| Metric | Value |
|---|---|
| Angle (degrees) | 5° |
| Angle (radians) | 0.0872664626 |
| Angle (gradians) | 5.5555555556g |
| Sine, sin(θ) | 0.087155742748 |
| Cosine, cos(θ) | 0.996194698092 |
| Tangent, tan(θ) | 0.087488663526 |
| Cotangent, cot(θ) | 11.430052302761 |
| Secant, sec(θ) | 1.003819837543 |
| Cosecant, csc(θ) | 11.47371324567 |
| Complementary angle | 85° |
| Supplementary angle | 175° |
| Reflex angle | 355° |
| Grade (tan × 100) | 8.7488663526% |
| Slope ratio (run per 1 rise) | 1 : 11.4300523028 |
| Item | Value |
|---|---|
| Known side type | Adjacent |
| Known side value | 10 |
| Adjacent (a) | 10 |
| Opposite (o) | 0.8748866353 |
| Hypotenuse (h) | 10.0381983754 |
| Item | Value |
|---|---|
| Radius (r) | 12 |
| Arc length (s = rθ) | 1.047197551197 |
| Sector area (A = ½r²θ) | 6.28318530718 |
| Chord length (c = 2r sin(θ/2)) | 1.046865296768 |
| Item | Value |
|---|---|
| Rotation matrix R(θ) | [[cosθ, −sinθ], [sinθ, cosθ]] |
| cosθ | 0.996194698092 |
| −sinθ | -0.087155742748 |
| sinθ | 0.087155742748 |
| Case | Angle | Given | Key outputs |
|---|---|---|---|
| Roof / ramp grade | 5° | — | tan≈0.08749, grade≈8.749% |
| Right triangle | 5° | Adjacent = 10 | Opp≈0.8749, Hyp≈10.0382 |
| Circle sector | 5° | Radius = 12 | Arc≈1.0472, Area≈6.2832 |
Angle conversions
- θ(rad) = θ(deg) × π / 180
- θ(deg) = θ(rad) × 180 / π
- θ(grad) = θ(deg) × 200 / 180
Trigonometry
- sin(θ), cos(θ), tan(θ)=sin(θ)/cos(θ)
- cot(θ)=cos(θ)/sin(θ), sec(θ)=1/cos(θ), csc(θ)=1/sin(θ)
Right triangle (θ at the base)
- tan(θ) = opposite / adjacent
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
Grade and slope ratio
- grade(%) = tan(θ) × 100
- run per 1 rise = 1 / tan(θ)
Circle sector (θ in radians)
- arc length: s = rθ
- sector area: A = ½ r² θ
- chord length: c = 2r sin(θ/2)
- Keep Use fixed 5° angle enabled for strict output.
- Optionally, switch it off to try other angles.
- To solve a triangle, enter one known side value.
- Select whether it is adjacent, opposite, or hypotenuse.
- For circle outputs, enter a radius in your unit.
- Click Calculate to update all computed values.
- Use Download CSV or Download PDF to save.
1) What this page calculates
This tool locks onto 5° by default and reports degrees, radians, gradians, and core trig values. For 5°, θ≈0.087266 radians and θ≈5.5556 gradians, so you can switch between engineering and surveying units without manual conversions.
2) Key trig data for 5°
When θ=5°, sinθ≈0.08716, cosθ≈0.99619, and tanθ≈0.08749. These numbers are small but useful: cos stays near one, while tan gives a reliable “rise over run” estimate for mild slopes.
3) Slope and grade interpretation
Grade is tanθ×100. At 5°, grade≈8.749%. That means an 8.749 unit rise per 100 units run. The inverse ratio is also helpful: run per 1 rise ≈ 11.43, so you need about 11.43 meters of run to climb 1 meter.
4) Right-triangle sizing with one side
If the adjacent side is 10, opposite≈0.8749 and hypotenuse≈10.0382. If the hypotenuse is 10, adjacent≈9.9619 and opposite≈0.8716. These outputs help with layout checks, bracing, and small-angle offsets.
5) Circle geometry at small angles
Sector calculations require radians. With radius 12 and θ=5°, arc length s=rθ≈1.0472 and sector area A=½r²θ≈6.2832. Chord length stays close to the arc: c≈1.0469, which is handy for quick chord spacing.
6) Rotation matrix insight
The tool also shows the 2D rotation matrix using cosθ and sinθ. For 5°, the matrix is near the identity matrix, which explains why small rotations preserve most of a vector’s original x-component while adding a modest y-component.
7) Practical accuracy tips
Keep units consistent for triangle and circle inputs. Use more decimals when values are tiny, especially with short lengths. If your problem is sensitive, measure twice and compute twice; small angle errors can dominate results over long runs. For long baselines, even a 0.5° setup drift changes grade noticeably. Over 20 m of run, 5° gives about 1.75 m rise, while 5.5° gives about 1.93 m. Use a level or inclinometer, and record the measured angle for audits and rechecks before cutting, drilling, or pouring.
1) Why is 5° used so often?
Five degrees is a gentle, repeatable angle that approximates many real ramps and pitches. It is steep enough to model drainage and motion, yet small enough for quick linear estimates in layouts.
2) What is the grade percentage at 5°?
Grade equals tan(θ) × 100. For θ = 5°, tanθ ≈ 0.08749, so grade ≈ 8.749%. This means about 8.749 units of rise per 100 units of run.
3) How do I use the right-triangle solver?
Enter a known side length, then choose whether it is adjacent, opposite, or hypotenuse. The calculator applies sine, cosine, and tangent relationships to compute the other two sides instantly.
4) Why do circle formulas need radians?
Arc length and sector area are derived from the radian definition, where angle equals arc divided by radius. Using radians keeps formulas simple: s = rθ and A = ½r²θ.
5) Can I calculate angles other than 5°?
Yes. Turn off the fixed 5° switch, enter your angle, and select the unit. The same outputs update, including trig values, grade, triangle sides, and circle geometry.
6) Why does “undefined” appear for some values?
Some functions require dividing by sinθ or cosθ. If sin or cos is extremely close to zero, values like tan, sec, or csc become unstable. The calculator labels these cases as undefined.