Mensuration: An overview

Mensuration is a branch of geometry concerned with measuring lengths, areas, and volumes.

• A shape's perimeter is its complete length or route.

• The whole area covered by a given shape is referred to as its area.

• The overall space filled by a given shape is referred to as its volume.

 

Geometric shapes can be divided into two categories:

There are two types of shapes: open and closed. Closed geometric shapes are further divided into two types: two-dimensional and three-dimensional shapes.

 

Volume of a three-dimensional object

The volume of a three-dimensional object is the amount of space it takes up. It is a three-dimensional measurement.

A cuboid's volume

Where the cuboid's length is l, breadth is b, and height is denoted by h.

A cube's volume

Where l is the length of each edge of the cube.

A cylinder's volume

Where r is the radius of the base and h is the height of the cylinder.

Area of a rectangle: l × b;  perimeter of rectangle: 2(l + b); where l is length and b is breadth.

The region occupied by a rectangle inside its four sides or borders is known as the area of a rectangle. A rectangle's area is determined by its sides. In general, the area of a rectangle is equal to the product of its length and width.

The whole distance of a rectangle's outer boundary is its perimeter. Perimeter = 2 (length + width) is twice the sum of its length and breadth and is computed using the formula: Perimeter = 2(length + width).

 

Area of a square: a × a (where a is side); perimeter of square: 4a

The total number of unit squares in the shape of a square is the area of a square. To put it another way, it's the area occupied by the square.

The distance covered by a square's four sides is known as its perimeter. In a two-dimensional plane, the perimeter surrounds or borders the shape. Because all of a square's sides are equal, the perimeter of the square will be 4 times its side length or 4 sides.

 

Area of a triangle: ½ × base × height; perimeter of triangle: a + b + c.

In a two-dimensional plane, the total space filled by the three sides of a triangle is called its area. A triangle's area is half of the product of its base and height, so A = ½ × b × h is the basic formula. This formula applies to any triangle, including scalene triangles, isosceles triangles, and equilateral triangles. It's crucial to remember that the base and height of a triangle are perpendicular to one another. The area of a triangle is the area enclosed by the triangle's sides. The area of a triangle changes depending on the length of the sides and the internal angles of the triangle.

 

Area of a parallelogram: base × height;

the perimeter of parallelogram: 2(a + b); where a is side and b is base.

A parallelogram's area is the space enclosed by its four sides. A parallelogram's area is equal to the sum of its length and height. The interior angles of a quadrilateral sum up to 360 degrees. A parallelogram is made up of two pairs of parallel sides that are of equal length.

A parallelogram's perimeter is the whole distance outside of the geometrical shape. A parallelogram's opposite sides are equal, hence the perimeter is equal to the sum of two parallel sides, say a and b. 

The perimeter of a Parallelogram = 2 (a + b) units is the formula for calculating the perimeter of a parallelogram.

 

Area of a circle: πr²; perimeter of circle: 2πr; where r is the radius.

The area of the circle formula can be used to calculate how much space a circular field or plot takes up. If you have a plot and want to fence it, the area formula can help you figure out how much fencing you will need. 

A circle has no volume because it is a two-dimensional object. It merely has a perimeter and an area. As a result, we lack the volume of a circle.

The perimeter of a circle is its boundary, or the total arc length of its peripheral. The circumference of a circle is the term used to describe its perimeter.

A trapezium’s area is calculated by dividing it into shapes with simpler area formulae. Consider the trapezium, which has parallel sides a and b and a height of h. The trapezium is made up of three parts: two triangles and one rectangle.

Here h is the height, a and b are two parallel sides.

Area of trapezium =½ × h (sum of 2 parallel sides)

By calculating the area of a triangle of the same area, you can calculate the area of a trapezium. By separating the trapezium into a triangle and a polygon, the area may be calculated.

Consider a trapezium with the coordinates W X Y Z. Join A Z and mark a midway A for side X Y. A Z Y is obtained by cutting the trapezium along A Z. Place the A Z Y in the position described below. A triangle is now the new polygon.

Area of a trapezium= ½ ×(a+b)×h

A closed two-dimensional object with four sides or edges, as well as four corners or vertices, is known as a general quadrilateral. There are many different forms of quadrilaterals, all of which have four sides and a sum of angles of 360 degrees.

A general quadrilateral's area

Consider the ABCD quadrilateral. Draw AC in a diagonal direction. Draw perpendiculars h 1 and h 2 from B and D to AC.

Area of a quadrilateral= ½ ×d×(h1+h2),

where d is diagonal and h1,h2 are perpendicular drawn to a diagonal.

Area of a rhombus

Area of a rhombus =½ ×d1×d2,

where d1 and d2 are the diagonals.

Area of a polygon

Cutting a polygon into shapes whose areas are known and summing the areas of these shapes yields the area of the polygon.

The following are some of the ways to locate the location.

Area of this polygon = area of 2 trapeziums

Area of this polygon = area of 2 triangles + area of the rectangle

Area of this polygon = area of 4 triangles.

Solid forms

Solid shapes, also known as solid figures, are three-dimensional figures with dimensions of length, width, and height. Surface areas and volumes of these figures are calculated using these. The total surface area of a solid is equal to the sum of the areas of all of its faces or surfaces. The tops and bottoms (bases), as well as the remaining surfaces, make up the faces. The surface area of a solid without the bases is called the lateral surface area.

Solids with at least two identical faces

Solids with two faces that are identical are:

Solid shapes' surface area

The total area occupied by the object's surface is known as the surface area or surface area is the total area of all flat surfaces (called faces).

Cuboid’s surface area

Total surface area of cuboid =2(lb+bh+lh)

The lateral surface of an object is all of the sides of the object, excluding its base and top (when they exist). The area of the lateral surface is referred to as the lateral surface area.

Lateral Surface area of cuboid =2h(l+b)

Where l is the length, b is the breadth and h is the height.

Cube’s surface area

Total surface area of a cube =6l²

Lateral surface area of a cube  =4l²

Where l is the length of each side of the cube.

Cylinder’s surface area

Curved surface area of cylinder (C.S.A) =2πrh

Total surface area of cylinder(T.S.A) =2πr(r+h)

Where r is the radius of the cylinder and h is the height of the cylinder.

Volume and capacity relationship

An object's volume is the amount of space it takes up. The greatest measure of an object's ability to hold a substance, such as a solid, a liquid, or a gas, is measured in cubic units. Litres, gallons, pounds, and other units can all be used to calculate capacity.

For example, if a bucket holds 9 litres of water, its capacity is 9 litres.