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Revision Notes on Mensuration

Mensuration

It is all about the measurement of area, perimeter and volume of the plane and solid figures.

Mensuration

Area

The surface covered by the border line of the figure is the area of the plain shape.

Unit of the area is square if the length unit.

Perimeter

The perimeter is the length of the boundary of the plane shape.

The unit of the perimeter is same as the length unit.

Perimeter

The green part is the area of the square and the distance all the way around the outside is the perimeter.

Area and Perimeter of Some 2D Shapes

Shape Image Area  Perimeter 
Square Square (Side)2 4 × Side
Rectangle Rectangle Length × Breadth 2(Length + Breadth)
Triangle Triangle (1/2) × Base × Height
where, a, b and c are the three sides of the triangle)
a + b + c
Parallelogram Parallelogram Base × Height 2(sum of adjacent sides)
Circle Circle πr2 2πr
Where, r = radius of the circle

Area of Trapezium

A trapezium is a quadrilateral whose two sides are parallel. And if its non-parallel sides are equal then it is said to be an isosceles trapezium.

Trapezium

Area of Trapezium can be found,

1. By Splitting the figure

One way to find the Area of trapezium is to divide it into two or three plane figures and then find the area.

Area of Trapezium

In the trapezium ABCD,

It can be divided into two parts i.e. a rectangle and a triangle.

Area of ABCD = Area of ABED + Area of DEC

2. By using formula

Another way is to calculate the area by using formula.

Formula of trapezium

Area of trapezium is half of the product of the summation of the parallel sides and the perpendicular distance between them.

Example

Find the area of the trapezium whose parallel sides are 6 cm and 16 cm, with a height of 5 cm. Calculate the area using both the methods.

Trapezium

Solution:

Area of trapezium

Splitting the trapezium we get –

Splitting the trapezium

Area of the trapezium = Area of rectangle + Area of a triangle

= (6 x 5) + (1/2) x 5 x 10

= 30 + 25

= 55 cm2

Remark: We should use the formula most of the time if possible as it is the quick and easy method.

Area of a General Quadrilateral

Quadrilateral

To find the area of any quadrilateral we can divide it into two triangles and then the area can be easily calculated by calculating the area of both the triangles separately.

Area of ABCD = Area of ∆ABC + Area of ∆ACD

= (1/2) × AC × h1 + (1/2) × AC× h2

The formula for the Area of a General Quadrilateral

Area of Quadrilateral

Where h1 and h2 are the height of both the triangles and d is the length of common diagonal i.e.AC.

Example

Find the area of quadrilateral ABCD.

Quadrilateral ABCD

Solution:

In the quadrilateral ABCD,

BD is the common diagonal so d = 5 cm.

Height of the two triangles are h1 = 2 cm and h= 1 cm.

Area of quadrilateral ABCD

Area of Special Quadrilaterals (Rhombus)

A rhombus is a quadrilateral with all the sides are equal and parallel but not the right angle. Its two diagonals are the perpendicular bisector to each other.

Area of Special Quadrilaterals

In this also we can split the rhombus into two triangles and can find the area of rhombus easily.

Formula of Area of Rhombus

Formula of Area of Rhombus

Area of rhombus is half of the product of its two diagonals.

Area of a Polygon

There is no particular formula for the area of the polygon so we need to divide it in a possible number of figures like a triangle, rectangle, trapezium and so on. By adding the area of all the split figures we will get the area of the required polygon.

Example

Find the area of the given octagon.

Octagon

Solution:

We can divide the given octagon into three parts.

Octagon into three parts

Two trapezium A and B and one rectangle shown by part B.

Two trapezium A and B and one rectangle shown by part B.
Area of A = Area of B = (1/2) × (a + b) × h

= (1/2) x (10 + 3) × 2

= 13 cm2.

Area of B = Length x Breadth

= 10 x 3

= 30 cm2.

So, the area of Octagon = 2A + B

= 2 × 13 + 30

= 56 cm2.

Solid Shapes

The 3-dimensional shapes which occupy some space are called solid shapes. Example- Cube, Cylinder, Sphere etc.

Solid Shapes

Surface Area

If we draw the net of the solid shape then we can see it’s all the faces clearly and if we add the areas of all the faces then we get the total surface area of that solid shape. The unit of surface area is a square unit.

Lateral or Curved Surface Area

If we leave the top and bottom faces of the solid shape then the area of the rest of the figure is the lateral surface of the shape. The unit of lateral surface area is a square unit.

Surface Area of Cube, Cuboid and Cylinder

Name Figure Lateral or Curved Surface Area Total Surface Area Nomenclature
Cube Cube 4l2 6l2 l = Edge of the cube
Cuboid Cuboid 2h(l + b) 2(lb + bh + lh) l = Length, b = Breadth, h = Height
Cylinder Cylinder 2πrh 2πr2+ 2πrh = 2πr(r + h)  r = Radius, h = Height

Volume

Volume is the space occupied by any solid figure i.e. the amount of capacity to carry something is the volume of that solid shape. The unit of volume is a cubic unit.

Volume of Cube, Cuboid and Cylinder

Name Volume Nomenclature
Cube l3 l = Edge of the cube
Cuboid lbh l = Length, b = Breadth, h = Height
Cylinder πr2h r = Radius, h = Height

Example 1

There is a shoe box whose length, breadth and height is 9 cm, 3 cm and 4 cm respectively. Find the surface area and volume of the shoe box.

shoe box

Solution:

Given,

length = 9 cm

Breadth = 3 cm

Height = 4 cm

Area of cuboid = 2(lb + bh + lh)

= 2(9 × 3 + 3×4 + 9 × 4)

= 2(27 + 12 + 36)

= 2(75)

= 150 cm2

Volume of cuboid = lbh

= 9 × 3 × 4

= 108 cm3

Example 2

If there is a cold drink can whose height is 7 cm and the radius of its round top is 3 cm then what will be the lateral surface area and volume of that cylinder? (π = 3.14)

Cylinder

Solution:

Given,

radius = 3 cm

Height = 7 cm

Lateral surface area of cylinder = 2πrh

= 2 × 3.14 × 3 × 7

= 131.88 cm2

Volume of cylinder = πr2h

= 3.14 × 3 × 3 × 7

= 197.82 cm3

Example 3

If there is a box of cube shape with the length of 4 cm then what will be the capacity of this box. Also, find the surface area of the box if it is open from the top.

Cube

Solution:

Given, side = 4 cm

Capacity or volume of the box = s3

= 43 = 64 cm3

The total surface area of the box = 6s2

But, if the box is open from the top then the surface area will be total surface area minus the area of one face of the cube.

Surface Area = Total Surface Area - Area of one face

= 6s2 – s2

= 5s2 = 5 × 42

= 80 cm2

Volume and Capacity

Volume and capacity are one and the same thing.

Volume is the amount of space occupied by a shape.

Capacity is the quantity that a container can hold.

 Capacity can be measured in form of liters.

We can see the relation between liter and cm3 as,

1 L = 1000 mL

1 mL = 1 cm3,

1 L = 1000 cm3.

Thus, 1 m3 = 1000000 cm3 = 1000 L.


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