In interior design, precise representation of spaces and forms is essential for successful project execution. Mastery of plane geometry enables design professionals to construct accurate drawings that communicate spatial relationships and design intent clearly. This chapter equips interior design students with the skills to produce geometric constructions fundamental for creating detailed floor plans, elevations, and decorative elements. Applying these principles within Kenya’s varied architectural contexts ensures designs are both functional and aesthetically pleasing.

3.1 Construction of Geometric Forms

Geometric forms form the foundation of all technical drawings in interior design. Whether drafting furniture layouts, partition shapes, or decorative motifs, understanding how to construct these forms accurately is critical. In Kenya, where interior spaces often blend modern and traditional influences, skillful geometric construction supports creativity while maintaining clarity and precision in drawings.

3.1.1 Basic geometric shapes

Basic geometric shapes are the building blocks for all interior design drawings. They include simple forms such as squares, rectangles, triangles and circles, all built from lines, the simplest geometric element used to define edges, boundaries, and connections. Lines must be drawn with precision using appropriate tools like rulers or drafting machines to ensure accurate dimensions. Mastery of these shapes allows designers to create floor plans, furniture outlines, and other components essential for space planning.

3.1.1.1 Squares

A square is a four-sided figure with all sides equal and all four angles at 90°. Squares are fundamental for walls, windows, doors, and furniture footprints, and are constructed using perpendicular lines and accurate measurements to maintain right angles. In interior drawings they represent floor tiles, ceiling panels, square tables, columns and grid layouts. Example: a designer drawing the floor of a bank's customer hall sets out a grid of 600 mm square tiles before placing the counters.

Properties of a square

• Equal sides: All four sides are the same length.
• Right angles: Every corner measures 90°.
• Equal diagonals: Both diagonals are equal and cross at right angles.
• Bisected diagonals: The diagonals cut each other into equal halves.
• Symmetry: A square has four lines of symmetry.

Steps for constructing a square of a given side

1. Draw the base: Draw line AB equal to the given side length.
2. Erect a perpendicular at A: Use a set square or compass construction to draw a line at 90° from A.
3. Mark the side on the perpendicular: Measure the side length from A and mark point D.
4. Erect a perpendicular at B: Draw a 90° line from B.
5. Mark point C: Measure the side length from B and mark C.
6. Join C and D: Complete the square and check that the diagonals are equal.

Worked example: diagonal of a square tile

Diagonal = side × √2

A floor tile measures 600 mm by 600 mm.

Diagonal = 600 × 1.414

Diagonal ≈ 848.5 mm

3.1.1.2 Rectangles

A rectangle is a four-sided figure with opposite sides equal and parallel and all four angles at 90°. Constructing a rectangle accurately is fundamental in drafting interior elements such as furniture or room layouts, and rectangles represent rooms, doors, windows, beds, desks and cabinets. Example: the outline of a reception desk for a dental clinic is drawn as a 2,400 mm by 800 mm rectangle before the details are added.

Properties of a rectangle

• Opposite sides equal: The two lengths are equal and the two widths are equal.
• Right angles: Every corner measures 90°.
• Equal diagonals: The diagonals are equal and bisect each other.
• Parallel sides: Opposite sides are parallel.
• Two lines of symmetry: One through the length and one through the width.

Step-by-step procedure for constructing a rectangle with given dimensions

1. Draw the base line: Use a ruler to draw a horizontal line equal to the given length.
2. Mark endpoints: Label the endpoints as points A and B.
3. Construct perpendicular lines: At points A and B, use a set square to draw perpendicular lines upwards.
4. Measure height: On each perpendicular line, mark the given width.
5. Connect top points: Join the two marks with a straight line parallel to the base.
6. Label corners: Mark the rectangle's corners A, B, C, and D clockwise or counterclockwise.

Worked example: checking a room is square-cornered

Diagonal = √(length² + width²)

A room set out for a bakery's sales area measures 2,400 mm by 1,800 mm.

Diagonal = √(2,400² + 1,800²)

Diagonal = √(5,760,000 + 3,240,000) = √9,000,000

Diagonal = 3,000 mm, so both measured diagonals should read 3,000 mm if the corners are true right angles

3.1.1.3 Triangles

A triangle is a three-sided figure whose interior angles add up to 180°. Triangles are often used in truss designs or decorative patterns, and construction requires accurate measurement of sides or angles to ensure geometric integrity. They also appear in roof and gable details, bracing, staircase profiles and tiling patterns. Example: a designer creating a feature wall for a fitness studio uses repeated equilateral triangles in two colours to form a dynamic geometric pattern.

Types of triangles

Steps for constructing an equilateral triangle

1. Draw the base: Draw line AB equal to the given side length.
2. Set the compass: Open the compass to the length of AB.
3. Draw the first arc: With the compass point on A, draw an arc above the line.
4. Draw the second arc: With the compass point on B, draw an arc to cross the first.
5. Mark the apex: Label the crossing point C.
6. Complete the triangle: Join AC and BC and check that all sides are equal.

Worked example: height of an equilateral triangle

Height = side × √3 ÷ 2

A triangular decorative panel has sides of 500 mm.

Height = 500 × 1.732 ÷ 2

Height ≈ 433 mm

3.1.1.4 Circles

A circle is a curved figure in which every point on the outline is the same distance from a fixed centre. Circles are used to represent columns, round tables, or decorative elements, and construction involves a compass to ensure perfect roundness and correct radius measurement. They also show light fittings, door swings and circular rugs on plans. Example: a designer planning a round meeting table for a cooperative society's boardroom draws a 1,200 mm diameter circle and arranges chairs around it.

Parts of a circle

• Centre: The fixed point from which all points on the circle are equally distant.
• Radius: A straight line from the centre to the circumference.
• Diameter: A straight line through the centre, equal to twice the radius.
• Circumference: The distance around the circle.
• Chord: A straight line joining two points on the circumference.
• Arc: Part of the circumference.
• Sector: The area between two radii and an arc.
• Tangent: A straight line touching the circle at one point only.

Steps for constructing a circle of a given diameter

1. Mark the centre: Mark the centre point O.
2. Find the radius: Divide the given diameter by two.
3. Set the compass: Open the compass to the radius using a ruler.
4. Place the point: Put the compass point on O.
5. Draw the circle: Rotate the compass in one smooth movement.
6. Check the size: Measure the diameter through the centre to confirm accuracy.

Worked example: circumference and area of a round table

Circumference = 3.142 × diameter

Area = 3.142 × radius²

The boardroom table has a diameter of 1,200 mm.

Circumference = 3.142 × 1,200 ≈ 3,770 mm

Area = 3.142 × 0.6² = 3.142 × 0.36 ≈ 1.131 m²

3.1.2 Complex geometric forms

Complex geometric forms arise from combining or modifying basic shapes to create more intricate designs. Interior designers use these forms to develop custom furniture, feature walls, or ornamental details that enhance spatial aesthetics.

3.1.2.1 Polygons,

A polygon is a closed figure with three or more straight sides. Regular polygons have all sides and angles equal, while irregular polygons have unequal sides or angles. Shapes like pentagons or hexagons appear in tiling patterns or bespoke furniture designs, and they require systematic construction using equal side lengths and angles. Irregular polygons are non-standard shapes that may arise in bespoke interior elements, and their construction involves careful measurement of unequal sides and angles. Example: a café uses hexagonal floor tiles in a honeycomb pattern behind the counter.

Common regular polygons

Steps for constructing a regular hexagon in a circle

1. Draw a circle: Draw a circle with radius equal to the hexagon's side length.
2. Mark a starting point: Mark point A anywhere on the circumference.
3. Step off the radius: Without changing the compass, step around the circle from A marking points B, C, D, E and F.
4. Check the steps: The sixth step should return exactly to A.
5. Join the points: Join A to B, B to C and so on to F and back to A.
6. Check accuracy: Confirm all sides are equal and each angle measures 120°.

Worked example: interior angle of a regular polygon

Interior angle = (n − 2) × 180° ÷ n

A designer draws an octagonal ceiling coffer, where n = 8.

Interior angle = (8 − 2) × 180 ÷ 8

Interior angle = 1,080 ÷ 8 = 135°

3.1.2.2 Curves

Curves are lines that change direction smoothly, without sharp corners. In interiors they soften spaces and appear in arches, curved walls, furniture, ceiling features and decorative mouldings. Arcs and segments are applied in archways, curved furniture, or decorative cornices, and are constructed by defining the radius and center point, then drawing the precise arc length. Ellipses are frequently used in furniture design and ceiling patterns, and require specific construction methods using focal points and string techniques to maintain proportion. Example: an oval conference table for a tea marketing agency is drawn as an ellipse 1,800 mm long and 1,000 mm wide.

Types of curves in interior drawing

• Arcs: Parts of a circle used for arches, door swings and curved counters.
• Ellipses: Oval shapes used for tables, mirrors and ceiling features.
• Spirals: Curves that wind around a point, used in staircases and decorative motifs.
• Ogee curves: S-shaped curves used in mouldings and arches.
• Freeform curves: Smooth curves drawn with French curves or flexible curves for organic shapes.

Steps for constructing an ellipse using pins and string

1. Draw the axes: Draw the major axis (length) and minor axis (width) crossing at right angles at the centre.
2. Find the focal points: With compass set to half the major axis, draw arcs from the end of the minor axis to cut the major axis at F1 and F2.
3. Fix pins: Place pins at F1, F2 and the end of the minor axis.
4. Tie the string: Tie a string tightly around all three pins.
5. Remove the third pin: Remove the pin at the end of the minor axis and place a pencil inside the string loop.
6. Draw the ellipse: Keep the string tight and move the pencil round to draw the ellipse.

Worked example: position of ellipse focal points

Focal distance from centre = √(a² − b²)

The oval table is 1,800 mm long and 1,000 mm wide, so a = 900 mm and b = 500 mm.

Focal distance = √(900² − 500²)

Focal distance = √(810,000 − 250,000) = √560,000 ≈ 748 mm from the centre on each side

3.1.3 Composite shapes and symmetrical patterns

Composite shapes are formed by combining basic and complex shapes into one figure, and symmetrical patterns repeat shapes in a balanced way. Both are common in windows, doors, furniture, panelling and floor designs.

• Composite Shapes: Created by combining two or more basic shapes, such as a rectangle topped with a semicircle for window designs. Accurate construction involves sequential building and intersection finding.
• Symmetrical Patterns: Achieved by replicating shapes across an axis or center point, common in wall paneling or flooring designs. Construction uses reflection and rotational symmetry techniques.
• Breaking down shapes: Complex outlines are divided into simple shapes for drawing and calculation.
• Construction order: Basic shapes are drawn first, then joined or trimmed to form the composite outline.
• Checking tangency: Where curves meet straight lines, they must join smoothly without bumps.

Worked example: area of an arched window

Area = rectangle area + semicircle area

An arched window is 1,200 mm wide with a rectangular part 1,500 mm high, topped by a semicircle of radius 600 mm.

Rectangle area = 1.2 × 1.5 = 1.8 m²

Semicircle area = 3.142 × 0.6² ÷ 2 = 3.142 × 0.36 ÷ 2 ≈ 0.565 m²

Total area = 1.8 + 0.565 ≈ 2.365 m²

3.1.4 Application of geometric construction in interior design projects

Geometric constructions translate design ideas into precise technical drawings that contractors and clients can understand.

• Space planning: Accurate shapes ensure that furniture and partitions fit within available space without conflicts.
• Custom furniture design: Complex geometric forms help create unique furniture that complements interior themes.
• Decorative motifs: Geometric patterns are often incorporated in wall, floor, or ceiling treatments to add aesthetic value.
• Lighting layouts: Circles and arcs aid in positioning fixtures and calculating light spread.
• Material estimation: Accurate geometric drawings help quantify materials, reducing waste and cost.

Chapter Summary

This chapter covers the production of plane geometry through the construction of both basic and complex geometric forms, including squares, rectangles, triangles, circles, polygons, and curves. It explains the classification of different types of angles and the application of trigonometric principles in their construction, highlighting the essential tools required for accurate angle creation. The chapter also addresses the measurement of angles, emphasizing the use of appropriate instruments, techniques for bisecting angles, and the importance of accuracy and precision. Construction of scales is explored through the development of diagonal and plain scales, providing practical methods for representing measurements. Various geometric construction techniques are introduced to enhance drawing skills, followed by guidelines on the correct labelling of constructed figures to ensure clarity. Finally, the chapter discusses the proper presentation of these figures according to technical drawing standards, ensuring professional and readable outputs.