Mensuration

In mathematics, measurement is the branch that deals with the measurement of geometrical figures and their parameters such as length, volume, area, perimeter, etc. It deals with both two-dimensional (2D) and three-dimensional (3D) figures.

Basic concepts of mensuration

Length

Length is a measure of distance. For example, the length of a pencil tells how long the pencil is.

Circumference

The perimeter of a shape is the total distance around that shape. Imagine you are putting a fence around a park, then the total length of the fence will be the perimeter.

Area

Area is the size of a surface. It measures the amount of space inside a shape. For example, if you want to cover a floor with tiles, the number of tiles needed depends on the area of the floor.

Volume

Volume measures the amount of space an object occupies. Consider a fish tank: its volume determines how much water it can hold.

Now, let's look at how we measure these properties for different shapes.

Measuring two-dimensional shapes

Rectangle

A rectangle is a four-sided shape whose opposite sides are equal and all angles are right angles.

Perimeter of a rectangle:

Perimeter = 2 × (Length + Width)

Example: If the length is 8 units and the width is 4 units, then:

Perimeter = 2 × (8 + 4) = 24 units

Area of a rectangle:

Area = Length × Width

Example: For the same rectangle with length = 8 units and width = 4 units:

Area = 8 × 4 = 32 square units

Square

A square is a rectangle whose all sides are equal.

Perimeter of a square:

Perimeter = 4 × Side

Example: If one side of the square is 6 units, then:

Perimeter = 4 × 6 = 24 units

Area of a square:

Area = Side × Side

Example: For a square with side 6 units:

Area = 6 × 6 = 36 square units

Triangle

A triangle has three sides and three angles. There are different types of triangles depending upon the length of the sides and angles. Here we will discuss the formula of a general triangle.

Perimeter of a triangle:

Perimeter = Side1 + Side2 + Side3

Example: If the lengths of the sides of a triangle are 3 units, 4 units and 5 units, then:

Perimeter = 3 + 4 + 5 = 12 units

Area of the triangle:

The most common formula for the area of a triangle uses the base and height:

Area = (Base × Height) / 2

Example: For a triangle with base = 5 units and height = 3 units:

Area = (5 × 3) / 2 = 7.5 square units

Circle

A circle is a group of points located at equal distance from a central point.

Circumference of a circle:

Circumference = 2 × π × Radius

Example: If the radius of the circle is 7 units, then assuming π is approximately 3.14:

Circumference = 2 × 3.14 × 7 = 43.96 units

Area of a circle:

Area = π × Radius × Radius

Example: For the same circle with radius = 7 units:

Area = 3.14 × 7 × 7 = 153.86 square units

Measuring three-dimensional shapes

Cuboid

A cuboid is a box-shaped 3D figure with six rectangular faces, and each pair of opposite faces is identical.

Surface area of cuboid:

Surface Area = 2 × (Length × Width + Width × Height + Height × Length)

Example: If the length is 3 units, width is 4 units, and height is 5 units:

Surface Area = 2 × (3 × 4 + 4 × 5 + 5 × 3) = 94 square units

Volume of a cuboid:

Volume = Length × Width × Height

Example: Using the same dimensions:

Volume = 3 × 4 × 5 = 60 cubic units

Cube

A cube is a special type of cuboid in which all sides are equal.

Surface area of a cube:

Surface Area = 6 × Side × Side

Example: If the side of the cube is 4 units:

Surface Area = 6 × 4 × 4 = 96 square units

Volume of a cube:

Volume = Side × Side × Side

Example: For the same cube:

Volume = 4 × 4 × 4 = 64 cubic units

Cylinder

A cylinder has a circular base and a certain height. It is like a can.

Surface area of a cylinder:

Total Surface Area = 2 × π × Radius × (Height + Radius)

Example: For a cylinder of radius 3 units and height 6 units:

Total Surface Area = 2 × 3.14 × 3 × (6 + 3) = 169.56 square units

Volume of a cylinder:

Volume = π × Radius × Radius × Height

Example: Using the same cylinder dimensions:

Volume = 3.14 × 3 × 3 × 6 = 169.56 cubic units

Applications of mensuration

Measurement has countless applications in everyday life, whether you are planning to paint a wall and want to calculate the amount of paint needed, or you are checking the space capacity of various objects and containers.

Example 1: Painting a wall

Let's say you want to paint a rectangular wall 10 meters long and 3 meters high. You need to calculate the area to know how much paint to buy:

Area = Length × Height = 10 × 3 = 30 square meters

Example 2: Filling a swimming pool

You have a rectangular swimming pool 25 m long, 10 m wide, and 2 m deep. To find the volume of the pool and the amount of water it can hold:

Volume = Length × Width × Depth = 25 × 10 × 2 = 500 cubic meters

Example 3: Construction of a shed

You want to build a shed with a rectangular floor plan. If the floor is 8 m long and 5 m wide, and you want to fence around it, you need to know the perimeter:

Perimeter = 2 × (Length + Width) = 2 × (8 + 5) = 26 meters

Conclusion

Measurement is an essential topic in math that has a lot of use in the real world. Understanding how to calculate the perimeter, area, and volume of different shapes makes it easier to tackle practical problems in many fields, from construction to logistics and beyond.

Comments