Stair Geometry and Design Calculations

Introduction and context

A stair is a circulation system that converts vertical movement into a sequence of safe horizontal and vertical movements. For a civil draftsman, stair design is not simply the drawing of attractive steps. It requires coordinated decisions about floor-to-floor height, rise, going, pitch, number of risers, landings, headroom, structural thickness, handrails, balustrades and the stairwell opening.

Good stair design balances three requirements:

1. Safety: users must be able to ascend and descend without excessive effort or slipping.
2. Ergonomics: the relationship between rise and going must suit normal human movement.
3. Compliance: dimensions must satisfy the applicable building code, accessibility requirements and project brief.

The numerical limits used in this chapter are common design guidance, not a substitute for the authority having jurisdiction. National and local codes differ, particularly for maximum rise, minimum going, handrail height, barrier openings, accessible stairs and fire exits.

Learning objectives

After studying this chapter, the learner should be able to:

• explain the geometric components of a stair;
• select a preliminary rise and going using ergonomic principles;
• calculate the number of risers and treads;
• determine pitch, stair width and landing dimensions;
• calculate headroom and estimate the required stairwell opening;
• prepare a dimensionally coordinated stair drawing;
• identify common drafting errors and verify a design against applicable regulations.

1. Essential stair terminology

A flight is a continuous series of steps between floors or landings. A riser is the vertical face or vertical interval between two consecutive treads. A tread is the horizontal surface on which the foot rests. The horizontal dimension of a tread measured in the direction of travel is the going.

The rise is the vertical height of one step. The total vertical distance between finished floor levels is the floor-to-floor height. A landing is a level platform at the top, bottom or intermediate position of a stair. A nosing is the projecting front edge of a tread.

The pitch line is an imaginary line joining the nosings, or their equivalent reference points. The pitch angle is the angle between the pitch line and the horizontal. The stair width is the clear width available for movement, normally measured between finished surfaces or between handrails as required by the code.

The stairwell is the opening through a floor or roof that accommodates the stair. Headroom is the clear vertical distance between the pitch line, tread nosing or walking surface and the obstruction above it.

2. Geometry of a stair

For a straight stair with uniform dimensions, the basic geometry is represented by a right triangle. If the rise of each step is R and the going is G, the pitch angle θ is obtained from:

where R and G must be expressed in the same units. If R and G are in millimetres, their ratio is dimensionless and θ is obtained using the inverse tangent function:

The horizontal going and vertical rise of one step define the stair slope. A larger rise or smaller going produces a steeper stair. A smaller rise or larger going produces a more comfortable but longer stair.

The horizontal run of a flight is:

where n_t is the number of treads in that flight. For a conventional stair rising from one finished floor to the next, the number of treads is usually one less than the number of risers because the upper floor acts as the final walking surface:

Here n_r is the number of risers. This relationship applies to a flight beginning at one floor and ending at the next floor, without counting an upper landing as an additional tread. Drawings must nevertheless be checked against the actual arrangement because intermediate landings alter the layout.

The sloping length of the pitch line is:

where L is the pitch-line length. This is useful for estimating stair length, handrail length and the position of the stairwell opening.

3. Ergonomic selection of rise and going

Human comfort depends strongly on the combined effect of rise and going. A commonly used preliminary relationship is the stair comfort formula:

The lower end generally produces a more generous stair, while the upper end may produce a more compact stair. This formula is a design guide rather than a universal legal rule. It assumes ordinary adult pedestrian use and uniform steps.

Another useful relationship is the alternate formula:

It should not be applied independently of code requirements or the comfort formula. The designer must first check the permitted minimum going and maximum rise. A stair that satisfies an ergonomic equation can still be illegal if, for example, its rise exceeds the code maximum.

For ordinary building stairs, a preliminary rise of approximately 150–180 mm and a going of approximately 250–300 mm is often practical. Public, institutional and accessible environments may require gentler proportions. Private residential stairs may be permitted to be steeper, subject to local regulations.

