angle

A straight angle changes the direction to point the opposite way.

Sometimes people say "You did a complete 180 on that!" ... meaning you completely changed your mind, idea or direction.

All the angles below are straight angles:

Reflex AnglesA Reflex Angle is one which is more than 180° but less than 360°

This is a reflex angle

 

All the angles below are reflex angles:

Notice that I have used the same angles as on the Obtuse Angles page. The obtuse angle is the other side of the lines. When naming the angles make sure that you know which angle is being asked for.

If you look at both pages and add the reflex and the obtuse angle for each shape you will always come to 360°

Parallel Lines, and Pairs of AnglesParallel Lines

Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember:

Always the same distance apart and never touching.

The red line is parallel to the blue line in both these cases:

Example 1 Example 2

Parallel lines also point in the same direction.

Pairs of Angles

When parallel lines get crossed by another line (which is called a Transversal), you can see that many angles are the same, as in this example:

These angles can be made into pairs of angles which have special names.

 

Click on each name to see it highlighted:

(If you can't see anything on the right, then you may need to install Flash Player)

Testing for Parallel LinesSome of those special pairs of angles can be used to test if lines really are parallel:

If Any Pair Of ... Example:

Corresponding Angles are equal, or a = e

Alternate Interior Angles are equal, or c = f

Alternate Exterior Angles are equal, or b = g

Consecutive Interior Angles add up to 180° d + f = 180°

... then the lines are Parallel

Examples

These lines are parallel, because a pair of Corresponding Angles are

equal.

These lines are not parallel, because a pair of Consecutive Interior Angles do not add up to 180° (81° + 101° =182°)

These lines are parallel, because a pair of Alternate Interior Angles are

equal

Transversals

A Transversal is a line that crosses at least two other lines.

The red line is the transversal in each example:

Transversal crossing two lines

this Transversal crosses two parallel lines

... and this one cuts across three lines

Triangles Contain 180° 

In a triangle, the three angles always add to 180°:

A + B + C = 180°

We can use that fact to find a missing angle in a triangle

Example: Find the Missing Angle "C"

Start With: A + B + C = 180°

Fill in what we know: 38° + 85° + C = 180°

Rearrange C = 180° - 38° - 85°

Calculate: C = 57°

Proof

This is a proof that the angles in a triangle equal 180°:

The top line (that touches the top of the triangle) is running parallel to the base of the triangle.

So:

angles A are the same angles B are the same

And you can easily see that A + C + B does a complete rotation from one side of the straight line to the other, or 180°

Supplementary Angles

Two Angles are Supplementary if they add up to 180 degrees.

These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°.

Notice that together they make a straight angle.

But the angles don't have to be together.

These two are supplementary because 60° + 120° = 180°

If the two angles add to 180°, we say they "Supplement" each other. Supplement comes from Latin supplere, to complete or "supply" what is needed.

Spelling: be careful, it is not "Supplimentary Angle" (with an "i")

Complementary vs Supplementary

A related idea is Complementary Angles, they add up to 90°

How can you remember which is which? Easy! Think:

"C" of Complementary stands for "Corner" (a Right Angle), and "S" of Supplementary stands for "Straight" (180 degrees is a straight line)

Complementary AnglesTwo Angles are Complementary if they add up to 90 degrees (a Right Angle).

These two angles (40° and 50°) are Complementary Angles, because they add up to 90°.

Notice that together they make a right angle.

But the angles don't have to be together.

These two are complementary because 27° + 63° = 90°

Right Angled Triangle

In a right angled triangle, the two acute angles are complementary, because in a triangle the three angles add to 180°, and 90° have been taken by the right angle.

 

If the two angles add to 90°, we say they "Complement" each other.

Complementary comes from Latin completum meaning "completed" ... because the right angle is thought of as being a complete (full) angle.

Spelling: be careful, it is not "Complimentary Angle" (with an "i") ... that would be an angle you get for free!

