Sarin has a string. The length is one metre. He likes to use the string to form different shapes. First, he ties the two ends of the string and makes the string into different shapes.
He can form the string to the shape of a square, a circle, a rectangle, a triangle, a hexagon and many… many…. different shapes…….
Perimeter and area
The string forms an enclosure or the boundary surrounding the formed shapes. It is called the perimeter of the shapes.
Since the string that Sarin used is measured one metre in length. The perimeter of the shapes that he formed will all measure one metre.
The space inside the surrounded or enclosed in the string is the size of the formed shapes and it is called the area of the shapes.
An area is a 2D measurement of a plane shape. It is use to indicate the size of a plane shape.
Units of measurement
Units of measurement of perimeter, length, width, height as well as radius and diameter can be kilo-meter (km), meter (m), centimeter (cm), millimeter (mm). These are 1D measurement.
Units of measurement of area can be kilo-meter square (km2), meter square (m2), centimeter square (cm2), millimeter square (mm2). These are 2D measurement.
Area of a rectangle = L × W
Sarin uses a string with a length of 1 meter to form the rectangle, so the perimeter of the rectangle is 1 m.
He makes the length of the rectangle as 1/3 m.
A rectangle has two equal Length and two equal Width.
Two times of the length = L + L = 1/3 + 1/3 = 2/3 m.
So two times of the width of the rectangle is 1 – 2/3 = 1/3 m
The width of the rectangle = 1/3 ÷ 2 = 1/6 m
The area of the rectangle he formed is L × W = 1/3 × 1/6 = 1/18 m2
A square is a special case of a rectangle,
Length of a square is equal to the width of the square. i.e. Length = Width or L = W
The perimeter of a square = L + L + L + L
= 4 × L
Area of a square = W × L = L × L
The perimeter of a triangle = L + L + L = 3 × L
The area of a triangle = ½ × L × W
Sarin also notice that, triangles that with same length and width have the same area in measurement and need not require be same in footprint.
Refer to the above diagram; the area of the orange triangle is equal to the area of the blue triangle.
Sarin now understands that the information of area needs to come together with the footprint to give an accurate and meaningful illustration of the school.
The area of a triangle = Half the area of the two triangles = ½ × L × W
Important concept: The areas of the triangles are the same as long as they have same length and same width as above diagrams illustrated.
The area of a circle = π × r × r = ¼ π × D × D
The perimeter of a circle = 2 × π × r = π × D
With r is the radius of the circle, or half the diameter (D) of the circle.
π is a constant value = 3.14159… or = 22/7
Diameter of a circle is a straight line that passing through the centre point of the circle and with both ends of the line end at the perimeter of the circle.
The perimeter of the circle also calls the circumference.
Question: Is the area of the different shapes having the same sizes or area with the same perimeter?
Sarin very much wanted to know the answer and he start his investigation by examine the rectangle and square that he formed using the 1 metre string.
A square form by 1 metre string. the perimeter = 1 m
Length of the square = ¼ m
The area of the square = ¼ × ¼ = 1/16 m2
The area of the rectangle = 1/18 m2
Sarin realized that the area of the square is not equal to the area of the rectangle even though they have the same perimeter.
1-unit square and grid table
We can also examine the question by using 1-unit square and the grid table.
A 1-unit square is a square with all sides are with length of 1 unit and the area of the square is 1 unit square.
A grid table is makes up of lots of square arranged in the array form.
If every single grid square in the grid table representing 1-unit square, we can determine the area of by counting the numbers of square.
We can use grid table to outline a shape and determine the area of the shape by counting the number of 1-unit squares.
The area of the shaded shape/figure on the above is 5-units square.
The perimeter of the shaded shape/figure is 9-units
The area of the shaded shape/figure on the above is 6-units square.
The perimeter of the shaded shape/figure is 9-unit
The areas are different but the perimeters are the same!!
You can also have same in areas but the perimeters are different.
Sarin needs a container at home to store water for washing. Before he goes to the hardware shop to purchase a container, he needs to make sure that he gets the container with the right size and shape.
He uses the string to form a square plane shape base on the shape and size of the base of the container so that he can determine the possibility of putting the container at the washing area. He also measures the height of the container so that to make sure he can know the volume of the water can be stored as well as fit the height limited at the washing area.
The size of the container that Sarin needs = L × W × H
The maximum volume of the water can be filled in the container = L × W × H
The volume is the quantity that fills or occupied a 3D object/figure. Example like solid shapes such as cube, prism, pyramid, cylinder etc… Liquid, gas, or, plasma can be express by volume as they occupy space. The volume of a container is generally understood to be the capacity or size of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Example: A container that half fill with water, the capacity or size of the container is 2 times the volume of the water in the container.
Units of measurement
Units of measurement of volume can be kilo-meter cubic (km3), meter cubic (m3), centimeter cubic (cm3), and millimeter cubic (mm3). These are 3D measurement.
We also use the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus
1 litre = (10 cm)3 = 1000 cm3 = 0.001 m3,
1 m3 = 1000 litres.
Small amounts of liquid are often measured in millilitres (ml), where
1 ml = 0.001 L = 1 cm3.
Volume of a Cuboid
The volume of a cuboid = L × W × H
Or, define B as area = L × W
The volume of a cuboid = B × H
Volume of a Cube
The volume of a cube = L × W × H
Since, L = W = H
The volume of a cube = L × L × L = 3 × L
A 1-unit cube, L = W = H = 1 unit
The volume of a 1-unit cube = 1 unit cubic
If a solid figure can be sub-divided into many 1-unit cubes, the volume of the figure can be determine by summation of the 1-unit cubes.
Number of 1-unit cubes = 5
If the volume of 1-unit cube = 1 cm3
The volume of the solid figure = 5 cm3
Volume of a Cylinder
A cylinder is a prism that both end is a same size circle.
Area B = π × r × r
Volume of cylinder = B × H = π × r × r × H
Volume of a irregular prism
Volume of a irregular prism = B × H
Take note that both end of the prism is same in shape and same in size.
If you have area (B) and the height (H), you can find the volume or capacity of the prism.
Important concept: The volume of a cuboid, a prism, or a cylinder is always the product of it base area and it measured height.
Volume of a pyramid
Volume of pyramid = 1/3 × L × W × H
= 1/3 × B × H
Note that the volume is always 1/3 of a prism with same base shape and size.
Volume of a Cone
Volume of a cone = 1/3 × B × H = 1/3 × π × r × r × r
Note that the volume of a cone is always 1/3 of the volume of a cylinder with same base shape and area
Important concept: The volume of a pyramid or a cone is always one third of the product of it base area and it measured height.
Volume of a sphere
Volume of a sphere = ¾ × π × r × r × r
= 3/32 × π × D × D × D