Uniformity is essential. Differences between successive rises or goings create unexpected changes in foot placement and are a significant tripping hazard. Finishes, nosings and floor build-ups must be included when determining finished dimensions.

4. Code and safety considerations

Before finalising a stair, consult the current code and approval requirements for the project location. Important provisions commonly regulate:

• maximum riser height;
• minimum tread going;
• permitted pitch or slope;
• minimum clear width;
• maximum number of risers in a flight;
• landing length and width;
• headroom;
• handrail height and continuity;
• balustrade or guard height;
• maximum opening in a barrier;
• slip resistance and nosing projection;
• fire escape width and travel requirements;
• accessibility and tactile warning requirements.

A useful preliminary design range is shown below. These values are illustrative and must be replaced by the governing code values during professional design.

Fire stairs and accessible routes may have more restrictive requirements. A stair is not an accessible route for a person who cannot use steps; ramps, lifts or other compliant means may be required.

5. Design calculation procedure

A reliable calculation sequence is as follows:

Step 1: Establish finished floor-to-floor height

Measure from the finished surface of the lower floor to the finished surface of the upper floor. Do not use structural slab-to-slab height unless finishes are separately incorporated.

Step 2: Select a trial rise

Choose a rise within the permitted code range and suitable for the building type. Divide the total height by the trial rise.

Step 3: Round to a whole number of risers

A stair cannot have a fractional riser. The result must be rounded to a practical whole number and then checked by recalculating the actual rise:

where H is the total floor-to-floor height.

Step 4: Select the going

Choose a going that satisfies the minimum code requirement and the comfort relationship. Calculate the pitch and verify that it is acceptable.

Step 5: Calculate treads and run

For a floor-to-floor flight:

Then calculate the horizontal run using Run = n_t × G. Add landings, wall thicknesses and clearances as appropriate.

Step 6: Arrange flights and landings

If the run is too long for the available room, divide the stair into two or more flights. A landing should provide a stable level area for turning, resting and opening doors. Its clear length in the direction of travel should normally be at least the stair width unless a local code requires more.

Step 7: Check headroom and stairwell opening

Draw the stair in section, including slab thickness and all overhead obstructions. Measure vertical clearance at the critical locations rather than assuming that a plan layout is adequate.

6. Worked example: straight residential stair

Problem. Design a preliminary straight stair for a finished floor-to-floor height of 3,000 mm. Use a target rise near 165 mm and a going of 270 mm. Determine the risers, actual rise, treads, run and pitch.

Solution.

First estimate the number of risers:

Select 18 risers. The actual rise is:

Thus, use 18 equal risers of approximately 166.7 mm each.

For a stair from one floor to the next:

Using a going of 270 mm:

Check the comfort relationship:

This falls within the common preliminary range of 600–650 mm. The pitch is:

The design is therefore geometrically comfortable as a preliminary solution. Final acceptance still requires checks against the local code, stair width, landings, headroom, handrails, barriers and structural design.

7. Worked example: two-flight stair with landing

Problem. A building has a floor-to-floor height of 3,360 mm. A two-flight stair is proposed with a 160 mm rise and a 280 mm going. Determine the number of risers, treads and approximate run if the risers are divided equally between two flights.

Solution.

The number of risers is:

Because 21 cannot be divided equally into two flights, the designer must either use unequal flight counts, revise the rise, or adopt a different arrangement. Unequal numbers of risers per flight are acceptable in many layouts, but every individual rise must remain identical.

A practical arrangement is 10 risers in the first flight and 11 in the second. The total number of treads associated with the complete floor-to-floor stair remains:

If the intermediate landing replaces one direction of travel, the treads are distributed as 9 treads in the first flight and 11 treads in the second, depending on whether the landing level and upper floor are counted as walking surfaces in the selected drawing convention. The drafter must dimension the actual plan, not rely only on a formula.

