Circle is one of familiar shape in daily life. It was studied since long time ago. There are many things that is circle such as clock, wheel, gear, ring, coins, etc..
Table of Contents
Definition of Circle
Circle is two-dimensional shape. It is set of points that has same distane from the center in a plane. In another words, there are unlimited points that has same distance from the center or unlimited radius in a circle. Total angle of a circle is 360°.
Parts of Circle
There are some terms in parts of circle:
Center of a Circle
Center of a circle means a point in the middle of circle.
Radius of a Circle
Radius of a circle means distance from the center outwards.
Diameter of a Circle
Diameter means a line that trough the center and it across the circle.
Look at radius and diameter. Diameter (d) is twice of radius (r).
In another words, radius (r) is a half of diameter (d).
d = 2.r ↔ r = ½.d
Besides that, circle is also identic with phi (p = 22/7 = 3,14159…).
It is because the ratio of perimeter (circumference) and diameter is 3,14159… (phi value).
Annulus
Annulus means area in between two circles. One of daily life thing is tire. It is area between the rubber and the rim of tire.
Tangent
Tangent is a line that touching the circle in a point.
Chord
Chord means a line that lying in a circle.
Secant (extended chord)
Secant is line that cut the circle at two points.
Arc
Arc means connected curve of a circle.
Sector
Sector is an area that bounded by two radii (radius) and an arc.
Circle divide the region to be two parts. There are interior and exterior.
Area of a Circle
Area of a circle means area that is bordered by points that has same distance to the center.
The area of a circle is
A = pr2
Because of diameter (d) = 2r, then
A = p.( ½ d)2
or
A = ¼.d2. p
Perimeter of a Circle
Perimeter is also called as circumference. Perimeter means total distance of a circle in around.
The perimeter of a circle is
P = 2p.r
Because of diameter (d) = 2r, then
P = p.d
Circle Equation
Circle can be drawn in cartesian coordinate system. The center is in point (a,b), radius is “r”, and (x,y) as the points that is through by the circle,
then the standard equation of the circle is
(x – a)2 + (y – b)2 = r2
graph for (x-2)2 + (y-1)2 = 16
center (2,1) and r = √16 = 4
There is special case if the circle lay on the center of cartesian coordinate system and the center is in (0,0), then the equation will be
(x – a)2 + (y – b)2 = r2
(x – 0)2 + (y – 0)2 = r2
x2 + y2 = r2
graph for x2 + y2 = 9
Besides that, there is another form of circle equation. It comes from first equation. It is called as general form.
(x – a)2 + (y – b)2 = r2
x2 – 2ax + a2 + y2 – 2by + b2 = r2
x2 + y2 -2ax – 2 by + a2 + b2 – r2 = 0
because of
-2a = A
-2b = B
a2 + b2 – r2 = C
then it will be
x2 + y2 + Ax + By + C = 0
the center is (-A/2, -B/2)
There is formula to determine the radius in general equation of a circle.
r = √[(- ½ a)2 + (- ½ b)2 – C] = √[ ¼ A2 + ¼ B2 – C]
Properties of a Circle
There are properties of a circle.
- Circle is divided by diameter to be two equal parts.
- Arc of a circle has same distance from the center.
- Circles will be congruent if they have equal radius.
- Circles that has different radius has different size.
- Diameter of a circle is twice of radius (radius is a half of diameter).
- Diameter of a circle is the largest chord.
Examples
1. Determine the area of a circle that has diameter 14cm.
d = 14
it will be easier using phi 22/7 than 3,14 because of d is multiple of 7.
A = p . r2
= (22/7). 142
= 616 cm2
2. Determine the differences of the perimeter of two circles if the first has r = 5 cm and second r = 8 cm.
r1 = 5
r2 = 8
then
P1 = 2p.r1
= 2(3,14).5
= 31,4 cm
P2 = 2p.r2
= 2(3,14).8
= 50,24 cm
Then the differences is 50,24 – 31,4 = 18,84 cm.
3. The perimeter of a circle is 31,4 cm then determine the radius.
P = 31,4
Then
P = 2pr
31,4 = 2.(22/7).r
219,8 = 44r
r = 4,9954 » 5 cm.
4. Determine the center and the diameter of circle (x-3)2 + (y+1)2 = 12.
Equation of the circle is (x-3)2 + (y+1)2 = 12
Let equation of a circle is (x-a)2 + (y-b)2 = r2.
Then the center is (a,b) = (3,-1)
And the radius is √12 = 2√3.
So the diameter is d = 2 . 2√3 = 4√3
5. There is a circle that has center in (-2,4) and the diameter is 6. Determine the general form of the circle.
Center (-2,4)
Diameter = 6 → r = 3.
Then
(x-(-2))2 + (y-4)2 = 32
(x+2)2 + (y-4)2 = 32
x2 + 4x + 4 + y2 – 8y + 16 = 9
x2 + y2 + 4x – 8y + 4 + 16 -9 = 0
x2 + y2 + 4x – 8y + 11 = 0
6. Determine the center and radius of the circle if the general equation is x2 + y2 – 12x – 16y + 19 = 0.
The general equation is x2 + y2 – 12x – 16y + 19 = 0
Let the general equation of circle is x2 + y2 + Ax + By + C = 0
Then
A = -12
B = -16
C = 19
So the center is (a,b) = (-A/2,-B/2) = (6,8)
The radius is
r = √[ ¼ A2 + ¼ B2 – C]
= √[ ¼ (-12)2 + ¼ (-16)2 – 19]
= 9.