What is meant by area computation?

The area is the mathematical estimation of the inner space occupied by any shape or object and area computation is the evaluation of how much inner space that particular shape or object has utilized. Simple applications include calculating the area of wall for the quantity of paint or plaster to be applied on it, area calculations for construction of a trapezoidal dam and how much material is required, calculating the area of a circle to determine the range or distance of wheels that they are capable of covering due to varied radii and so on.

The SI unit of area is square meters (m²) and the dimensional unit of area is [L²].

Formulae used for area calculations based on geometry

Shape	Formula	Annotation
Area of a Triangle	Area = ½ x ab sin C Area = ½ x B x H Area =	B = base of the triangle H = height of the triangle a, b, c = known lengths of the triangle C = angle between the lengths a and b s = ½ (a + b + c)
Area of a Rectangle	Area = L x B	L = length of the rectangle B = breadth of the rectangle
Area of a Trapezoid	Area = ½ (a + b) h	h = vertical height of trapezoid a, b = parallel sides of the trapezoid
Area of an Ellipse	Area = πab	a = half of minor axis b = half of major axis
Area of a Square	Area = S2	S = side square length
Area of a Parallelogram	Area = b x h	b = base of parallelogram h = vertical height of the parallelogram
Area of a Circle	Area = πr2 or (π d2/4)	r = radius of the circle d = diameter of the circle
Area of a Sector	Area = ½ r2 Ѳ	r = radius of sector Ѳ = angle (in radians)

Grid method for area calculations

Determine the area by dividing the polygon into smaller equal squares

Suppose a question says that each grid in the square box is 1m², determine the area of the square box. We know that the area of each grid is 1m², thus by counting the number of grids and multiplying them with the area of one grid gives the area of the entire square box.

For instance, in the below figure, the total area will be 1m² x 16 number of grids = 16m² (where, 1 grid = 1m², 16 grids)

The area for a shaded region is calculated by calculating the area of the unshaded part and the area of the whole object, then subtracting the latter from the former.

Area of the blue shaded region = area of the unshaded region – area of the whole square field

Determine the area by dividing the polygon into smaller equal triangles

The most suitable method is to divide the figure into a number of triangles. The base and height of every triangle are scaled and its area is calculated.

Determine the area by dividing the polygon into trapezoids

A number of parallel lines equally spaced are drawn on a tracing paper and it is placed over the plan so that the proposed area is accurately in between the two parallel lines. The area is then divided into a number of strips. The curved ends are replaced by perpendicular lines. The sum of the base length of the rectangles computed and those multiplied by the common distance between all the parallel lines gives the area of the proposed plan.

Plan Area = [length of rectangles] x [constant distance common breadth]

Graphical method for area calculation

Mid-ordinate method

It is calculated graphically using coordinates, where the ordinates are measured at each division’s mid-point.

$Area=\left(h_1+h_2+h_3+\dots +h_n\right)\frac{L}{n}$

$Area=\left(h_1+h_2+h_3+\dots +h_n\right) d$

where,
Ordinates (vertical distance) at the midpoint of each division = h₁, h₂, h₃,…,h_n
Length of the base = L
Number of equal parts into which the baselines are divided = n
Common distance between the ordinates (L/n) = d

Average-Ordinate method

$Strip Area=\left(\frac{O_1+O_2+O_3+\dots +O_n}{n+1}\right) L$

where,

Height of vertical ordinates = O₁, O₂, O₃,…,O_n

Number of ordinates = n

Length of the base = L

Trapezoidal method

$Area=\frac{d}{2}\left(O_0+2O_1+2O_2+\dots +2O_{n-1}+O_n\right)$

where,

Length of the ordinates = O₁, O₂, O₃,…,O_n

d = Common distance between the ordinates

Simpson’s rule method (Parabolic rule)

The boundaries between the edges of the coordinates are considered to be an area of a parabola.

$A=\frac{d}{3}\left[\right(O_1+O_n)+4\left(O_2+O_4+\dots +O_{n-2}\right)+2(O_3+O_5+...+O_{n-1}\left)\right]$

where,

d = common distance between the ordinates

Height of vertical ordinates = O₁, O₂, O₃,…,O_n

Simpson’s rule is applicable only if ordinates are odd and the number of division parameters are even. If the number of ordinates is even, the area value of the last division may be computed separately and added to the result by applying Simpson’s rule to two remaining ordinates. In case the first or last ordinate is zero, they cannot be removed from Simpson’s rule.

Instrumentation method

This method comprises of calculating the area of a proposed property by using a planimeter. It is the most premier method that produces precise conclusions and observations than any other method.

Triangulation method

The triangulation method in surveying is the procedure of finding the position of a point by measuring angles from the vertices to the point from the points known at either edge of a secured baseline, by measuring the distances to the point directly like in trilateration. The point may then be braced as the third vertex of a triangle with one side and two angles already known.

Context and Applications

• Bachelors in Technology (Civil Engineering)
• Masters in Science (Surveying and GIS)
• Masters in Commerce (Statistic and Economics)

Practice Problems

1. The dimensional unit of area is?

1. [L]
2. m²
3. [L²]
4. Meters

Correct option- c

Explanation: The dimensional unit of area is [L²].

2. Which one of the following is the correct rectangle area computation?

1. To multiply length and breath
2. To multiply radius and width
3. To offset the sides
4. None of these

Correct option- a

Explanation: The correct rectangle area computation is to multiply length and breadth.

3. Which method is applied when the ordinates are odd and a number of division parameters are even?

1. Approximation method
2. Offset method
3. Trapezoidal method
4. Simpson’s rule method

Correct option- d

Explanation: Simpson’s rule method is applied when the ordinates are odd and a number of division parameters are even.

4. In the given rectangular block, find the area of the green space for laying the turf over it.

where, length of the outer rectangle = 8m, breadth of the outer rectangle = 5m, length of the inner rectangle = 5m, breadth of the inner rectangle = 3m

1. 40 m²
2. 15 m²
3. 25 m²
4. 55 m²

Correct option- c

Explanation: Area of the inner unshaded rectangle = 5m x 3m = 15 m²

Area of the outer rectangle = 8m x 5m = 40 m²

Area of the shaded greenspace = 40 m² - 15 m² = 25 m²

Thus, the area of the green space for laying the turf over the block is 25 m².

5. Which instrument is used in the instrumentation method for precise results?

1. Hydrometer
2. Planimeter
3. Thermometer
4. None of these

Correct option- b

Explanation: The planimeter instrument is used in the instrumentation method for precise results.

Related Concepts

• The perimeter and volume of figures
• Offset method in surveying
• Cartesian coordinate system