The total horizontal tread run is:

This run is divided between the two flights, with the landing length added separately. For example, if the flight arrangement uses 9 treads and 11 treads, their tread runs are 2,520 mm and 3,080 mm. If the landing is 1,200 mm long, the overall plan length and width depend on whether the flights are parallel, opposite or perpendicular.

The comfort relationship is:

The pitch is approximately:

The result is a relatively gentle stair. The final drawing must clearly show which level each landing serves and must avoid implying a different rise at the landing.

8. Landings, width and circulation

A landing performs several functions: it interrupts a long climb, provides a resting place, permits a change in direction and can allow a door to be approached safely. A landing should not be reduced merely to the minimum possible area if users must turn, carry objects or pass one another.

For preliminary planning, use a landing clear length at least equal to the stair width. Where a wheelchair turning space, stretcher movement, door swing or fire evacuation requirement applies, the landing may need to be substantially larger.

The clear stair width should be measured consistently. A drawing may show structural width, finished width and handrail-to-handrail width; these are not necessarily identical. The code definition must be used. Doors should not swing across a landing in a way that reduces the required escape width or obstructs stair users.

For a stair with two flights, the plan dimensions are governed by:

This simple expression applies to a straight or dog-legged arrangement in the direction being measured. Wall thickness, stringer projections, handrails and required clearances must be added or deducted according to the drawing convention.

9. Headroom and stairwell size

Headroom is checked in section. The measurement is vertical, not perpendicular to the pitch line. A common preliminary target is at least 2,000 mm of clear vertical headroom, but the applicable code must determine the final value.

For a stair rising at a constant rate, the pitch line at a horizontal distance x from the start has elevation:

where z(x) is the rise above the starting level, x is horizontal distance, R is rise per step and G is going per step. This expression assumes a straight flight with uniform rise and going.

If an overhead slab or beam underside is represented by an elevation z_o(x), the available headroom is:

The design is acceptable only if the minimum value of C(x), measured from the required walking reference, is at least the code minimum. In practice, the critical position is often near the underside of the upper floor or near the inside of a stair opening.

The stairwell opening must extend far enough in plan for the stair to pass beneath the floor without reducing headroom. A preliminary opening length can be estimated by locating the point where the stair reaches the underside of the upper slab. However, the final opening must account for:

• slab thickness;
• beams and downstands;
• floor finishes;
• stair nosing and handrail projections;
• the required headroom reference;
• edge protection and balustrade construction;
• construction tolerances.

A section drawing is indispensable. A plan-only stair layout cannot prove headroom compliance.

10. Drafting method and documentation

A professional stair drawing normally includes plan, elevation or section, enlarged details where necessary, dimensions, levels, direction arrow and notes. The drawing should identify:

• finished floor levels;
• total floor-to-floor height;
• number of risers;
• rise and going;
• clear stair width;
• landing dimensions;
• headroom;
• handrail and balustrade heights;
• nosing treatment;
• stairwell opening;
• materials and finishes;
• direction of ascent.

In plan, draw the lower treads first, show the break line where the stair passes above the cutting plane, and indicate the upper flight or landing. In section, show the true rise sequence, slab thickness and the headroom dimension. Dimension repeated elements using a clear notation such as “17 treads @ 270” only when the total and individual dimensions are unambiguous.

CAD and building information modelling systems can automate repeated steps and generate sections, but automation does not replace checking. A model may repeat an incorrect rise perfectly. Use independent calculations, visible levels and clash detection to verify the result.

11. Common design errors

Fractional or inconsistent risers

Rounding the number of risers without recalculating the actual rise can produce a mismatch with the floor level. Every riser in a flight should be equal after finishes are considered.

Confusing treads and risers

The number of treads is not always equal to the number of risers. In a floor-to-floor flight, the upper floor commonly acts as the final walking surface, giving one fewer tread than risers.

Ignoring finishes

If the structural stair is calculated before floor finishes are established, the first or last riser may differ from the others. Finished levels must control the calculation.

