Develop and use formulas to find the areas of circles and sectors. Develop and use formulas to find the areas of special quadrilaterals

1. Circles, Sectors, & Quads
The student is able to (I can):
• Develop and use formulas to find the areas of circles and
sectors
• Develop and use formulas to find the areas of special
quadrilaterals

2. areaareaareaarea – the number of square units that will completely cover
a shape without overlapping
rectangle arearectangle arearectangle arearectangle area formulaformulaformulaformula – one of the first area formulas you
learned was for a rectangle: A = bh, where b is the
length of the base of the rectangle and h is the height
of the rectangle.
b
h A = bh

3. We can take any parallelogram and make a rectangle out of
it:
parallelogram formulaparallelogram formulaparallelogram formulaparallelogram formula – the area formula of a parallelogram
is the same as the rectangle: A = bh
(Note: The main difference between these formulas is that
for a rectangle, the height is the same as the length of a side;
a parallelogram’s side is not necessarilynot necessarilynot necessarilynot necessarily the same as its
height.)

4. If you cut a circle into wedges, you can arrange the
wedges into a parallelogram-shaped figure:
radius
1
2
circumference
A = bh
1
circumference radius
2
A = i
( )
1
2 radius radius
2
A = πi i
2
A r= π

5. Examples
1. Find the exact area of a circle whose diameter is 18 in.
2. Find the diameter and area of a circle whose
circumference is 22π cm.
3. Find the radius of a circle whose area is 81π sq. ft.

6. Examples
1. Find the exact area of a circle whose diameter is 18 in.
A = πr2 = π(92) = 81π in2
2. Find the diameter and area of a circle whose
circumference is 22π cm.
22π = πd
d = 22 cm
A = π(112) = 121π cm2
3. Find the radius of a circle whose area is 81π sq. ft.
81π = πr2
81 = r2
r = 9 ft

7. sector of asector of asector of asector of a circlecirclecirclecircle – a region bounded by a central angle.
The area of a sector is proportional to the area of the circle
containing the sector.
Formula:
•
R
A
G
Area of sector central angle
Area of circle 360
=
°
2
360
S m
r
°
=
π °
2
360
m
S r
° 
= π  
° 

8. Examples: Find the area of each sector. Leave answers in
terms of π.
1.
2.
•
•
120º 2"2"2"2"
72º
10m10m10m10m

9. Examples: Find the area of each sector. Leave answers in
terms of π.
1.
2.
•
•
120º 2"2"2"2"
72º
10m10m10m10m
( )2 120
2
360
S
°
= π
°
4 120
360
 
= π 
 
i
24
in.
3
= π
( )2 72
10
360
S
° 
= π  
° 
7200
360
 
= π 
 
2
20 m= π

10. Like making a rectangle out of a parallelogram, we can use a
similar process to find out that the area of a triangle is one-
half that of a parallelogram with the same height and base:
triangle formulatriangle formulatriangle formulatriangle formula –
1
or
2 2
bh
A bh A= =

11. A trapezoid is a little more complicated to set up, but its
formula also can be derived from a parallelogram:
trapezoidstrapezoidstrapezoidstrapezoids –
b1 + b2
b2
h
b1
b1
b2
h
( )
( )1 2
1 2
1
or
2 2
h b b
A h b b A
+
= + =
( )1 2A h b b= +

12. A rhombus or kite can be split into two congruent triangles
along its diagonals (since the diagonals are perpendicular):
Area of one triangle =
Two triangles =
(Since they are rhombi, squares can use the same formula.)
( )1 2 1 2
1 1 1
2 2 4
d d d d
 
= 
 
1 2 1 2
1 1
2
4 2
d d d d
 
= 
 
Rhombi and KiteRhombi and KiteRhombi and KiteRhombi and Kite
formulaformulaformulaformula

13. Example: Find the d2 of a kite in which d1 = 12 in. and the
area = 96 in2.
1 2
2
2
2
2
12
96
2
6 96
16 in.
d d
A
d
d
d
=
=
=
=