What is mensuration?

Mensuration is a branch of mathematics that deals with the measurement of areas and volumes of various geometrical figures. Figures such as cubes, cuboids, cylinders, cones, and spheres have volume and area. Mensuration deals with the development of formulas to measure their areas and volumes.

 

 

CUBES

A cube is a solid of uniform cross-section. It is formed by squares and has eight vertices. An example is processed cubed sugar.

The length of one side of a cube is ‘e’ which is the length of all sides since a cube is formed with squares.

 

TOTAL SURFACE AREA OF A CUBE

A cube has six faces. The surface area of each side = e² as each side is a square.

Therefore, Total surface area of a cube (all 6 sides) = 6es

The surface area of a cube is gotten by the formula.

Surface area= 6e² sq. units                 

                                                                                                                                         

VOLUME OF A CUBE

In a cube all sides are equal. Length=e, height=e and width=e

Therefore, Volume of a cube= length× width× height

Volume of a cube= e³ cubic units

CYLINDERS

A Cylinder is a uniform circular cross-section. Examples of cylinders are unsharpened pencils like HB or 2B pencils, garden rollers, tins of milk or tomato etc.

TOTAL SURFACE AREA OF A CYLINDER

There are two types of cylinders.

1. A closed cylinder
2. An open cylinder

TOTAL SURFACE AREA OF A CLOSED CYLINDER

The total surface area of a closed cylinder consists of a sum of the areas of (i) the curve surface and (ii) The two circular end faces.

The curved surface when opened out is a rectangle. This rectangle has length equal to the length of the Cylinder and the width are equal to the circumference of the circular end face.

Area of curved surface of a cylinder = area of rectangle of dimensions length (h) and width

(Circumference of base) = 2πrh

Area of the two circular end faces= twice the area of one circular face = πr2

Hence the total surface area of the closed cylinder = 2πrh + 2πr2 . Unit  = m3

THE TOTAL SURFACE AREA OF AN OPEN CYLINDER 

The total surface area of an open cylinder is the area of the curved surface which is the area of the rectangle the cylinder forms when spread.

Sometimes we are given a thick hollow cylinder. The total surface area is the sum of.

1. The area of the external curved surface
2. The area of the internal curved surface and
3. The area of the end annular faces which will be shaded.

VOLUME OF A CYLINDER

A right circular cylinder is a solid of uniform cross-section. If a paper is wrapped round a cylinder, on opening it, a rectangle will be found.

Thus, if the height of a cylinder = h units

And the base radius = r units

Then the volume of a cylinder = Area of base by height = πr2h. Unit = m3

TRIANGULAR PRISMS

 A prism is a solid with uniform cross-section of a shape of a triangle or a trapezium or any other polygon.

TOTAL SURFACE AREA OF A TRIANGULAR PRISM

In the case of a triangular shaped prism, the total surface area is the sum of the surface areas of the five faces that make up the prism.

VOLUME OF A TRIANGULAR PRISM

The volume of a prism is the area of its cross-section multiplied by the distance between the end faces. Examples are funnel, Chinese hat, cut periwinkle shell etc.

CONES

A cone is a figure with circular base and sides slanting to a common point or vertex. There are two types of cones (i) a right circular cone, where the line joining the vertex is symmetrical and perpendicular to the base of the cone and (ii) the non-right circular cone but this isn’t in the syllabus.

SURFACE AREA OF A CONE

Since a cone is formed from a sector of circle, then the surface area of a cone is equal to the area of the sector that formed it. Let L be the radius of the sector, then L becomes the slant height of the cone. If r is the radius of the base of the cone, then the length of arc of the sector is equal to 2pi.r which equals the circumference of the base of the cone.

If the sector subtends an angle which it always does, then the area of the sector will be  = θ /360 ×πl2 = curved surface of a cone but 2πr = length of arc of a sector.

Therefore, 2πr = θ /360 x 2πl

Finally, surface area of a cone (SA) = πrl

TOTAL SURFACE AREA OF A CONE

The total surface area of cone is the sum of

1. The curved surface area (CSA) πrl  
2. The area of the base of the cone πr2

ASSESSMENT

1. What is mensuration?
2.  What are the two types of cylinders?
3. What is the formula for the total surface area of a cone?
4. What is the formula for the volume of a cylinder?

Answer

1. Mensuration is a branch of mathematics that deals with the measurement of areas and volumes of various geometrical figures
2. 1. A closed cylinder
   2. An open cylinder
   
   
3. TSA = πr(r + l)
   
4.  V =  πr2h