Basic Mensuration and Geometry
Mensuration is the branch of mathematics that measures the boundary line and the surface of flat shapes and the capacity of solid objects, and in the ADRE Grade IV paper it appears as a small but dependable set of questions on the perimeter and area of rectangles, squares, triangles and circles, and sometimes on the volume of simple solids. The whole chapter rests on a handful of formulas, and once these are fixed in memory the questions become direct substitution. The method followed in the Master Book: Maths Masti (D. Hurina) is to write the given measurements with their units, identify the shape, and then pick the single formula that fits that shape, because almost every mensuration question is decided the moment the correct formula is chosen. A candidate who also keeps the units correct rarely loses a mark here.
Basic Terms
- Perimeter: the total length of the boundary of a closed figure.
- Area: the amount of surface enclosed by a figure, measured in square units.
- Rectangle: a four-sided figure with opposite sides equal and every angle a right angle.
- Square: a rectangle with all four sides equal.
- Triangle: a three-sided closed figure.
- Circle: a set of points at the same distance from a fixed centre.
- Radius and diameter: the distance from the centre to the edge, and twice that distance.
- Circumference: the perimeter of a circle.
- Volume: the space occupied by a solid, measured in cubic units.
Two quantities recur in every question, the linear measurement and the square measurement, and the candidate must always keep straight which one the question asks for. Perimeter is a length, area is a surface, and volume is a capacity.
Perimeter of Standard Figures
The perimeter of a rectangle is twice the sum of its length and breadth. The perimeter of a square is four times its side. The perimeter of a triangle is the sum of its three sides, and for an equilateral triangle it is three times the side. The circumference of a circle is twice pi times the radius, or pi times the diameter. The value of pi is taken as twenty-two sevenths or as three point one four, as the question requires.
Areas of Rectangles, Squares and Triangles
The area of a rectangle is length multiplied by breadth, and the area of a square is the side multiplied by itself. The area of a triangle is half the base multiplied by the height. For an equilateral triangle the area is root three by four multiplied by the side squared. In a right-angled triangle the two perpendicular sides serve as base and height.
Circle: Circumference and Area
The area of a circle is pi times the radius squared. The circumference is twice pi times the radius. A semicircle has half the area and a perimeter that adds the diameter to half the circumference. A ring between two circles has an area equal to pi times the difference of the squares of the outer and inner radii.
Quadrilaterals and Other Flat Figures
The area of a parallelogram is base multiplied by height. The area of a trapezium is half the sum of the parallel sides multiplied by the distance between them. The area of a rhombus is half the product of its diagonals. These three results cover nearly every flat figure that appears in the paper beyond the rectangle, square, triangle and circle.
Volume and Surface Area of Solids
The volume of a cuboid is length multiplied by breadth multiplied by height, and the volume of a cube is the side cubed. The volume of a cylinder is pi times the radius squared multiplied by the height. The curved surface of a cylinder is twice pi times the radius multiplied by the height, and its total surface adds the two circular ends. A sphere has volume four-thirds pi times the radius cubed, and a cone has volume one-third pi times the radius squared times the height.
Units and Conversions
Lengths are measured in millimetres, centimetres, metres and kilometres, areas in square units, and volumes in cubic units. One metre is one hundred centimetres, one kilometre is one thousand metres, one hectare is ten thousand square metres, and one litre is one thousand cubic centimetres. Converting the given measurements to a single unit before calculating prevents most mistakes.
Applications in Fields, Rooms and Materials
Mensuration questions are usually dressed as practical problems. Fencing a field asks for the perimeter, painting a wall or tiling a floor asks for the area, and filling a tank asks for the volume. Questions on the cost of ploughing, the number of tiles needed, or the length of wire around a garden are all direct applications of the same few formulas.
How This Chapter Appears in the Examination
In ADRE Grade IV, APSC and Assam Police papers the mensuration questions usually ask for the perimeter or area of a given figure, the circumference or area of a circle, the volume of a cuboid or cylinder, or the cost of covering a surface at a given rate. Data interpretation occasionally reports a figure per unit area. Because the formulas are few and fixed, this is one of the surest scoring sections.
Shortcut Methods for Rapid Calculation
Square a number ending in five by multiplying the leading part by the next integer and writing twenty-five after it. Use twenty-two sevenths for pi when the radius is a multiple of seven, and three point one four otherwise. Keep the common squares and cubes handy. In cost questions find the area first and multiply by the rate only at the end, so that a single rounding is needed.
Common Mistakes to Avoid
- Confusing perimeter with area, or reporting a length when a surface is asked.
- Forgetting to halve the base and height in the triangle formula.
- Using the diameter where the radius belongs, or the reverse.
- Mixing units such as centimetres and metres in the same calculation.
- Reporting the area in linear units instead of square units.
How to Prepare This Topic
- Learn the perimeter and area formulas for rectangle, square, triangle and circle.
- Memorise pi as twenty-two sevenths and three point one four and know when to use each.
- Practise parallelogram, trapezium and rhombus areas.
- Work on cuboid, cube and cylinder volume questions.
- Finish with unit-conversion and cost problems, and revise the formula list weekly.
Frequently Asked Questions
What is the difference between perimeter and area?
Perimeter is the length of the boundary and is measured in linear units. Area is the surface enclosed and is measured in square units. A fence needs the perimeter, while paint needs the area.
How do I decide the value of pi to use?
Use twenty-two sevenths when the radius or diameter is a multiple of seven, because the fraction cancels neatly. Otherwise use three point one four, as directed by the question.
Why is the area of a triangle half of base times height?
A triangle is half of a parallelogram on the same base and between the same parallels, so its area is half the base multiplied by the height.
How do I convert square metres to square centimetres?
One metre is one hundred centimetres, so one square metre is one hundred multiplied by one hundred, that is ten thousand square centimetres. Every length conversion is squared for area and cubed for volume.
What is the safest way to solve a cost question?
First convert every measurement to one unit, then find the total area or volume, and multiply by the given rate only in the last step. This keeps the arithmetic short and reduces rounding errors.