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 and pyramids. Plane Geometry Plane 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). Plane
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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.
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.
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
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°).
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)
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.
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"):
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:
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
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:
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
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:
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.
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:
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!