Interior design professionals in Kenya rely heavily on precise geometric constructions to create functional and aesthetically pleasing spaces. Mastery of plane geometry is essential for translating design concepts into accurate technical drawings, enabling clear communication with contractors and clients. This chapter equips students with the skills to construct both basic and complex geometric forms, as well as accurately measure and draw angles, foundational competencies for producing detailed interior design plans.

3.1 Construction of Geometric Forms

Interior spaces often incorporate a variety of geometric shapes to enhance visual appeal and functionality. Whether designing custom furniture, floor patterns, or decorative elements, understanding how to construct these shapes accurately is fundamental. This section covers the techniques to draw basic geometric shapes and extend these skills to more complex forms frequently encountered in interior design projects.

3.1.1 Basic geometric shapes

Basic geometric shapes form the building blocks of interior design drawings. These include squares, rectangles, triangles and circles, each with unique properties and construction techniques essential for accurate representation. Nearly every plan, elevation and furniture detail is built up from these shapes, so drawing them precisely with a T-square, set squares, compass and scale ruler is the first skill a designer develops. Example: a kitchen layout for a restaurant is drawn as rectangles for counters, squares for floor tiles, circles for sinks and triangles for the work triangle between cooker, sink and fridge.

3.1.1.1 Squares

A square is a four-sided polygon with equal sides and four right angles. Squares are common in tile layouts, furniture footprints, ceiling grids and window panes, and an inaccurate square causes tiles to drift out of line across a floor. A square is constructed by drawing one side and then erecting perpendicular lines to create right angles, and it is checked by confirming that its two diagonals are equal.

Steps for constructing a square

1. Draw the base: Draw a horizontal base line using a ruler to the required length.
2. Erect a perpendicular: At one endpoint of the base, use a set square to draw a perpendicular line upwards.
3. Mark the side: Mark the length of the side on this perpendicular line equal to the base length.
4. Draw the top side: From the top point, draw a line parallel to the base using a ruler and set square.
5. Close the square: Erect a perpendicular at the other end of the base and mark the side length to meet the top line.
6. Check: Check all sides with a ruler to confirm they are equal and measure both diagonals, which must be equal.

Worked example: diagonal of a square

Diagonal = side × √2

A square floor tile for a hotel lobby has sides of 600 mm.

Diagonal = 600 × √2

Diagonal = 600 × 1.414

Diagonal = 848.5 mm

3.1.1.2 Rectangles

A rectangle is a four-sided figure with four right angles and opposite sides equal. Rectangles are ubiquitous in room layouts, doors, window designs, cabinets, beds and tables. A rectangle is constructed by drawing one length and then perpendicular sides of the required width, and it is checked for squareness by measuring its diagonals.

Steps for constructing a rectangle

1. Draw the length: Draw the base line to the required length using a scale ruler.
2. Erect perpendiculars: At both ends of the base, draw perpendicular lines using a set square or compass construction.
3. Mark the width: Measure the width along each perpendicular.
4. Join the top: Join the two marks to form the top side, which should be parallel to the base.
5. Check the diagonals: Measure both diagonals; equal diagonals confirm a true rectangle.
6. Line in: Darken the outline and lightly erase construction lines.

Worked example: diagonal of a rectangle

Diagonal = √(length² + width²)

A wardrobe front for a school staffroom is 1,800 mm by 1,200 mm.

Diagonal = √(1,800² + 1,200²)

Diagonal = √(3,240,000 + 1,440,000)

Diagonal = √4,680,000 = 2,163.3 mm

3.1.1.3 Triangles

A triangle is a three-sided figure whose interior angles add up to 180°. Triangles are the most rigid shape, which is why they appear in roof trusses, shelf brackets and bracing, and they are also used in decorative patterns. Triangles are classified by their sides as equilateral, isosceles or scalene, and by their angles as acute, right-angled or obtuse.

Types of triangles

• Equilateral triangle: Three equal sides and three 60° angles, used in truss designs and decorative patterns.
• Isosceles triangle: Two equal sides and two equal base angles, common in gable ends and pediments.
• Scalene triangle: All sides and angles different, found in irregular plots and sloping ceilings.
• Right-angled triangle: One 90° angle, used in stair profiles, brackets and setting out right angles with the 3:4:5 rule.
• Obtuse triangle: One angle greater than 90°, seen in shallow roof pitches and angled wall junctions.

Steps for constructing an equilateral triangle

1. Draw the base: Draw the base line AB to the required length.
2. Set the compass: Open the compass to the length of AB.
3. Swing the first arc: With the point on A, draw an arc above the base.
4. Swing the second arc: With the point on B, draw a second arc to cut the first at C.
5. Join the apex: Join A to C and B to C.
6. Check: Measure all three sides to confirm they are equal.

Worked example: height of an equilateral triangle

Height = side × √3 ÷ 2

A decorative wall panel for a bank's waiting area is an equilateral triangle with 900 mm sides.

Height = 900 × 1.732 ÷ 2

Height = 1,558.8 ÷ 2

Height = 779.4 mm

3.1.1.4 Circles

A circle is a set of points equidistant from a central point. Circles are used in designing round tables, ceiling lights, columns, sinks and decorative motifs. A circle is constructed using a compass set to the desired radius, while large circles on site are set out with a trammel or a string fixed at the centre.

Parts of a circle

• Centre: The fixed point from which every point on the circle is the same distance.
• Radius: The distance from the centre to the circumference, used to set the compass.
• Diameter: A straight line through the centre from one side to the other, equal to twice the radius.
• Circumference: The distance round the circle.
• Chord and arc: A chord is a straight line joining two points on the circle, and an arc is the part of the circumference between them.

