Geometry

Geometry 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, cylinders and spheres.

Hint: Try drawing some of the shapes and angles as you learn ... it helps.

Solid Geometry

Solid Geometry is the geometry of three-dimensional space,
the kind of space we live in ...

Three Dimensions

It is called three-dimensional, or3D because there are three dimensions: width, depth andheight.

Simple Shapes

Let us start with some of the simplest shapes:

Common 3D Shapes

Properties

Solids have properties (special things about them), such as:

volume (think of how much water it could hold)
surface area (think of the area you would have to paint)
how many vertices (corner points), faces and edges they have

Polyhedra and Non-Polyhedra

There are two main types of solids, "Polyhedra", and "Non-Polyhedra":

Polyhedra :
(they must have flat faces)

	Cubes and Cuboids (Volume of a Cuboid)
	Platonic Solids
	Prisms
	Pyramids

Non-Polyhedra:
(if any surface is not flat)

	Sphere		Torus
	Cylinder		Cone

Euler's Formula

(There is another "Euler's Formula" about complex numbers,
this page is about the one used in Geometry and Graphs)

Euler's Formula

For any polyhedron that doesn't intersect itself, the

Number of Faces
plus the Number of Vertices (corner points)
minus the Number of Edges

always equals 2

This can be written: F + V − E = 2

Try it on the cube:

A cube has 6 Faces, 8 Vertices, and 12 Edges,

so:

6 + 8 − 12 = 2

To see why this works, imagine taking the cube and adding an edge (say from corner to corner of one face). We get an extra edge, plus an extra face: 7 + 8 − 13 = 2

Likewise when we include another vertex (say halfway along a line) we get an extra edge, too. 6 + 9 − 13 = 2. "No matter what we do, we always end up with 2" (But only for this type of Polyhedron ... read on!)

Example With Platonic Solids

Let's try with the 5 Platonic Solids (Note: Euler's Formula can be used to prove that there are only 5 Platonic Solids):

Name	Faces	Vertices	Edges	F+V-E
Tetrahedron	4	4	6	2
Cube	6	8	12	2
Octahedron	8	6	12	2
Dodecahedron	12	20	30	2
Icosahedron	20	12	30	2

The Sphere

All Platonic Solids (and many other solids) are like a Sphere ... we can reshape them so that they become a Sphere (move their corner points, then curve their faces a bit).

For this reason we know that F + V − E = 2 for a sphere

(Be careful, we can not simply say a sphere has 1 face, and 0 vertices and edges, for F+V−E=1)

Point, Line, Plane and Solid

A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)

2D Shapes

Regular Polygons

A polygon is a plane (2D) shape with straight sides.
To be a regular polygon all the sides and angles must be the same:

Triangle - 3 Sides	Square - 4 Sides
Pentagon - 5 Sides	Hexagon - 6 sides
Heptagon - 7 Sides	Octagon - 8 Sides
Nonagon - 9 Sides	Decagon - 10 Sides

Other Common Polygons

Quadrilateral Any 4 sided 2D shape	Rectangle - 4 Sides All right angles
And many more!

Curved Shapes

These 2D shapes have curves, so are not polygons:

Circle - 1 Side
Ellipse - 1 Side

And many more!

Quadrilaterals

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

A Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join up), and has straight sides.

Properties

Four sides (edges)

Four vertices (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 andrectangle 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) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles.

A rhombus is sometimes called a rhomb or a diamond.

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

A parallelogram has opposite sides parallel and equal in length. Also 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 a 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.

And a trapezium (UK: trapezoid) is a quadrilateral with NO parallel sides:

Trapezoid Trapezium

In the US: a pair of parallel sides NO parallel sides

In the UK: NO parallel sides a pair of parallel sides

(the US and UK definitions are swapped over!)

The Kite

Hey, it looks like a kite. It has two pairs of sides.

Each pair is made up of adjacent sides (they meet) 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 "Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.")

This may seem odd, as 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.

Perimeter

Perimeter is the distance around a two-dimensional shape.

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

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

The perimeter of a circle is called the circumference:

Circumference = 2π × radius

Perimeter Formulas

		Triangle Perimeter = a + b + c
		Square Perimeter = 4 × a a = length of side
		Rectangle Perimeter = 2 × (w + h) w = width h = height
		Quadrilateral Perimeter = a + b + c + d
		Circle Circumference = 2πr r = radius
		Sector Perimeter = r(θ+2) r = radius θ = angle in radians
		Ellipse Perimeter = very hard!