Measuring headroom incorrectly

Headroom is a vertical clearance. Measuring normal to the pitch line gives a different and usually larger value and may falsely suggest compliance.

Insufficient landing space

A landing that is only graphically shown but not dimensioned may be too small for turning, door operation or evacuation.

Overlooking obstructions

Beams, ducts, sprinkler pipes and light fittings can reduce the available clearance. Coordination must include building services.

Treating guidance as law

The 2R + G relationship is ergonomic guidance. It does not replace code limits, accessibility provisions or fire regulations.

12. Applied case: illustrative apartment stair redesign

Background. An illustrative apartment project has a finished floor-to-floor height of 2,940 mm. The initial sketch shows 16 risers, a 270 mm going and a narrow intermediate landing.

Problem. The initial rise is:

This may exceed a common residential preliminary range and may fail a local maximum-rise requirement. The 16-riser arrangement also creates a long and relatively steep flight if used without a suitable landing.

Analysis. The design team tests 18 risers:

With a 270 mm going:

This is slightly below the commonly cited 600 mm lower guide, so a 275 mm going is tested:

The pitch becomes:

Approach and solution. The 18-riser stair is divided into two flights of 9 risers each. The stair is drawn with 17 treads overall, distributed around the intermediate landing according to the selected layout. A landing equal to the clear stair width is provided initially, then enlarged where required for door clearance. A section confirms that the upper slab does not intrude into the required headroom envelope.

Outcome. The revised stair has a lower rise, a comfortable pitch and a more usable landing. Before approval, the designer must still confirm the governing code, minimum width, handrail geometry, barrier requirements, fire strategy and structural support.

Learning points. A whole-number riser decision should be made early. The actual rise must be recalculated from the total height. Ergonomic checks, plan arrangement and section headroom checks must be performed together rather than as separate drafting tasks.

13. Best-practice checklist

Before issuing a stair drawing, the drafter should:

1. confirm finished floor levels;
2. identify the governing code and occupancy type;
3. choose a whole number of equal risers;
4. recalculate the actual rise;
5. select a compliant going;
6. check 2R + G and pitch;
7. calculate treads and horizontal runs;
8. dimension every landing;
9. verify clear width and door operation;
10. draw and measure headroom in section;
11. include slab, beam and service obstructions;
12. show handrails, guards and nosings;
13. coordinate finishes and structural support;
14. have the design independently checked before submission.

Chapter Summary

Stair design is a coordinated exercise in geometry, ergonomics, safety and regulation. The key dimensions are floor-to-floor height, rise, going, number of risers, number of treads, flight run, landing size, stair width, pitch and headroom. The preliminary comfort relationship 2R + G ≈ 600 to 650 mm assists selection of rise and going, but code requirements always control the final design.

The number of risers is obtained by dividing floor-to-floor height by a trial rise and selecting a whole number. The actual rise is then calculated from the total height. For a typical floor-to-floor flight, the number of treads is one less than the number of risers. Pitch is determined from θ = tan⁻¹(R ÷ G). Landings must provide safe circulation, and headroom must be verified vertically in section. A complete professional drawing communicates levels, dimensions, clearances and safety components and is checked against the current jurisdictional requirements.

Key Terms

Rise: Vertical height of one stair step.

Going: Horizontal depth of one tread measured in the direction of travel.

Riser: Vertical face or vertical interval between consecutive treads.

Tread: Horizontal walking surface of a step.

Nosing: Projecting front edge of a tread.

Flight: Continuous series of steps between landings or floors.

Landing: Level platform at the end or between flights.

Pitch line: Imaginary line connecting tread nosings or equivalent reference points.

Pitch angle: Angle of the pitch line above the horizontal.

Stair width: Clear width available for stair movement.

Headroom: Vertical clearance above the walking surface or pitch reference.

Stairwell: Floor or roof opening accommodating a stair.

Floor-to-floor height: Finished vertical distance between two floors.

Riser count: Number of vertical rises required between floor levels.

Tread count: Number of horizontal walking surfaces in a flight or floor-to-floor stair.