Worked example: circumference and area of a round table

Circumference = π × diameter

Area = π × radius²

A round meeting table for a clinic has a diameter of 1,200 mm.

Circumference = 3.142 × 1,200 = 3,770.4 mm

Area = 3.142 × 600² = 3.142 × 360,000 = 1,131,120 mm² = 1.13 m²

3.1.2 Complex geometric forms

Complex geometric forms are combinations or modifications of basic shapes, often required in advanced interior design elements such as arches, ellipses, and irregular polygons. Mastery of these forms allows designers to innovate while maintaining precision. They are constructed by building on the methods for basic shapes, using compass divisions, centre points and plotted points.

3.1.2.1 Polygons,

A polygon is a closed plane figure with three or more straight sides. A regular polygon has all sides and angles equal, such as a pentagon, hexagon or octagon, and is applied in decorative floor tiles, wall panel designs, gazebo plans and mirrors. Irregular polygons have unequal sides and angles and are common in bespoke cabinetry, odd-shaped rooms and floor patterns.

Types of polygons

• Regular polygons: Equal sides and angles, such as hexagonal tiles or an octagonal bay window, constructed by dividing a circle into equal arcs and joining the points.
• Irregular polygons: Unequal sides and angles, constructed by connecting points defined by measurements or coordinates.
• Star polygons: Shapes with intersecting sides, often used in decorative motifs, constructed by drawing a regular polygon and joining every second or third point.
• Convex polygons: All interior angles less than 180°, such as most tile shapes.
• Concave polygons: At least one interior angle greater than 180°, such as an L-shaped room plan.

Steps for constructing a regular hexagon in a circle

1. Draw the circle: Draw a circle with a radius equal to the side of the hexagon.
2. Mark a starting point: Draw a diameter and mark one end as the first point.
3. Step off the radius: Without changing the compass, step the radius round the circumference from the first point.
4. Complete the divisions: Continue until six equal points are marked.
5. Join the points: Join adjacent points with straight lines.
6. Check: Confirm all six sides are equal and erase construction lines.

Worked example: interior angle of a regular polygon

Interior angle = (number of sides − 2) × 180° ÷ number of sides

An octagonal reading pod for a library is drawn as a regular octagon.

Interior angle = (8 − 2) × 180° ÷ 8

Interior angle = 6 × 180° ÷ 8

Interior angle = 1,080° ÷ 8 = 135°

3.1.2.2 Curves

Curves are smooth lines that change direction continuously, and in interior design they include arcs, ellipses, arches and freehand profiles. They soften interiors and appear in doorways, niches, mirrors, furniture tops, reception counters and ceiling features. Curves are constructed from centre points with a compass, from focal points with a string, or by plotting points and joining them with a French curve or flexible curve.

Common curves in interior design

• Ellipse: An elongated circle used in window shapes, mirrors, and furniture tops. Constructed using the string-and-pin method or by applying the focal point definition.
• Arches: Curved structures often used in doorways or niches. Constructed by defining the radius and centre points, then drawing the arc using a compass or flexible curve.
• Arcs and blends: Parts of circles used to round the corners of counters and tables, joined smoothly to straight lines.
• Spirals: Curves that wind outwards from a centre, used in staircases and decorative motifs.
• Free curves: Irregular, flowing curves drawn through plotted points with a French curve, used in furniture and wall features.

Steps for constructing an ellipse by the string method

1. Mark the focal points: Mark two focal points on the drawing surface based on the desired ellipse dimensions.
2. Prepare the string: Tie a string into a loop longer than the distance between the two focal points.
3. Place the pins: Place pins at each focal point and loop the string around them.
4. Trace the curve: Using a pencil, pull the string taut and trace the curve, keeping the string tight.
5. Complete the ellipse: Continue moving the pencil around the focal points until the ellipse is complete.
6. Check: Erase any construction lines and check symmetry with a ruler.

Worked example: locating the focal points of an ellipse

Distance from centre to each focus = √(a² − b²)

where a is half the major axis and b is half the minor axis.

An elliptical mirror for an office washroom is 1,000 mm by 600 mm, so a = 500 mm and b = 300 mm.

Distance to each focus = √(500² − 300²)

Distance to each focus = √(250,000 − 90,000)

Distance to each focus = √160,000 = 400 mm from the centre

3.1.3 Composite shapes in interior design

Composite shapes are formed by combining basic shapes, such as a rectangle topped with a semicircle, common in window heads or furniture profiles. They are drawn by constructing each shape in turn from shared centre lines so that they join exactly. Example: a church window is drawn as a 900 mm by 1,500 mm rectangle with a semicircle of 450 mm radius centred on its top edge.

Area = rectangle area + semicircle area

Area = (900 × 1,500) + (3.142 × 450² ÷ 2)

Area = 1,350,000 + 318,128

Area = 1,668,128 mm² = 1.67 m²

Chapter Summary

This chapter focused on the production of plane geometry through systematic construction of geometric forms, beginning with basic shapes such as squares, rectangles, triangles, and circles, and extending to more complex forms like polygons and curves. It then covered the construction of angles, detailing different types, the application of trigonometric principles, and the appropriate tools required for accurate creation. The measurement of angles was explored next, emphasizing the use of specific instruments, techniques for bisecting angles, and ensuring accuracy and precision in all measurements. The chapter also explained the construction of scales, including diagonal and plain scales, which are essential for precise technical drawing. Various geometric construction techniques were presented to enhance drawing skills and ensure proper figure formation. Attention was given to the correct labelling of constructed figures to maintain clarity and communication in technical drawings. Finally, the chapter addressed the professional presentation of constructed figures, highlighting standards that improve readability and overall quality.