The perimeter of a plane shape is the length of its outside boundary or the distance around its edges.

**Irregular shape**

An irregular shape does not have a definite shape. To determine the perimeter of such shape, string or thread can be used to measure it. Place the string around the edge, then straighten it out and measure it with a ruler from the mark part.

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**Regular Shape**

A regular shape has a well-defined edge which may be straight lines or smooth curves. Examples are regular polygon and circles

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**The Unit of Measurement**

Perimeter is measured in length units. These are kilometres (km), metres (m), centimetres (cm) and millimetres (mm).

__Example 1 __

Use a ruler to measure the perimeter of triangle ABC.

Solutions

By measurement: AB: AB = 21mm, BC = 30mm, AC = 14mm

Perimeter =Total length of sides

= AB + BC +AC

=21mm+ 30mm +14mm

= 65mm

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**Using formulae to calculate perimeter**

Rectangles

The longer side of a rectangle is called the length and is usually represented by letter l. The shorter side is called the width or breadth and it may be represented by w ( or b).

AB = DC = lcm and AD = BC = bcm

Perimeter (P) = AB + BC + CD + DA = l + b + l + b

= 2l + 2b = 2(l + b)

P = 2 ( l + b)

Note: This is also used to determine the perimeter of a parallelogram

Example 1

The length of a rectangular room is 10m and the width is 6cm. Find the perimeter of the room.

Solution

Length of the room, l = 10m ; width/breadth of the room, w (or b) = 6m

Perimeter = 2(l +b) = 2 (10m + 6m)

= 2 ( 16m) = 32m

Example 2

Calculate the perimeter of a square whose length is 8cm.

Solution

A square has all its four sides equal, so each length is l cm.

The perimeter = l +l + l + l = 4l

= 4 8 = 32m

In general, perimeter of a square, P = 4l. This is also used to determine the perimeter of a rhombus

__Example 3__

A rectangle has a perimeter of 74m. Find: (a) the length of the rectangle if its breadth is 17m, (b) the breadth of the rectangle if its length is 25m.

__Solution__

Note: since perimeter of a rectangle = 2( l + b)

Length = ; Breadth =

So, to find the length

(a) Length =

= = 37m – 17m = 20m

(b) breadth=

= = 37m – 25m = 12m

**Evaluation**

- The perimeter of a square is 840cm. Find the length of the square in metres.
- A rectangle has sides of 9cm by 7.5cm. Find its perimeter
- Esther fences a 3m by 4m rectangular plot to keep her chickens in. The fencing costs N 200 per metre. How much does it cost to fence the plot?

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**Perimeter of triangles**

Isosceles triangle

The perimeter = a +a +b = 2a +b

Equilateral triangle

Perimeter = a + a + a = 3a

Example 4

An isosceles triangle has a perimeter of 250mm. If the length of one of the equal sides is 8cm, calculate the length of the unequal side.

Solution

First convert to the same unit of measurement

250mm = 25cm

Sum of equal sides = 8cm + 8cm = 16cm

The length of the unequal side = 25cm – 16cm = 9cm

** ****Trapezium**

The perimeter = p + q + r + s |
Isosceles trapezium The perimeter = a + b + a + c = 2a + b + c |

Example 5

An isosceles trapezium has a perimeter of 50cm if the sizes of the unequal parallel sides are 12cm and 8cm. Calculate the size of one of the equal sides.

Solution

Perimeter = 50cm

Perimeter of an isosceles triangle = 2 (equal sides) + b + c = 2x + 8 + 12

50 = 2x + 20

50 -20 = 2x + 20 – 20

2x = 30 ; x = 15cm

Therefore, one of the equal sides = 15cm

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**Perimeter of Circles **

The circumference (C) of a circle is the distance around the circle. This means that the circumference of a circle is the same as its perimeter.

AB = diameter, OA = OB = radii

But AB = OA + OB i.e. d = r + r

diameter , d = 2 radius (r) or radius, r = diameter (d)/ 2

The circumference, C of a circle is given by C =D, where D is the diameter of the circle. If R is the radius of the circle, then C = 2R.

Therefore, C = D or C = 2R

__Example 6__

Calculate the perimeter of a circle if its (a) diameter is 14cm (b) radius is 4.9cm (Take ).

__Solution__

- Diameter = 14cm

Perimeter , C = D = 14 = 44cm

- Radius= 4.9cm

Perimeter = 2R

= 2 4.9 = 30.8cm

__Example 7__

Calculate the perimeter of these figures. (Take ).

__Solution__

- A semicircle is half of a circle. The diameter = 3.15 cm

The perimeter of a circle = D = 3.15

= = 9.9cm

The length of the curved edge = = 4.95cm

The perimeter of the shape = 4.95cm + 3.15 cm = 8.1cm

- A quadrant is a quarter of a circle

The perimeter of a circle = 2R = 2 0.63 = 3.96m

The length of the curved edge = = 0.99m

Perimeter of the shape = 0.99m + 0.63m + 0.63m = 2.25m

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**Evaluation**

- Calculate the perimeter of a circle with radius 42cm. If a square has the same perimeter as the circle, calculate the length of one side of the square. (Take )
- The three sides of a triangle are ( x + 5)cm, ( 2x + 4 )cm and ( 2x -3)cm.