Clear landing: Usable landing area excluding obstructions.

Balustrade: Protective barrier alongside a stair or landing.

Handrail: Graspable support provided alongside a stair.

Comfort formula: Ergonomic guide expressed as 2R + G ≈ 600 to 650 mm.

Finished level: Elevation measured to the completed floor or tread surface.

Review Questions

Short Answer Questions

1. Define rise, going and pitch angle.
2. Why is the number of treads commonly one less than the number of risers in a floor-to-floor flight?
3. State the purpose of the relationship 2R + G.
4. Why must headroom be checked in section?
5. List four items commonly controlled by building codes for stairs.

Long Answer Questions

1. Explain the complete procedure for designing a stair from a known floor-to-floor height.
2. Discuss how rise, going and pitch influence safety and user comfort.
3. Explain the difference between structural stair dimensions and finished stair dimensions.
4. Describe the information that should appear on a professional stair plan and section.
5. Discuss the functions of intermediate landings and the factors affecting their size.

Analytical / Application Questions

1. A floor-to-floor height is 3,240 mm. Select a whole number of risers near 165 mm, calculate the actual rise, choose a 280 mm going, and check the comfort relationship and pitch.
2. A proposed stair has 17 risers and 16 treads, each 175 mm and 250 mm respectively. Calculate the total rise, horizontal run and pitch, then comment on whether the proportions require further investigation.
3. A stair plan appears compliant, but a beam crosses above the upper flight. Explain the sectional checks required to determine whether the stairwell opening provides adequate headroom.

Multiple Choice Questions with Answers

Question 1

A stair has a rise of 165 mm and a going of 280 mm. What is the value of 2R + G?

A. 445 mm

B. 610 mm

C. 725 mm

D. 890 mm

Correct Answer: B. 610 mm

Question 2

A floor-to-floor height is 3,000 mm and 18 equal risers are used. What is the actual rise?

A. 150.0 mm

B. 160.0 mm

C. 166.7 mm

D. 180.0 mm

Correct Answer: C. 166.7 mm

Question 3

For a conventional floor-to-floor stair with 20 risers, how many treads are normally counted?

A. 18

B. 19

C. 20

D. 21

Correct Answer: B. 19

Question 4

Which equation gives the pitch angle of a uniform stair?

A. θ = tan⁻¹(R ÷ G)

B. θ = R × G

C. θ = G ÷ R

D. θ = R + G

Correct Answer: A. θ = tan⁻¹(R ÷ G)

Question 5

If a stair has 17 treads, each 270 mm going, what is the tread run?

A. 3,990 mm

B. 4,320 mm

C. 4,590 mm

D. 4,860 mm

Correct Answer: C. 4,590 mm

Question 6

Which dimension must be measured vertically when checking stair headroom?

A. The stair width

B. The landing diagonal

C. The distance normal to the pitch line

D. The clear vertical distance above the walking reference

Correct Answer: D. The clear vertical distance above the walking reference

Question 7

A 3,360 mm floor-to-floor height is divided into 21 equal risers. What is the rise?

A. 150 mm

B. 160 mm

C. 168 mm

D. 176 mm

Correct Answer: B. 160 mm

Question 8

What is the principal reason for requiring equal risers in a flight?

A. To reduce wall thickness

B. To prevent unexpected changes in foot movement

C. To increase the stair pitch

D. To eliminate the need for a landing

Correct Answer: B. To prevent unexpected changes in foot movement

Question 9

Which drawing is indispensable for verifying stair headroom?

A. A roof plan only

B. A door schedule only

C. A sectional drawing through the stair

D. A furniture layout only

Correct Answer: C. A sectional drawing through the stair

Question 10

The formula 2R + G ≈ 600–650 mm should be understood as:

A. A universal replacement for building codes

B. An ergonomic preliminary design guide

C. A formula for calculating stair width

D. A structural load equation

Correct Answer: B. An ergonomic preliminary design guide