Mensuration is fundamental in architectural design and construction, enabling precise measurement and calculation of lengths, areas, and volumes. Accurate mensuration ensures correct material estimation, cost control, and compliance with building codes in Kenya's dynamic construction sector. This chapter focuses on calculating perimeters and areas of regular figures, fundamental for site layout, floor planning, and structural detailing.

4.1 Perimeter and Areas of Regular Figures

4.1.1 Addition

Addition in mensuration involves combining lengths or areas to find total values, such as adding lengths of walls or areas of adjoining rooms. This operation is essential when calculating total boundary lengths or total floor areas from multiple segments.

The general formula for addition of lengths or areas is the sum of individual measurements:

$$ P_{total} = P_1 + P_2 + \cdots + P_n $$

$$ A_{total} = A_1 + A_2 + \cdots + A_n $$

Worked Examples

Example 1: Calculate the total perimeter of a rectangular courtyard with two sides measuring 10 m and 15 m, and an adjoining square garden with side 5 m.

Given:
Rectangle sides: 10 m and 15 m
Square side: 5 m

Rectangle perimeter:$$ P_{rect} = 2 \times (10 + 15) $$$$ P_{rect} = 2 \times 25 $$$$ P_{rect} = 50 \text{ m} $$

Square perimeter:$$ P_{square} = 4 \times 5 $$$$ P_{square} = 20 \text{ m} $$

Total perimeter (assuming separate boundaries):$$ P_{total} = 50 + 20 $$$$ P_{total} = 70 \text{ m} $$

Answer: 70 m

Example 2: Find the total floor area of two adjacent rectangular rooms measuring 6 m by 4 m and 5 m by 3 m.

Given:
Room 1: 6 m by 4 m
Room 2: 5 m by 3 m

Area room 1:$$ A_1 = 6 \times 4 $$$$ A_1 = 24 \text{ m}^2 $$

Area room 2:$$ A_2 = 5 \times 3 $$$$ A_2 = 15 \text{ m}^2 $$

Total area:$$ A_{total} = 24 + 15 $$$$ A_{total} = 39 \text{ m}^2 $$

Answer: 39 m²

Example 3: A rectangular wall has two sections with lengths 8 m and 12 m and height 3 m. Calculate the total surface area.

Given:
Lengths: 8 m, 12 m
Height: 3 m

Area first section:$$ A_1 = 8 \times 3 $$$$ A_1 = 24 \text{ m}^2 $$

Area second section:$$ A_2 = 12 \times 3 $$$$ A_2 = 36 \text{ m}^2 $$

Total area:$$ A_{total} = 24 + 36 $$$$ A_{total} = 60 \text{ m}^2 $$

Answer: 60 m²

4.1.2 Subtraction

Subtraction is used in mensuration to find remaining lengths or areas after removing parts, such as openings, courtyards, or cutouts in walls or floors. This operation helps in calculating net usable spaces or material requirements.

The formula for subtraction of areas or lengths is:

$$ A_{net} = A_{total} - A_{cutout} $$

$$ P_{net} = P_{total} - P_{removed} $$

Worked Examples

Example 1: Calculate the net floor area of a rectangular hall measuring 12 m by 8 m with a rectangular courtyard opening of 3 m by 4 m inside it.

Given:
Hall: 12 m by 8 m
Courtyard: 3 m by 4 m

Hall area:$$ A_{hall} = 12 \times 8 $$$$ A_{hall} = 96 \text{ m}^2 $$

Courtyard area:$$ A_{courtyard} = 3 \times 4 $$$$ A_{courtyard} = 12 \text{ m}^2 $$

Net area:$$ A_{net} = 96 - 12 $$$$ A_{net} = 84 \text{ m}^2 $$

Answer: 84 m²

Example 2: A rectangular fence has a total length of 50 m but there is a 6 m gate opening. Find the effective perimeter enclosing the compound.

Given:
Total fence length: 50 m
Gate length: 6 m

Effective perimeter:$$ P_{effective} = 50 - 6 $$$$ P_{effective} = 44 \text{ m} $$

Answer: 44 m

Example 3: A rectangular wall measures 20 m by 3 m. It has two rectangular windows measuring 2 m by 1.5 m and 1.5 m by 1 m. Calculate the net wall surface area to be painted.

