maths

1. Circles and sphere

2. 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

3. 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

4. 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

5. Surface area of hemisphere
Total 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.

6. Answer
Therefore,
3 π r2 = 1846.32
3 ( 3.14 ) r2 =1846.32
9.42 r2 = 1846.32
r2 = 1814.929.42
r2 = 192.67
Radius of the hemisphere, r = 14 cm

7. 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

8. 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

9. Circumference
C π = = 3.14
22
7

10. AREA
π = = 3.14
22
7
• The AREA is the amount inside the shape.
Example :
Find the Area
A = πr
A = 3.14 x 3
= 28.26 cm
2
2
2
12cm x 6cm = 72cm
2
72cm + 28.26cm = 100.26cm2 2 2

11. 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.

12. Arc Length (Radian)

13. Arc Length (Degree)
When θ is in degree form
The arc length of circle:
S = r x (θ x )
Convert to
radian form

14. 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

15. 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.

16. 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
angle of the
sector in
degree
Area of Sector
where θ is angle
of sector
in radian form

17. Area of Segment
Definition:
The segment of a circle is the region bounded by a
chord and the arc subtended by the chord.

18. 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.

19. Solution:
Start by finding the area of the sector: