PERIMETERS AND AREAS
1.
PERIMETER OF POLYGONS
The Perimeter of a polygon is the distance around the outside of the polygon.
It is the sum of the lengths of all the sides.
Examples:
The perimeter of this rectangle is 7 + 3 + 7 + 3 = 20
The perimeter of that polygon is 1 + 5 + 4 + 2 + 7 = 19
The perimeter of this regular pentagon is
3 + 3 + 3 + 3 + 3 = 5×3 = 15
► EXERCISES :
1. Find the perimeter of that shapes:
a)
b)
c)
d)
e)
f)
g)
h)
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2. Find the perimeter of that shapes (Be careful, sometimes all the measurements are not
needed):
a)
b)
c)
e)
f)
g)
d)
h)
3. The perimeter of that hexagon is 48 cm. Find the length of the side of that hexagon.
4. The diagram shows a rectangle and a square.
If they have equal perimeters, what is the length of one side of the square?
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2.
AREA OF POLYGONS
Area is the size of a surface
Area is the measurement of the amount of space occupied by a closed flat surface and is
measured in square units. The most widely used units of area are mm2, cm2 and m2.
Area by Counting Squares
You can also put your shape on a grid and count the number of squares:
This rectangle has an area of 15
If each square was 1 cm on a side, then the area would be 15 cm2 (15 square cm)
Example:
These shapes all have the same area of 9
(But they do not have the same perimeter)
Sometimes the squares may not match the shape exactly, so you will need to "approximate" an
answer.
How can we find the area of shapes that are not regular?
At best we can only estimate the answer. One method of doing this is
to draw grid lines across the figure. Then we count all the full squares
and, as we do so, cross them out. Then count squares which are more
than half square unit as 1 (●), and those less than half square unit as 0.
So our estimate for the total area is 26 square units.
AREA OF A RECTANGLE
Different letters can be used:
A=b·a
Examples:
1. What is the area of this rectangle?
A = 5 · 3 = 15 cm2
This rectangle has an area of 15 cm2
2. A rectangle is 6 m wide and 3 m high, what is its Area?
Area = 6 m · 3 m = 18 m2
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► EXERCISES :
5. What is the area of this rectangle?:
6. What is the area of a rectangle having a length of 12 m and a width of 3'2 m?
AREA OF A SQUARE
A square is a special kind of rectangle, where the length is equal to the width.
A = a·a
A = a2
The letter s can be used for the side
Examples:
1. What is the area of this square?
Area = a2 = 32 = 9
This square has an area of 9 cm2
2. A square has a side length of 6 m, what is its Area?
Area = 6 m × 6 m = 62 m2 = 36 m2
► EXERCISES :
7. What is the area of this square?
8. A square has a side of 12 cm. Find its area.
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AREA OF A PARALLELOGRAM
LOOK:
We can cut a triangle from one side and paste it to the other side to make a rectangle with sidelengths b and h. The area of this rectangle is b · h.
Beware!
h is the height, not the side.
(h is at right angles to b)
A=b·h
Examples:
1. What is the area of this parallelogram?
Area = b · h = 8 · 5 = 40
2. A parallelogram has a base of 6 m and is 3 m high, what is its Area?
Area = 6 m × 3 m = 18 m2
► EXERCISES :
9. How much is the area of this parallelogram?
10. Find the area of this parallelogram:
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AREA OF A TRIANGLE
LOOK:
Take a second triangle identical to the first, then rotate it and "paste" it to the first triangle as
pictured below:
The figure formed is a parallelogram with base length b and height h, and has an area of b · h .
This area is twice that the area of the triangle.
So the area of the triangle is
A=
b⋅h
2
Examples:
1. What is the area of this triangle?
(Note: 6 is the height, not the length of the left-hand side)
Height = h = 6
Base = b = 5
Area =
b⋅h 5· 6
=
= 15
2
2
The base can be any side, just be sure the "height" is measured at
right angles to the "base":
2. What is the area of this triangle?
The height is 4
The base (at right angle) is 3 + 1 = 4
Area =
b⋅h 4 · 4
=
=8
2
2
The perpendicular height can be inside the triangle, one of its sides or outside of the triangle as
can be seen in the picture.
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► EXERCISES :
11. How much is the area of that triangle?
12. How much is the area of that triangle?
AREA OF A RHOMBUS
The area of a rhombus can be calculated as the area of a parallelogram
as we have seen before.
But it can also be calculated if we know the length of the diagonals.
The area of the rectangle is D · d
There are four equal triangles inside and outside the rhombus.
So the area of the rhombus is half the area of the rectangle.
A =
D· d
2
Examples:
1. What is the area of this rhombus?
Area =
D⋅d
30· 16
=
= 240
2
2
2. One of the diagonals of a rhombus is 10 m and the other one is 6 m. What is its Area?
Area =
D⋅d
10 · 6
=
= 30 m2
2
2
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► EXERCISES :
13. Find the area of the rhombus
AREA OF A TRAPEZIUM
LOOK:
To demonstrate this, consider two identical trapezoids, and "turn" one around and "paste" it to
the other along one side as it is drawn below:
The figure formed is a parallelogram having an area of
of one trapezium.
A=
A = (a + b)·h , which is twice the area
(a+b)⋅h
2
Examples:
1. What is the area of this trapezium?
Area =
(7.5+3.5)⋅4 11 · 4
=
= 22
2
2
2. The length of the parallel sides of a trapezium are 5 cm and 8 cm, and its height is 3 cm.
What is its Area?
(5+8)⋅3 13 · 3
Area =
=
= 19.5 cm2
2
2
► EXERCISES :
14. Find the area of a trapezium having bases 13 and 9 and a height of 6?
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AREA OF A REGULAR POLYGON
Breaking into Triangles
We can learn a lot about regular polygons by breaking them into triangles like this:
Notice that:
• the "base" of the triangle is one side of the polygon.
• the "height" of the triangle is the "Apothem" of the polygon.
Now, the area of a triangle is half of the base times height, so:
base · height
Side · Apothem
=
2
2
Area of one triangle =
To get the area of the whole polygon, just add up the areas of all the little triangles ("n" of them):
Area of Polygon =
n ·
And since the perimeter is
Area of Polygon =
n ·
Side · Apothem
2
n × side , we get:
Area of Polygon =
Perimeter · Apothem
2
Side · Apothem
Perimeter · Apothem
=
2
2
Example:
What is the area of this polygon?
The area of one triangle is
AT =
2' 5⋅3
= 3' 75 cm2
2
The area of the polygon is A P = 8 · AT = 8 · 3' 75 = 30 cm
2'5 cm
( A different way:
2'5 · 8 · 3 : 2 = 30 )
► EXERCISES :
15. Find the area of a that polygon
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2
► EXERCISES :
16. Find the area of the following shapes (you can break them in small shapes):
a)
e)
h)
b)
c)
f)
d)
g)
i)
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j)
17. How far will a runner travel if he runs 5 times around a triangular block with sides 320
m, 480 m and 610 m?
18. Find the cost of fencing a square block of land with side length 75 m if the fence costs
$14'50 per metre.
19. What is the perimeter of an equilateral triangle with 35'5 mm sides?
20. If the perimeter of a regular pentagon is 1'35 metres, what is the length of one side?
21. Find the length of the sides of a rhombus which has a perimeter of 72 metres.
22.
Find the area of the triangle
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23. Find the area of a rhombus if we know the length of the largest diagonal that is 12 cm
and the smaller is half of it.
24. Find the shaded area .
25. Find the perimeter and the area of that trapezium:
26. This trapezium has an area of 55 cm2 . Find the height
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3.
CIRCUMFERENCE AND CIRCLE
LENGTH OF A CIRCUMFERENCE
The circumference of a circle is the perimeter ... the distance around the outer edge.
Circumference = 2 π r
Circumference = π D
or
r is the radius
D is the Diameter
π = 3'141592.... ≈ 3'14
Examples:
1. What is the circumference of a circle with radius 5?
Circumference = 2 π r = 2 · 3'14 · 5 = 31'4
2. What is the perimeter of that circle ?
Perimeter = π · D = 3'14 · 12 = 37'68 cm
► EXERCISES :
27. Find the circumference of a circle with a radius of 3 cm
LENGTH OF AN ARC OF CIRCUMFERENCE
The length of an arc of the circumference of n degrees is
a =
r is the radius
2π r
· n
360
and
π ≈ 3'14
Example:
What is the length of an arc of 90º in a circumference with a radius of 10 cm?
a =
2π r
· n =
360
2 π 10
· 90 = 15 ' 7 cm
360
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► EXERCISES :
28. Find the length of an arc of 120º in a circumference with a radius of 5 cm
AREA OF A CIRCLE
The area of a circle is
r is the radius
and
Area = π r
π ≈ 3'14
Examples:
1. What is the area of a circle with a radius of 5 m ?
Area = π r 2 = 3'14 · 52 = 3'14 · 25 = 78'5 m2
2. What is the area of that circle ?
Area = π r 2 = 3'14 · 22 = 3'14 · 4 = 12'56 cm2
► EXERCISES :
29. Find the area of a circle with a radius of 7 cm
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2
30. Find the shaded area
AREA OF A SECTOR
The area of a sector of n degrees of a circle is
π r2
As =
· n
360
r is the radius
and
π ≈ 3'14
Example:
What is the area of a sector of 120º in a circle with a radius of 3 cm?
2
2
πr
3 '14 · 3
As =
· n =
· 120 = 9' 42cm 2
360
360
► EXERCISES :
31. Find the area of a sector of 200º in a circle of a radius of 10 m.
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32. Find the area and the perimeter of the shapes:
a)
b)
c)
d)
33. Find the area of the shaded area
34. Find the area of the shaded area. The diameter of the big circumference is 6 cm.
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