How do I convert slope distance to horizontal distance?

Multiply the slope distance by the cosine of the vertical angle: H = S × cos α. With a zenith angle (the angle a total station shows, 90° = level) use H = S × sin z. Example: 1,474.00 ft at zenith 86°06′36″ gives H = 1,474 × sin 86.11° = 1,470.60 ft.

How do I calculate slope distance from horizontal distance and elevation difference?

Use the Pythagorean theorem: S = √(H² + V²). A 250 ft run with a 12 ft rise has a slope distance of √(62,500 + 144) = 250.29 ft, a grade of 4.80% and a vertical angle of 2.75°.

What is the difference between a zenith angle and a vertical angle?

A vertical angle is measured up or down from the horizontal (positive uphill, negative downhill). A zenith angle is measured down from straight overhead, so a level sight reads 90°. They are complementary: α = 90° − z. A zenith angle of 86° is a 4° uphill sight; 94° is a 4° downhill sight.

What is the slope correction for a taped distance?

The slope correction is the amount to subtract from the slope distance: S − H = S × (1 − cos α), or approximately V² ÷ 2S. A 30.000 m tape on an 8% grade reads 0.0955 m long, so the horizontal distance is 29.9045 m.

When do I need the curvature and refraction correction?

On long sights. The combined correction is (c + r) = 0.0675 K² metres with K in kilometres (0.0206 F² ft with F in thousands of feet), using a refraction coefficient of 0.14. It is under 1 mm at 100 m, about 7 mm at 300 m, 68 mm at 1 km and 1.7 m at 5 km. It is added to the trigonometric elevation difference.

How do I convert percent grade to an angle?

Angle = arctan(grade ÷ 100). A 10% grade is 5.71°, 100% is 45°. A 1:n ratio (1 unit of rise per n units of run) is a grade of 100 ÷ n percent, so 1:12 is 8.33% or 4.76°.

How do I enter degrees, minutes and seconds?

Type the three numbers separated by spaces or symbols, for example 86 06 36 or 86°06'36". The tool converts to decimal degrees (86.11°) and shows the angle back in both forms.

Slope Distance Calculator

Horizontal distance = slope distance × cos(vertical angle) = slope distance × sin(zenith angle); elevation difference = slope distance × sin(vertical angle). Knowing horizontal distance and rise, slope distance = √(H² + V²).

Input Parameters

What do you know?

Distance unit

Slope distance (S)

Inclination given as

From horizontal; negative downhill. Decimal degrees or D M S (e.g. 12 30 15).

Trigonometric leveling (optional)

Instrument height (HI)

Target / prism height (HR)

Apply curvature & refraction (k = 0.14)

Results

Horizontal distance (H)

—

Elevation difference (V)

—

Slope distance (S)

—

Vertical angle

—

Zenith angle

—

Grade

—

Ratio (rise : run)

—

Slope correction (S − H)

—

Visual Diagram

Grade, angle and slope correction per 100 units of slope distance

Works in any unit: multiply by S ÷ 100. Correction = S − H.

Grade	Angle	Zenith	H per 100	V per 100	S − H per 100
1%	0.573°	89.427°	99.9950	1.0000	0.0050
2%	1.146°	88.854°	99.9800	1.9996	0.0200
3%	1.718°	88.282°	99.9550	2.9987	0.0450
5%	2.862°	87.138°	99.8752	4.9938	0.1248
8%	4.574°	85.426°	99.6815	7.9745	0.3185
10%	5.711°	84.289°	99.5037	9.9504	0.4963
12%	6.843°	83.157°	99.2877	11.9145	0.7123
15%	8.531°	81.469°	98.8936	14.8340	1.1064
20%	11.310°	78.690°	98.0581	19.6116	1.9419
25%	14.036°	75.964°	97.0143	24.2536	2.9857
30%	16.699°	73.301°	95.7826	28.7348	4.2174
50%	26.565°	63.435°	89.4427	44.7214	10.5573
100%	45.000°	45.000°	70.7107	70.7107	29.2893

Curvature and refraction correction by sight length

(c + r) = 0.0675 K² m, K in km (= 0.0206 F² ft, F in thousands of feet); refraction coefficient 0.14, Earth radius 6,371 km.

Horizontal distance	Feet	c + r (mm)	c + r (ft)
100 m	328	0.68	0.0022
300 m	984	6.08	0.0199
500 m	1,640	17	0.0554
1,000 m	3,281	68	0.2215
2,000 m	6,562	270	0.8858
5,000 m	16,404	1,688	5.5364
10,000 m	32,808	6,750	22.1457

Worked examples

Total station shot. Slope distance 1,474.00 ft, zenith angle 86°06′36″ (86.11°), HI 3.50 ft, HR 3.00 ft. H = 1,474 × sin 86.11° = 1,470.60 ft. V = 1,474 × cos 86.11° = 100.00 ft. Ground elevation difference = 100.00 + 3.50 − 3.00 = 100.50 ft, or 100.54 ft with curvature and refraction (0.0206 × 1.4706² = 0.045 ft).

Tape on a grade. A 30.000 m tape length on an 8% grade: α = arctan 0.08 = 4.574°, H = 30 × cos 4.574° = 29.9045 m, slope correction 0.0955 m.

Ramp or roof length from rise and run. Run 250 ft, rise 12 ft: S = √(250² + 12²) = 250.29 ft, grade 4.80%, angle 2.75°.

Method: right-triangle reduction as in Ghilani & Wolf, Elementary Surveying (slope reduction, trigonometric leveling and the 0.0675 K² curvature and refraction formula). The horizontal distance here is at instrument height; geodetic reduction to the ellipsoid or a grid (scale factor) is a separate step.

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Summary

A slope distance is measured along the line of sight or along the ground, for example by a tape on a hillside or a total station EDM. Plans and areas need the horizontal distance, and levels need the elevation difference. This calculator solves the right triangle from any two of its parts: slope distance with an inclination, horizontal distance with an inclination, horizontal distance with an elevation difference, or slope distance with an elevation difference. The inclination can be a vertical angle, a zenith angle (as total stations display it), a percent grade or a 1:n ratio, in decimal degrees or degrees-minutes-seconds. For trigonometric leveling it adds instrument and target heights and the combined curvature and refraction correction for long sights.

How it works

Choose what you know: slope distance and inclination, horizontal distance and inclination, horizontal distance and elevation difference, or slope distance and elevation difference.
Enter the inclination as a vertical angle (from horizontal, negative downhill), a zenith angle (from straight up, 90° is level), a percent grade or a 1:n ratio. Angles accept decimal degrees or D M S such as 86 06 36.
H = S × cos α = S × sin z, V = S × sin α = S × cos z, grade % = V ÷ H × 100, and S = √(H² + V²).
For trigonometric leveling, add instrument height (HI) and target or reflector height (HR): ground elevation difference = V + HI − HR.
Curvature and refraction (c + r) = 0.0675 K² metres, K = horizontal distance in km (k = 0.14). It is about 6.8 mm at 300 m and 6.8 cm at 1 km, so it matters only on long sights.

Use cases

Reducing total station or EDM slope distances and zenith angles to horizontal distances and elevation differences.
Converting tape measurements taken on sloping ground to horizontal distances (slope correction).
Finding the slope length of a ramp, roof, road or pipe from its horizontal run and rise.
Setting out a horizontal distance on a known grade by computing the slope distance to measure.
Trigonometric leveling between two points with instrument and prism heights.
Converting a clinometer reading and a slope measurement to map distance for forestry or geology.

Frequently Asked Questions

Last updated: 2026-10-10 · Reviewed by Nham Vu