This document discusses formulas for calculating the areas of various shapes, including circles, sectors, triangles, quadrilaterals, and other special shapes. It provides the area formulas, explains how to derive some formulas from rectangles or parallelograms, and includes examples of calculating areas using the appropriate formulas.

1. 5.13.1 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. area
rectangle area
formula
The number of square units that will
completely cover a shape without
overlapping
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. parallelograms
We can take any parallelogram and make a
rectangle out of it:
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 necessarily the
same as its height.)

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

5. 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

6. sector of a
circle
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
AAAA
G
Area of sector central angle
Area of circle 360
=
°
°
=
π °2
S m
r 360
° = π  
° 
2 m
S r
360

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

8. triangles
Like the 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:
= =
1 bh
A bh or A
2 2

9. trapezoids
A trapezoid is a little more complicated to
set up, but it also can be derived from a
parallelogram:
b1 + b2
b2
h
b1
b1
b2
h
( )
( )+
= + = 1 2
1 2
h b b1
A h b b or A
2 2
( )= +1 2A h b b

10. Rhombi,
squares, and
kites
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 =
(Squares can use the same formula.)
( )  = 
 
1 2 1 2
1 1 1
d d d d
2 2 4
  = 
 
1 2 1 2
1 1
2 d d d d
4 2

11. Example Find the d2 of a kite in which d1 = 12 in. and
the area = 96 in2.
= 1 2d d
A
2
= 212d
96
2
=26d 96
=2d 16 in.