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Type of Angle Description

Acute Angle

an angle that is less than 90°

Right Angle

an 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°

AS THE ANGLE INCREASES, THE NAME CHANGES

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.

TYPES OF ANGLES

1. Supplementary AnglesTwo 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°

2.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 TriangleIn 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.

3. 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, we can find an unknown angle.

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

4. Angles On One Side of A Straight LineAngles on one side of a straight line will always add to 180 degreesThis 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°

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

A triangle has three sides and three angles

The three angles always add to 180°

1. Triangles

Equilateral TriangleThree equal sidesThree equal angles, always 60°

Isosceles TriangleTwo equal sidesTwo equal angles

Scalene TriangleNo equal sidesNo equal angles

Equilateral, Isosceles and ScaleneThere 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:

What Type of Angle?Triangles can also have names that tell you what type of angle is inside:

Acute TriangleAll angles are less than 90°

Right TriangleHas a right angle (90°)

Obtuse TriangleHas an angle more than 90°

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

Right Isosceles TriangleHas a right angle (90°), and also two equal angles

Can you guess what the equal angles are?

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

Area

Example: What is the area of this triangle?

Height = h = 12Base = b = 20Area = bh/2 = 20 × 12 / 2 = 120

DefinitionIn fact the definition of a circle is:

The set of all points on a plane that are a fixed distance from a center.

2. circle

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

3. PolygonIs it a Polygon?

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Polygon(straight sides)

Not a Polygon(has a curve)

Not a Polygon(open, not closed)

Types of PolygonsSimple or ComplexA simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself!

Simple Polygon(this one's a Pentagon)

Complex Polygon(also a Pentagon)

Regular Irregular

Regular or IrregularIf all angles are equal and all sides are equal, then it is regular, otherwise it is irregular

Complex Polygon(a "star polygon",

inthis case, a pentagram)

Concave Octagon Irregular Hexagon

More Examples

    If it is a Regular Polygon...Name Sides Shape Interior Angle

Triangle (or Trigon) 3 60°

Quadrilateral (or Tetragon) 4 90°

Pentagon 5 108°Hexagon 6 120°

Heptagon (or Septagon) 7 128.571°

Octagon 8 135°Nonagon (or Enneagon) 9 140°

Decagon 10 144°Hendecagon (or

Undecagon) 11 147.273°

Dodecagon 12 150°

Names of Polygons

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

PolyhedronsA polyhedron is a solid with flat faces (from Greek poly- meaning "many" and -edron meaning "face").Each face is a polygon (a flat shape with straight sides).So, to be a polyhedron there should be no curved surfaces.Examples of Polyhedra:

Triangular Prism Cube Dodecahedron

Cube (Hexahedron) FactsNotice these interesting things:  It has 6 Faces  Each face has 4 edges, and is actually a square  It has 12 Edges  It has 8 Vertices (corner points)  and at each vertex 3 edges meet   And for reference:  Surface Area = 6 × (Edge Length)2

  Volume = (Edge Length)3

Counting Faces, Vertices and EdgesIf you count the number of faces (the flat surfaces), vertices (corner points), and edges of a polyhedron, you can discover an interesting thing:The number of faces plus the number of vertices minus the number of edges equals 2This can be written neatly as a little equation:F + V - E = 2It is known as the "Polyhedral Formula", and is very useful to make sure you have counted correctly!Let's try some examples:

This cube has: 6 Faces 8 Vertices

(corner points)

12 EdgesF + V - E = 6+8-12 = 2

Let's try some examples:

This prism has: 5 Faces 6 Vertices

(corner points) 9 Edges

F + V - E = 5+6-9 = 2

Cylinder FactsNotice these interesting things:  It has a flat base and a flat top

  The base is the same as the top, and also in-between

  It has one curved side

  Because it has a curved surface it is not a polyhedron.

   And for reference:  Surface Area = 2 × π × r × (r+h)

 

Surface Area of One End = π × r2

Surface Area of Side = 2 × π × r × h

  Volume = π × r2 × h

1. Cylinder

Volume of a CylinderJust multiply the area of the circle by the height of the cylinder:•Area of the circle: π × r2

•Height: h•Volume = Area × Height = π × r2 × h

Cylinder

Sphere FactsNotice these interesting things: It is perfectly symmetrical It has no edges or vertices (corners) It is not a polyhedron All points on the surface are the same distance from the center

2.Sphere

And for reference: Surface Area = 4 × π × r2 Volume = (4/3) × π × r3

Cone FactsNotice these interesting things:  It has a flat base  It has one curved side  Because it has a curved surface it is not a polyhedron.   And for reference:  Surface Area of Base = π × r2   Surface Area of Side = π × r × sor Surface Area of Side = π × r × √(r2+h2)     Volume = π × r2 × (h/3)

3. Cone

Torus FactsNotice these interesting things:

  It can be made by revolving a small circle along a line made by another circle.

  It has no edges or vertices  It is not a polyhedron And for reference:  Surface Area = 4 × π2 × R × r  Volume = 2 × π2 × R × r2

  Note: Area and volume formulas only work when the torus has a hole!

4. Torus