 

Complementary vs Supplementary

Note: A related idea is Supplementary Angles - those add up to 180°

How can you remember which is which? Easy! Think:

"C" of Complementary stands for "Corner" (a Right Angle), and "S" of Supplementary stands for "Straight" (180 degrees is a straight line)

Angles Around a PointAngles around a point will always add up to 360 degrees.

The angles above all add to 360°

53° + 80° + 140° + 87° = 360°

 

Because of this, if there is an unknown angle we can always find it.

Example: What is angle "c"?

To find angle c we take the sum of the known angles and take that from 360°

Sum of known angles = 110° + 75° + 50°  + 63° Sum of known angles = 298°

Angle c = 360° − 298° Angle c = 62°

Angles On One Side of A Straight LineAngles on one side of a staight line will always add to 180 degrees.

If a line is split into 2 and you know one angle you can always find the other one.

30° + 150° = 180°

 

Example: If we know one angle is 45° what is angle "a" ?

Angle a is 180° − 45° = 135°

 

This method can be used for several angles on one side of a straight line.

Example: What is angle "b" ?

Angle b is simply 180° less the sum of the other angles.

Sum of known angles = 45° + 39° + 24° Sum of known angles = 108°

Angle b = 180° − 108°Angle b = 72°

Interior AnglesAn Interior Angle is an angle inside a shape.

Exterior AngleThe Exterior Angle is the angle between any side of a shape, and a line extended from the

next side.

Interior Angles of PolygonsAn Interior Angle is an angle inside a shape.

Triangles

The Interior Angles of a Triangle add up to 180°

90° + 60° + 30° = 180° 80° + 70° + 30° = 180°

It works for this triangle! Let's tilt a line by 10° ...

It still works, because one angle went up by 10°, but the other went down by 10°

Quadrilaterals (Squares, etc)

(A Quadrilateral is any shape with 4 sides)

90° + 90° + 90° + 90° = 360° 80° + 100° + 90° + 90° = 360°

A Square adds up to 360°Let's tilt a line by 10° ... still adds up to 360°!

The Interior Angles of a Quadrilateral add up to 360°

Because there are Two Triangles in a Square

The internal angles in this triangle add

up to 180°

(90°+45°+45°=180°)

... and for this square they add up to 360°

... because the square can be made from two triangles!

Pentagon

A pentagon has 5 sides, and can be made from three triangles, so you know what ...

... its internal angles add up to 3 × 180° = 540°

And if it is a regular pentagon (all angles the same), then each angle is 540° / 5 = 108°

(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's internal angles add up to 540°)

The General Rule

So, each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:

(Note: it is a Regular Polygon when all sides are equal, all angles are equal.)

If it is a Regular Polygon...

Shape SidesSum of

Internal AnglesShape Each Angle

Triangle 3 180° 60°

Quadrilateral 4 360° 90°

Pentagon 5 540° 108°

Hexagon 6 720° 120°

Heptagon (or Septagon)

7 900° 128.57...°

Octagon 8 1080° 135°

... ... .. ... ...

Any Polygon n (n-2) × 180° (n-2) × 180° / n

That last line can be a bit hard to understand, so let's have one example:

Example: What about a Regular Decagon (10 sides) ?

Sum of Internal Angles = (n-2) × 180°

= (10-2)×180° = 8×180° = 1440°

And it is a Regular Decagon so:

Each internal angle = 1440°/10 = 144°

Exterior Angles of Polygons

The Exterior Angle is the angle between any side of a shape, and a line extended from the next side.

Note: If you add up the Interior Angle and Exterior Angle you get a straight line, 180°. (See Supplementary Angles)

Polygons

A Polygon is any flat shape with straight sides

The Exterior Angles of a Polygon add up to 360°

In other words the exterior angles add up to one full revolution

Think of it this way: the lines change direction and eventually return back to the start.

(Exercise: try this with a square or some odd-shaped polygon)