Circles and sphere

Sphere & Hemisphere

sphere

• A 3-dimensional object shaped like a ball.

• Every point on the surface is the same distance from the center.

Hemisphere

• half of a sphere

Surface area of spheres

surface area = 4 π r2

• Example 1

If 'r' = 5 for a given sphere, and π = 3.14, then the surface area of the sphere is:

surface area = 4 π r2

= 4 × 3.14 × 52

= 314

Volume of spheres

Question 1: Calculate the volume of a sphere of radius 13 cm ? Solution:Given,r = 13 cm

Volume of a sphere= (4/3)πr3

= (4/3) π (13 cm)3

= 9202.772 cm3

Volume = (4/3) × π × r3

Surface area of hemisphereTotal surface area of a hemisphere:

S = (2πr2) + (πr2)

S = 3πr2

Curve area

Circle

Question 1: Find the radius of a hemisphere where the total surface area is 1846.32 cm

Solution:The total surface area of the hemisphere = 1846.32 cm2

We know, total surface area of a hemisphere = 3 π r2 square units.

Answer

Therefore,

3 π r2 = 1846.323 ( 3.14 ) r2 = 1846.32

9.42 r2 = 1846.32

r2 = 1814.929.42

r2 = 192.67

Radius of the hemisphere, r = 14 cm

Volume of hemisphere

• Volume of a hemisphere:V = (1/2)(4/3)πr3

= (2/3)πr3

• Example: Calculate the volume of the hemisphere with a radius of 3cm.Answer:Volume of Hemisphere: = (2/3)πr³= 56.55 cm

SECTOR

CIRCLES• The circle is the shape with the largest area for a given length of perimeter.

•A circle is a plane figure bounded by one line, and such that all right lines

drawn from a certain point within it to the bounding line, are equal.

• The Bounding line is called circumference.

c

Center

Center

Circumference

C π = = 3.14 22 7

AREA

π = = 3.14 22 7

• The AREA is the amount inside the shape.

Example :

Find the Area

A = πrA = 3.14 x 3

= 28.26 cm

2

2

2

12cm x 6cm = 72cm2

72cm + 28.26cm = 100.26cm2 2 2

Trigonometry:Arc Length and Radian Measure

An arc of a circle is a “portion” of the circumference of

the circle.

The arc length is the length of its “portion” of the

circumference.

Arc Length (Radian)

Arc Length (Degree)

When θ is in degree form

The arc length of circle:

S = r x (θ x )

Convert to

radian form

How to convert?

2π radians = 360 degrees

π radians=180 degrees

1 radian= 180/ π degrees 1 degree= π/180 rad

To convert

From degrees to radians

x

To convert

From radians to degrees

x

Area of Sector

Sector of a Circle

Definition:

The part of a circle enclosed by two radii of a circle and

their intercepted arc. A pie-shaped part of a circle.

Semi-circle(half of circle = half of area)

Quarter-Circle(1/4 of circle = 1/4 of area)

Any Sector(fractional part of the area)

where n is the number of

degrees in the central angle of

the sector.

where CS is the arc length of the sector.

Area of Sector

Area of Segment

Definition:

The segment of a circle is the region bounded by a

chord and the arc subtended by the chord.

Example:

Find the area of a segment of a circle with a central angle of 120 degrees and a radius of 8 Express answer to nearest integer.

Solution:

Start by finding the area of the sector: