Hip and Valley Rafter Calculator
Free Hip valley rafter Calculator for roofing projects. Enter dimensions to get material lists and cost estimates. Enter your values for instant results.
Calculator
Adjust values & calculateShortening and Bevels
Formula
The hip rafter runs diagonally at 45 degrees in plan view. Its horizontal run is the common rafter run times the square root of 2. The hip rafter length is the hypotenuse of a triangle formed by the hip run and the common rise. The hip pitch per 12 inches of run equals the common pitch divided by the square root of 2. Jack rafters decrease in length by a constant difference based on spacing and pitch.
Last reviewed: December 2025
Worked Examples
Example 1: Standard Hip Roof โ 6/12 Pitch
Example 2: Steep Hip Roof โ 10/12 Pitch
Background & Theory
The Hip and Valley Rafter Calculator applies the following established principles and formulas. Structural and construction engineering is governed by fundamental load analysis, material science, and regulatory standards that ensure the safety and durability of built structures. The primary distinction in load analysis is between dead loads โ the permanent self-weight of structural elements, finishes, and fixed equipment โ and live loads, which represent variable occupancy, furniture, and environmental forces such as wind and snow. These are combined using factored load equations, such as the ASCE 7 formula U = 1.2D + 1.6L, where D is dead load and L is live load. Concrete mix design is governed by the water-cement (w/c) ratio, which is the primary determinant of compressive strength and durability. A w/c ratio of 0.40โ0.45 typically yields concrete with 28-day compressive strengths of 30โ40 MPa. Common mix ratios by weight for structural concrete are approximately 1 part cement : 1.5โ2 parts sand : 3 parts coarse aggregate. Structural steel is characterized by its yield strength (the stress at which permanent deformation begins, typically 250โ350 MPa for mild steel) and ultimate tensile strength (typically 400โ500 MPa). Mid-span deflection of a simply supported beam under a central point load is given by ฮด = FLยณ / (48EI), where F is force, L is span length, E is Young's modulus, and I is the second moment of area. Building insulation is rated by R-value, a measure of thermal resistance in units of mยฒยทK/W (SI) or ftยฒยทยฐFยทh/BTU (imperial). Higher R-values indicate greater resistance to heat flow. Foundation design depends on the allowable bearing capacity of the underlying soil, which ranges from approximately 75 kPa for soft clay to over 10,000 kPa for bedrock. Drainage gradients for surface water are typically specified as a minimum of 1โ2% slope away from building foundations to prevent hydrostatic pressure and water infiltration.
History
The history behind the Hip and Valley Rafter Calculator traces back through the following developments. The history of construction engineering spans thousands of years of accumulated empirical knowledge and, more recently, rigorous scientific analysis. The ancient Egyptians built the Great Pyramid of Giza around 2560 BCE using an estimated 2.3 million stone blocks, demonstrating sophisticated logistics, geometry, and workforce organization. Roman engineers advanced the field dramatically through the use of pozzolanic concrete โ a mixture of volcanic ash, lime, and seawater โ enabling the construction of the Pantheon dome (43.3 m diameter, completed around 125 CE) and a vast network of aqueducts and roads across the empire. Cast iron emerged as a structural material during the Industrial Revolution, first used prominently in the Iron Bridge at Coalbrookdale, England, completed in 1779. Wrought iron and later steel allowed far greater spans and heights. The Eiffel Tower, completed in 1889, demonstrated the structural possibilities of wrought iron at scale and influenced the development of steel-frame skyscraper construction in Chicago and New York. Reinforced concrete was systematically developed by Joseph Monier, a French gardener, who patented iron-reinforced concrete pots and panels in the 1860s, and later by engineers including Franรงois Hennebique who created the first comprehensive reinforced concrete framing system in the 1890s. The 1906 San Francisco earthquake caused widespread devastation and galvanized the engineering profession to develop seismic design provisions. Subsequent earthquakes โ including the 1971 San Fernando and 1994 Northridge events โ drove successive improvements in seismic codes, base isolation technology, and ductile detailing of reinforced concrete and steel frames. Building codes became increasingly standardized in the twentieth century, with the International Building Code (IBC) first published in 2000 providing a unified model code adopted across much of the United States. Building Information Modeling (BIM) emerged in the 2000s as a digital workflow integrating architectural, structural, and MEP design into a unified three-dimensional model, fundamentally changing coordination practices across the industry.
Frequently Asked Questions
Formula
Hip Length = sqrt((Run x sqrt(2))ยฒ + Riseยฒ)
The hip rafter runs diagonally at 45 degrees in plan view. Its horizontal run is the common rafter run times the square root of 2. The hip rafter length is the hypotenuse of a triangle formed by the hip run and the common rise. The hip pitch per 12 inches of run equals the common pitch divided by the square root of 2. Jack rafters decrease in length by a constant difference based on spacing and pitch.
Frequently Asked Questions
How do you calculate a hip rafter length?
A hip rafter runs diagonally from the ridge to the corner of the building at 45 degrees in plan view. Its horizontal run is the common rafter run multiplied by the square root of 2 (approximately 1.414). The hip rafter length equals the square root of the hip run squared plus the common rise squared. For a building with a 12-foot common run and 6/12 pitch, the common rise is 6 feet, the hip run is 16.97 feet, and the hip rafter length is approximately 18.0 feet before shortening adjustments.
What is the difference between a hip rafter and a valley rafter?
A hip rafter runs from the ridge down to an outside corner of the building where two roof planes meet at a convex angle. A valley rafter runs from the ridge down to an inside corner where two roof planes meet at a concave angle, such as where a dormer or wing intersects the main roof. Mathematically, hip and valley rafters have the same length and angle calculations when both roof sections have the same pitch. The key difference is in the backing bevel direction and the connection details at the ridge and bearing points.
What is the jack rafter difference and how is it calculated?
The jack rafter difference is the consistent length change between consecutive jack rafters spaced equally along the hip or valley. For equal-pitch hip roofs, each jack rafter is shorter than the previous one by a fixed amount that depends on the rafter spacing and roof pitch. The difference equals the square root of the spacing squared plus the rise-per-spacing squared. For 16-inch on-center spacing with a 6/12 pitch, the difference is approximately 17.9 inches. This makes layout efficient because you cut each subsequent jack rafter shorter by the same amount.
What is the hip rafter backing angle?
The backing angle is the bevel cut applied to the top edges of a hip rafter so the roof sheathing sits flat against it. Without backing, the sheathing would sit on the sharp top corners of the hip rafter, creating gaps at the hip line. The backing angle depends on the roof pitch and gets steeper as the pitch increases. For a 6/12 pitch, the backing angle is approximately 14.5 degrees. Some framers drop the hip rafter instead of backing it, which achieves the same result by lowering the rafter seat cut rather than beveling the top edges.
How do you shorten a hip rafter at the ridge?
The hip rafter must be shortened at the ridge because it meets at the intersection of two ridge boards rather than butting directly to the end of one. The shortening amount equals half the diagonal thickness of the ridge board divided by the cosine of the hip rafter angle. For a standard 1.5-inch thick ridge board, the shortening is half of 1.5 times the square root of 2 (about 1.06 inches) measured horizontally, then adjusted along the rafter slope. Always measure the shortening perpendicular to the plumb cut line at the ridge end of the hip rafter.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
References
Reviewed by Abdullah, Technical Content Specialist ยท Editorial policy