Maps for Math

What angle should solar panels be tilted?

A panel catches the most sun when it points straight at it. The share it gets is the dot product of two unit arrows, one toward the sun and one straight out of the panel, which equals the cosine of the angle between them. At noon, the best tilt is your latitude minus the sun's seasonal angle.

faces southwest ←panel faces71.5% of full sunyour paneldotted: a flat panel
Share of full sunlight on the panel
71.5%
Sun height above the horizon
73.4°
Whole-day sunlight vs. a flat panel
×0.744
Best tilt at noon this season
16.6°

This is the angle only. Clouds, haze, heat, dust, and shade cut real output, often by a lot.

Challenge: In a December at 47°N or farther north, collect at least 3× a flat panel's sunlight over the day.

Solar noon is when the sun is highest, often around 1 pm with daylight saving time.
More settings
Miami is about 26°, Denver 40°, Seattle 47°, London 51°.

Play

Start with the Denver roof, then move the time and the season. Compare a flat panel in summer and winter.

Challenge: In a December at 47°N or farther north, collect at least 3× a flat panel's sunlight over the day. The box under the picture turns green when you get it.

Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”

Understand

cos⁡θi=n^⋅s^=nxsx+nysy+nzsz\cos\theta_i = \hat n \cdot \hat s = n_x s_x + n_y s_y + n_z s_z

Point one arrow at the sun, s^\hat s, and one straight out of the panel, n^\hat n. The share of full sunlight the panel catches is their dot product:

cos⁡θi=n^⋅s^=nxsx+nysy+nzsz\cos\theta_i = \hat n \cdot \hat s = n_x s_x + n_y s_y + n_z s_z

When they line up, it's 1, full sun. When the sun hits at a slant, it's less, and behind the panel it's 0. The chart shows that share over the day, next to a flat panel.

Use

Every input has a unit menu, so you can type values in the units you already have. Results follow your units.

Show the work

  1. Arrow toward the sun (east, north, up), from latitude, season, and time\hat s = (0,\ -0.285,\ 0.959)
  2. Arrow straight out of the panel: tilt 55°, facing southwest\hat n = (\sin\beta\sin A,\ \sin\beta\cos A,\ \cos\beta) = (-0.579,\ -0.579,\ 0.574)
  3. Dot product: multiply matching parts and add\hat n \cdot \hat s = (-0.579)(0) + (-0.579)(-0.285) + (0.574)(0.959) = 0.715
  4. That’s the cosine of the angle between them\cos\theta_i = 0.715 \Rightarrow \theta_i = 44.4^\circ

Export

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Set the tilt, time, season, and which way the panel faces. "More settings" changes the latitude.

  • This counts direct sunlight only. Clouds, haze, heat, and shade lower real output.

For learning and estimation. Verify with applicable codes, standards, and a qualified professional before using in design, construction, or safety-critical work.

Cheat card

cos⁡θi=n^⋅s^=nxsx+nysy+nzsz\cos\theta_i = \hat n \cdot \hat s = n_x s_x + n_y s_y + n_z s_z
best noon tilt≈φ−δ\text{best noon tilt} \approx \varphi - \delta
n^=(sin⁡βsin⁡A, sin⁡βcos⁡A, cos⁡β)\hat n = (\sin\beta\sin A,\ \sin\beta\cos A,\ \cos\beta)
SymbolMeaningUnit
n^\hat nunit arrow straight out of the panel
s^\hat sunit arrow toward the sun
φ\varphilatitude°
δ\deltasun’s seasonal angle (+23.4° in June, −23.4° in December)°
  • Face the panel toward the equator, south in the northern hemisphere.
  • Tilt steeper in winter, when the sun is low, and flatter in summer.
  • A year-round compromise tilt is close to your latitude.

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Where it’s used

  • Electrical & Electronics
    Solar installers choose tilt and direction to maximize a year of energy.
  • Physics
    The same cosine rule explains why winter sunlight is weaker, since it hits the ground at a slant.
  • Home Projects
    Decide whether your roof’s angle and direction work for panels.

Questions people ask

What does the dot product have to do with sunlight?

Multiply matching parts of two arrows and add them up. For unit arrows that's the cosine of the angle between them, which is exactly the share of full sunlight a slanted surface catches.

Is a tilted panel always better than a flat one?

No. In summer at mid-latitudes, the sun rises and sets toward the north, behind a south-tilted panel, so a flat panel collects about as much or more over the whole day.

What tilt is best year-round?

About your latitude, facing the equator. Steeper helps in winter, and flatter helps in summer.

Sample VEC-0005 · rev 1 · model VEC-M05 · Level 4 · advanced algebra · updated 2026-09-29