Measurement 1: Area and Perimeter
Mathematics
Topic Five

### Converting Units

 Kilometre Metres 1 1000 Metre Centimetre 1 100 Centimetre Millimetres 1 10 Metre Millimetres 1 1000 Kilograms Grams 1 1000 Grams Centigrams 1 100 Centigram Milligrams 1 10 Grams Milligrams 1 1000 Kilolitre Litres 1 1000 Litres Centilitres 1 100 Centilitre Millilitres 1 10 Litre Millilitres 1 1000

### Polygon

A polygon is a plane shape (two-dimensional) with straight sides. Examples include triangles, quadrilaterals, pentagons and hexagons. A regular polygon has:

• all sides equal and
• all angles equal

A quadrilateral is a closed figure made up of four straight edges and four corners.

The special types of quadrilaterals are listed below:

1. Parallelogram - A two-dimensional flat shaped closed figure made up of four sides where both pairs of opposite sides are parallel with same length is termed as parallelogram.
2. Rectangle - A two-dimensional flat shaped four sided closed figure made up of four sides and four right angles.
3. Rhombus - A two-dimensional flat shaped closed figure made up of four congruent sides.
4. Square - A two-dimensional flat shaped four sided closed figure made up of four equal sides and four right angles.
5. Trapezoid - A two-dimensional flat shaped four sided closed figure with at least one pair of parallel sides and each angle measures lesser than 180 °.
6. Kite - A two-dimensional flat shaped closed figure made up of four sides such that each pair of consecutive sides is congruent.

### Area

Area is a measure of how much space there is on a flat surface.

Rectangle

A rectangle is a four-sided shape whose corners are all ninety degree angles.  In a rectangle, two of the sides are equal (length) and the two other sides are equal (width).

The area of a rectangle is its length multiply by its width

A = Length x Width

Example

Find the area of a rectangle with length of 8cm and width of 6 cm.

Area = 8 x 6 = 48cm2

Square

A square is a flat shape defined by four points at the four corners. A square has four sides all of equal length, and four corners, all right angles (90 degree angles). All the sides of a square are equal.

The area of a square = side x side

Example

Find the area of a square with side of 7cm

Area = 7 x 7 = 49cm2

Triangle

A triangle is a polygon with three edges and three vertices.  A polygon is a plane shape with straight sides.

The area of a triangle = ½ base times height,

Example

Find the area of a  triangle with height of 10 cm and base of 6 cm.

Area = 1 /2 x 6  x 10 = 30cm2

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides.

The area of a parallelogram, A = bh

where, b is the length of the base of the parallelogram and h is the perpendicular height of the parallelogram.

Example

Find the area of a parallelogram with base of 9cm and height of 5cm.

Area = bh

= 9 x 5 = 45cm2

Trapezium

A trapezium is a quadrilateral with only one pair of parallel sides.

The area of a trapezium = 1/2 (a + b) h

where, a is the length of one parallel side of the trapezium and b is the length of the second parallel side of the trapezium.

### Kite

The area of a kite is given by the following formula where x and y are the lengths of the kite's diagonals:

A = 1/ 2 xy

where

AC = x

BD = y

Example

Find the area of a kite where:

AC = 24cm

and

BD = 16 cm

Solution

1/ 2 . (24) . (16)  = 384 / 2 = 192.

### Rhombus

A rhombus is a four-sided shape where all sides have equal length. Also opposite sides are parallel and opposite angles are equal.  It is a quadrilateral all of whose sides have the same length.  A rhombus is actually just a special type of parallelogram. Many of the area calculations can be applied to them also. Choose a formula based on the values you know to begin with.

1. The "base times height" method

First pick one side to be the base. Any one will do, they are all the same length. Then determine the altitude - the perpendicular distance from the chosen base to the opposite side. The area is the product of these two, or, as a formula:

Area = b x a

where
b is the length of the base
a is the altitude (height).

2. The "diagonals" method

Another simple formula for the area of a rhombus when you know the lengths of the diagonals. The area is half the product of the diagonals. As a formula:

Area = 1/ 2 d1d2

where
d1 is the length of a diagonal
d2 is the length of the other diagonal.

Example

Find the area of a rhombus with d1 being 8cm and d2 being 9 cm.

Solution

Area = 1/ 2 d1d2

= (8 x 9)/2

= 72/2

= 36 cm2

Circle

The area of a circle, A = πr2

where, r is the radius of the circle

and, π is 3.142 or 22/ 7

Example

Find the area of a circle with radius of 14cm.

Area = 22/ 7 x 14 x 14 = 616cm2

### Perimeter

Perimeter deals with the total distance around an object such as square or rectangle.

Rectangle

The perimeter of a rectangle, P = L + w + L + w

= 2(L + w)

where, L is the length of the rectangle and w is the width of the rectangle.

Square

Perimeter = s + s + s + s

Triangle

The perimeter of a triangle, P = the sum of all the sides.

Perimeter = a + b + c

Circle

The perimeter of a circle is called its circumference. The circumference of a circle, C = 2πr or πd where, r is the radius of the circle and d is the diameter of the circle with π being 22/ 7 or 3.142.

Example

Find the perimeter of a square with radius of 7cm.

Solution

Perimeter, P = 2 x 22/ 7 x 7

= 44cm

### Area of a Sector and Arc Length

The sector of a circle is the portion enclosed by two radii and an arc. The smaller area is called the minor sector and the larger area, the major sector.

The arc length of the minor sector (minor arc), is the portion of the circumference of the circle which spans the minor sector.

The area of the sector of a circle, A = πr2Ѳ/360

The length of the arc, L = 2πr x Ѳ/360

Example

If radius is 7cm and Ѳ = 80, find:

(i)  the area of the sector and,

(ii) the length of the arc.

Solution

(i)

Area = 22/7 x 7 x 7 x 80/360 = 34.2cm2

(ii) L = 2 x 3.14 x 7 x 80/360 = 9.8cm