Given:
Wall: 20 m by 3 m
Window 1: 2 m by 1.5 m
Window 2: 1.5 m by 1 m

Wall area:$$ A_{wall} = 20 \times 3 $$$$ A_{wall} = 60 \text{ m}^2 $$

Window 1 area:$$ A_{w1} = 2 \times 1.5 $$$$ A_{w1} = 3 \text{ m}^2 $$

Window 2 area:$$ A_{w2} = 1.5 \times 1 $$$$ A_{w2} = 1.5 \text{ m}^2 $$

Total window area:$$ A_{windows} = 3 + 1.5 $$$$ A_{windows} = 4.5 \text{ m}^2 $$

Net wall area:$$ A_{net} = 60 - 4.5 $$$$ A_{net} = 55.5 \text{ m}^2 $$

Answer: 55.5 m²

4.1.3 Multiplication

Multiplication in mensuration is primarily used in scaling figures and calculating areas and perimeters by multiplying lengths. It is essential for converting units, scaling drawings, and computing quantities from dimensions.

Area of rectangles and squares is calculated by multiplying length and width:

$$ A = l \times w $$

Perimeter of regular polygons involves multiplying side length by the number of sides:

$$ P = n \times s $$

Worked Examples

Example 1: Calculate the area of a rectangular room measuring 7 m by 5 m.

Given:
Length = 7 m
Width = 5 m

Formula:$$ A = l \times w $$

Substitution:$$ A = 7 \times 5 $$

Calculation:$$ A = 35 \text{ m}^2 $$

Answer: 35 m²

Example 2: Find the perimeter of a regular hexagonal flower bed with each side measuring 4 m.

Given:
Number of sides \( n = 6 \)
Side length \( s = 4 \) m

Formula:$$ P = n \times s $$

Substitution:$$ P = 6 \times 4 $$

Calculation:$$ P = 24 \text{ m} $$

Answer: 24 m

Example 3: A rectangular plot is scaled on a drawing by a factor of 1:100. If the length on the drawing is 12 cm, find the actual length in meters.

Given:
Drawing length = 12 cm
Scale = 1:100

Formula:$$ L_{actual} = L_{drawing} \times scale $$

Substitution:$$ L_{actual} = 12 \times 100 $$

Calculation:$$ L_{actual} = 1200 \text{ cm} = 12 \text{ m} $$

Answer: 12 m

Example 4: Calculate the area of a square courtyard whose side length is 8 m.

Given:
Side length \( s = 8 \) m

Formula:$$ A = s^2 $$

Substitution:$$ A = 8^2 $$

Calculation:$$ A = 64 \text{ m}^2 $$

Answer: 64 m²

4.1.4 Division

Division in mensuration is used to find unknown lengths or areas when total values and number of segments are known. It is useful for partitioning spaces or calculating average dimensions.

The formula to find the length or area per segment is:

$$ \text{Value per segment} = \frac{\text{Total value}}{\text{Number of segments}} $$

Worked Examples

Example 1: A rectangular wall 24 m long is divided into 6 equal panels. Find the length of each panel.

Given:
Total length = 24 m
Number of panels = 6

Formula:$$ L_{panel} = \frac{24}{6} $$

Calculation:$$ L_{panel} = 4 \text{ m} $$

Answer: 4 m

Example 2: A rectangular garden has an area of 180 m² and length 15 m. Find the width.

Given:
Area \( A = 180 \text{ m}^2 \)
Length \( l = 15 \text{ m} \)

Formula:$$ w = \frac{A}{l} $$

Substitution:$$ w = \frac{180}{15} $$

Calculation:$$ w = 12 \text{ m} $$

Answer: 12 m

Example 3: A square floor has an area of 225 m². Find the length of one side.

Given:
Area \( A = 225 \text{ m}^2 \)

Formula:$$ s = \sqrt{A} $$

Calculation:$$ s = \sqrt{225} $$$$ s = 15 \text{ m} $$

Answer: 15 m

Example 4: A rectangular plot is divided into 3 equal parts for construction. The total area is 450 m². Find the area of each part.

Given:
Total area = 450 m²
Number of parts = 3

Formula:$$ A_{part} = \frac{450}{3} $$

Calculation:$$ A_{part} = 150 \text{ m}^2 $$

Answer: 150 m²

Chapter Summary

This chapter covered the fundamental mensuration skills necessary for architectural calculations, beginning with the determination of perimeters and areas of regular figures. It included essential arithmetic operations such as addition, subtraction, multiplication, and division applied in solving mensuration problems. The chapter then progressed to methods for calculating areas of irregular figures, highlighting techniques to handle complex shapes. Finally, it addressed the computation of volume and surface area of various solids, providing the mathematical foundation required for understanding three-dimensional architectural elements. Throughout, emphasis was placed on accurate formula application and step-by-step problem-solving to ensure precision in measurement. These skills are critical for architects to estimate materials, costs, and spatial layouts effectively.