# circles

A circle is a plane curve consisting of all points that have the same distance from a fixed point called center the common distance of the points on the curve from the center is called the radius. The region bounded by the circle is also often referred to as a circle (as when we speak of the area of a circle), the curve is referred to as the perimeter or the circumference of the circle.
Note:   that when the number of sides of a polygon is increased without limit, the sides merge into one line and polygon becomes a circle.

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# Words associated with circles

Here is a list of words associated with circles. How many do you know?

Put your mouse on each word to highlight an example.

You need to be familiar with the following abbreviations:

• C = circumference
• A = area
• d = diameter
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• The radius of the circle is a straight line drawn from the center to the boundary line or the circumference. The plural of the word radius is radii.

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## Diameter.

• The diameter is the line crossing the circle and passing through the center. It is the twice of the length of the radius.

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## Circumference.

• The circumference of a circle is the boundary line or the perimeter of the circle.

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## Chord

• The chord is a straight line joining two points on the circumference points of a circle. The diameter is a special kind of the chord passing through the center.

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## Arc

• An arc is a part of the circumference between two points or a continuous piece of a circle. The shorter arc between  and  is called the minor arc. The longer arc between  and  is called the major arc.

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## Semi-circle

• A semi-circle is an arc which is half of the circumference.

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## Tangent.

• A tangent is a straight line which touches the circle. It does not cut the circumference. The point at which it touches, is called the point of contact.

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## Area of shapes and their formulas

Rectangle:
Area = Length X Width
A = lw

Perimeter = 2 X Lengths + 2 X Widths
P = 2l + 2w

Parallelogram
Area = Base X Height
a = bh

Triangle
Area = 1/2 of the base X the height
a = 1/2 bh
Perimeter = a + b + c
(add the length of the three sides)

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# Area of a circle

There is only one formula for the area of a circle:

A = π r2

We must therefore remember to use the radius each time.

Cones
Volume = 1/3 pr2 x height
V= 1/3 pr2h
Surface = pr2 + prs
S = pr2 + prs
=
pr2 + pr

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## Area of shapes

Pyramid
V = 1/3 bh
b is the area of the base
Surface Area: Add the area of the base to the sum of the areas of all of the triangular faces. The areas of the triangular faces will have different formulas for different shaped bases

Sphere
Volume = 4/3 pr3
V = 4/3 pr3
Surface = 4pr2
S = 4pr2

Cylinder
Volume = pr2 x height
V = pr2 h
Surface = 2p radius x height
S = 2prh + 2pr2

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## Area of shapes.

Rectangular Solid
Volume = Length X Width X Height
V = lwh
Surface = 2lw + 2lh + 2wh

Prisms
Volume = Base X Height
v=bh
Surface = 2b + Ph (b is the area of the base P is the perimeter of the base)

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## simple area.

square = a 2

The area of a parallelogram:

To find the area of a parallelogram, just multiply the base length (b) times the height (h):

Area = b x h

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Area of rectangle= L x W

The area of a circle:

Area =
where r = the radius of the circle
and pi = 3.141592...

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# Angle properties of triangles

We already know that the angles in a triangle add up to 180°. Angles on a straight line also add up to 180°.

In the diagram, we see that:

a + b + c = 180° (angles in a triangle)

and

c + d = 180° (angles on a straight line).

If we rearrange both equations (subtract c from both sides), we get:

a + b = 180° - c and d = 180° - c.

Therefore, a + b and d must be the same (they are both equal to 180° - c):

a + b = d

Now look again at the diagram.

The exterior angle is equal to the sum of the two opposite interior angles.

This is true for any triangle.

Question

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## example

Find each angle marked with a letter, giving reasons for your answer.

Did you get the answer a = 50°? The exterior angle (120°) is equal to the sum of the two opposite interior angles (70° + a).

Therefore, 70° + a = 120°, so a = 50°.

b = 60°, because the angles on a straight line add up to 180°.

b + 120° = 180°, so b = 60°.

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## Regular and Irregular polygons.

The simplest polygon is a triangle (a 3-sided shape). Polygons of all types can be regular or irregular.

A regular polygon has sides of equal length, and all its interior angles are of equal size.

Irregular polygons can have sides of any length and angles of any size.

Examples of polygons:

• Triangle
• Pentagon
• Hexagon
• octagon
• Decagon
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## Angle properties of polygons

In your exam, you might be asked to find angles of polygons.

The formula for calculating the sum of the interior angles of a regular polygon is: (n - 2) × 180° where n is the number of sides of the polygon.

This formula comes from dividing the polygon up into triangles using full diagonals.

We already know that the interior angles of a triangle add up to 180°. For any polygon, count up how many triangles it can be split into. Then multiply the number of triangles by 180.

This quadrilateral has been divided into two triangles, so the interior angles add up to 2 × 180 = 360°.

This pentagon has been divided into three triangles, so the interior angles add up to 3 × 180 = 540°.

In the same way, a hexagon can be divided into 4 triangles, a 7-sided polygon into 5 triangles etc.

Can you see the pattern forming? The number of triangles is equal to the number of sides minus 2.

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## Angle properties of polygons

The exterior angle of a polygon and its corresponding interior angle always add up to 180° (because they make a straight line).

For any polygon, the sum of its exterior angles is 360°.

(You can see this because if you imagine 'walking' all the way round the outside of a polygon you make one full turn.)

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# Calculating the interior and exterior angles of regular polygons

## Finding the interior angle

We already know how to find the sum of the interior angles of a polygon using the formula

(n - 2) × 180°

We also know that all the interior angles of a regular polygon are equal.

Interior angle of a regular polygon = sum of interior angles ÷ number of sides

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# Calculating the number of sides in a regular polygon, given the interior angle

• The interior and exterior angles add up to 180° (a straight line - eg, a + f = 180°), and
• The sum of the exterior angles is 360° (a + b + c + d + e = 360°).
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