Stairs Stringer Calculator
Stringer Calculation Results:
Number of Risers:
Actual Riser Height:
Number of Treads:
Tread Run (Effective Depth):
Total Run (Horizontal Length):
Required Stringer Length (Approx.):
Cut Angle (for level tread): degrees
Notch Angle (for riser): degrees
Understanding Stairs and Stringer Calculations
Building a safe and functional staircase involves precise measurements and calculations, particularly when it comes to cutting the stringers. Stair stringers are the structural, often triangular, beams that support the treads and risers of a staircase. Their proper dimensioning ensures stability, consistent step height, and comfortable usability.
Key Measurements and Their Importance:
- Total Rise: This is the total vertical distance the stairs must cover, typically from the finished floor of one level to the finished floor of the next. Accuracy here is paramount for achieving the correct overall height.
- Desired Riser Height: This is the target vertical dimension for each individual step. Building codes often specify a range for riser height (e.g., 4 inches to 7 inches) to ensure comfortable climbing.
- Average Tread Depth: This refers to the usable horizontal surface of each step. Similar to riser height, codes dictate minimum tread depths (e.g., 9 inches) for safety.
- Stringer Board Thickness: The thickness of the lumber used for the stringers affects the structural integrity and the depth of the cuts made.
The Math Behind Stringer Calculation:
The core of stair stringer calculation lies in determining the number of steps and their exact dimensions based on the total rise and desired step heights. The relationship between riser height and tread depth is often described by the "6-8 rule" or "2R+T rule," which suggests that for comfortable stairs, twice the riser height plus the tread depth should be between 24 and 25 inches (approximately 60-63 cm).
- Calculate Number of Risers: The total rise is divided by the desired riser height to get an initial estimate of the number of risers. Since you can't have a fraction of a riser, this number is usually rounded to the nearest whole number.
Formula:
Number of Risers = Total Rise / Desired Riser Height - Determine Actual Riser Height: Once the number of risers is fixed, the actual riser height is calculated by dividing the total rise by the actual number of risers. This ensures the stairs reach the exact total rise.
Formula:
Actual Riser Height = Total Rise / Number of Risers - Calculate Number of Treads: For a standard staircase, there is typically one less tread than the number of risers (e.g., 10 risers means 9 treads). This is because the top landing or floor acts as the final "tread."
Formula:
Number of Treads = Number of Risers - 1 - Determine Effective Tread Run: This is the horizontal distance each step covers. It's usually dictated by the desired tread depth, but can be adjusted slightly to fit code or comfort guidelines, keeping the 2R+T rule in mind.
Formula:
Tread Run = Average Tread Depth(often adjusted slightly for code, aiming for 2R + T = ~25) - Calculate Total Run: This is the total horizontal length the stairs will occupy.
Formula:
Total Run = Number of Treads * Tread Run - Calculate Stringer Length: The stringer length is the hypotenuse of a right triangle formed by the total rise and the total run. The Pythagorean theorem is used here.
Formula:
Stringer Length = sqrt( (Total Rise)^2 + (Total Run)^2 ) - Determine Cut Angles:
- Cut Angle (for level tread): This angle is used to ensure the tread sits level. It's calculated using the arctangent of the ratio of the actual riser height to the tread run.
Formula:
Cut Angle = atan(Actual Riser Height / Tread Run)(in degrees) - Notch Angle (for riser): This angle is complementary to the cut angle, related to the angle of the riser. Often, this is 90 degrees minus the cut angle if considering the angle relative to the stringer's edge. However, in practice, it's often derived from the pitch of the stairs. A simpler interpretation for cutting is the angle used for the housing cut where the tread sits. For simplicity, we'll calculate the angle relative to the horizontal.
Formula:
Notch Angle = atan(Tread Run / Actual Riser Height)(in degrees), or more commonly derived from 90 – Cut Angle in some contexts, or calculated based on framing square markings.Note: Angle calculations can vary based on terminology and the tools used (e.g., framing square markings). This calculator provides angles relative to the horizontal for general understanding.
- Cut Angle (for level tread): This angle is used to ensure the tread sits level. It's calculated using the arctangent of the ratio of the actual riser height to the tread run.
Example Calculation:
Let's assume:
- Total Rise: 108 inches
- Average Tread Depth: 10 inches
- Desired Riser Height: 7 inches
- Stringer Board Thickness: 1.5 inches
Step 1: Number of Risers = 108 / 7 ≈ 15.42. Rounded to 15 risers.
Step 2: Actual Riser Height = 108 / 15 = 7.2 inches.
Step 3: Number of Treads = 15 – 1 = 14 treads.
Step 4: Tread Run = 10 inches (assuming this meets the 2R+T rule: 2*7.2 + 10 = 14.4 + 10 = 24.4, which is within the typical 24-25 range).
Step 5: Total Run = 14 * 10 = 140 inches.
Step 6: Stringer Length = sqrt( (108)^2 + (140)^2 ) = sqrt(11664 + 19600) = sqrt(31264) ≈ 176.8 inches.
Step 7: Cut Angle = atan(7.2 / 10) ≈ atan(0.72) ≈ 35.75 degrees.
Step 7b: Notch Angle = atan(10 / 7.2) ≈ atan(1.389) ≈ 54.25 degrees. (Note: 35.75 + 54.25 = 90 degrees)
Using the Calculator:
Enter the Total Rise, desired Riser Height, and Average Tread Depth. The calculator will automatically adjust the actual riser height and determine the number of treads and the total run. It also provides the approximate stringer length required and the critical angles for cutting the stringer accurately.