- Find the perimeter of the triangle in terms of x
- If x = 10, find the perimeter of the triangle

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**AREA OF PLANE SHAPES**

The area of a plane shape is a measure of the amount of surface it covers or occupies. Area is measured in square units, e.g. square metre (m^{2}), square millimetres (mm^{2}).

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**Finding the areas of regular shapes**

Area of Rectangles and Squares

A rectangle 5cm long by 3cm wide can be divided into squares of side 1cm as shown below.

By counting, the area of the rectangle is 15cm^{2}. If we multiply the length of the rectangle by its width the answer is also 15cm^{2} i.e. length X width = 5cm X 3cm = 15cm^{2}

In general, if A = area, l = length and w= width,

Area of a rectangle = length X width

Example 1

Calculate the area of a rectangle of length 6cm and width 3.5cm.

Solution

Area = length X width = 6cm X 3.5cm = 21cm^{2}

Example 2

The area of a rectangular carpet is 30m^{2}. Find the length of the shorter side in metres if the length of the longer side is 6000mm.

Solution^{ }

First convert the length i.e. 6000mm to metres

6000mm= = 6m

If A= area, l = length and b = breadth

Using breadth = ; breadth = = 5m

The length of the shorter side is 5m

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**Square**

A square has all its sides equal.

Area = ( length of one side)^{2} i.e. A = l^{2}

If Area, A is given then the length, l can be found by taking the root of both sides i.e. l = .

Example 3

Calculate the area of a square advertising board of length 5m.

Solution

Area of square board = l X l = 5m X 5m =25m^{2}

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**Area of shapes made from rectangles and squares**

Example 1

Calculate the area of the shape below. All measurements are in metres and all angles are right angles.

The shape can be divided into a 3X3 square, 6X10 and 2X4 rectangle.

Area of shape = Area of square + area of 2 rectangles

= ( (3X3) + (6X10) + (2 X4))m^{2}

= 9 + 60 + 8 = 77m^{2}

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**Area of parallelograms**

Area of a parallelogram = base X height

Example2

Calculate the area of a parallelogram if its base is 9.2cm and its height is 6cm.

Solution

Area of parallelogram = base X height = 9.2cm X 6cm = 55.2cm^{2}

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**Area of Triangles**

In general: Area of any triangle = base height i.e the area of a parallelogram (or rectangle that encloses it).

Example 1

Calculate the area of the triangle with base 6cm and height 4cm.

Solution

Base (b) = 6cm, Height (h) = 4cm

Area = base height = 5 4 = 10cm^{2}

__Example 2 __

Given that the area of triangle XYZ is 120cm^{2} and its height YD is 12cm. Find the length XZ.

Solution

Let the base XZ be bcm; Height, YD (i.e. h)= 12cm

Area of triangle XYZ= base height

120 = b 12

120= 6b

b = 20cm

the length XZ is 20cm.

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**Area of trapezium**

Area of trapezium =

Where (a + b) is the sum of the parallel sides and h, the height of trapezium.

Example

Calculate the area of trapezium with the dimensions shown in the figure below.

Solution

Area of trapezium =

= = = 168cm^{2}

**Area of Circles**

Area, A =r^{2} or A =

Example1

Find the area of a circle with radius 4.9cm (Take ).

Solution

Area of a circle = = 4.9^{2} cm^{2}

= 75.46cm^{2}

The area of the circle is 75.46cm^{2}r^{2}

Example 2

Find the area of a semicircle with diameter 20mm. (Take = 3.14)

Solution

Diameter, d = 20mm; Radius, r = 20/2 = 10mm

Area of a semicircle = = r^{2}

= 10^{2} = 157mm^{2}

Area of the semicircle = 157mm^{2}

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**Evaluation**

- A string is wound 30 times around a cylindrical object of diameter 7m. Calculate the length of the string. ( Take )
- A rectangular garden is 20m by 18m. Calculate the area of a path 1m wide going round the outer edge of the garden.

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**General Evaluation/Revision Questions**

- A regular polygon has all its sides …………… and all its angles …………..
- The distance around the circle is ………………………..
- What is the perimeter of a rhombus if the length of one side is 8cm?
- A circle of diameter 21cm has a perimeter of 66cm. If the circle is halved. Determine the perimeter of the half.

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**Weekend Assignment**

- What is the perimeter of a rectangle that measures 11cm by 3cm. (a) 39cm (b) 28cm (c) 36cm (d) 26cm
- The diameter of a circle is 13.8cm long. Find the length of its radius (a) 27.6cm (b) 7.6cm (c) 6.9cm (d) 6.4cm

- Two sides of an isosceles triangle are 3cm and 10cm. What must be the length of the third side? (a) 10cm (b) 6cm (c) 4cm (d) 8cm
- If the width of a rectangle is the equal to the length of a square and the rectangle measures 6cm by 4cm. What is the difference perimeter of the square? (a) 26cm (b) 16cm (c) 24cm (d) 36cm

- What is the difference in the perimeter of the rectangle and the square in question 4 above? (a) 4cm (b) 6cm (c) 8cm (d) 2cm

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**Theory**

- The diameter of a car wheel is 28cm, find its circumference. How far does the car move in metres when the wheel makes 150 turns? ( Take )
- (a) The longer side of a rectangle is 25cm and its perimeter is 80cm. Find the length of the shorter side. Determine its area (b) The area of a parallelogram is 8.5m
^{2}and its base is 500cm. Find its height.

**See also**