Degrees are used to measure several different things: angles, temperature, geographic coordinates, and slope. The method depends entirely on what you’re measuring. A protractor measures angles in geometry, a thermometer measures degrees of temperature, and a digital inclinometer measures degrees of tilt or slope. Here’s how each type works and what you need to know to get accurate results.
Measuring Angles With Common Tools
The most basic way to measure degrees is with a protractor, the semicircular tool marked from 0 to 180. Place the flat edge along one line of the angle, align the center point with the vertex (the corner where the two lines meet), and read where the second line crosses the numbered scale. For angles larger than 180 degrees, measure the smaller angle first and subtract from 360.
A digital inclinometer measures tilt angles electronically. Inside, a tiny weighted plate is suspended between fixed electrodes. When the device tilts, the plate shifts position, changing the electrical signal between the electrodes. The sensor converts that change into a degree reading on a screen. Consumer-grade versions are accurate to within about 1 degree, which is fine for hanging shelves or checking if a surface is level. Professional-grade sensors used in construction and engineering use arrays of precise electrodes to push accuracy well below that threshold.
For quick estimates without any tools, your hand works surprisingly well. At arm’s length, your pinky finger covers roughly 1 degree of arc. A closed fist spans about 10 degrees. The distance from your outstretched thumb to your pinky covers about 25 degrees. Astronomers use these hand measurements to locate objects in the night sky. The full moon, for reference, is only about half a degree wide.
Measuring Temperature in Degrees
Temperature degrees quantify how hot or cold something is, using one of three main scales. Fahrenheit places water’s freezing point at 32°F and boiling point at 212°F. Celsius sets freezing at 0°C and boiling at 100°C. Kelvin starts at absolute zero, the coldest possible temperature: negative 273.15°C, the point where atoms have essentially no thermal energy. Notably, the kelvin is the official scientific unit for temperature and is written without the word “degrees” before it (just “300 kelvin,” not “300 degrees Kelvin”).
To convert between the two everyday scales:
- Celsius to Fahrenheit: Multiply by 9/5, then add 32. So 37°C becomes (37 × 9/5) + 32 = 98.6°F.
- Fahrenheit to Celsius: Subtract 32, then multiply by 5/9. So 72°F becomes (72 − 32) × 5/9 = 22.2°C.
Which Thermometer Placement Is Most Accurate
If you’re measuring body temperature, where you place the thermometer matters more than most people realize. A study comparing common methods against a deep-body reference found that ear (tympanic) thermometers had the highest agreement with core body temperature, with a kappa coefficient of 0.88 for the right ear and 0.85 for the left. Oral thermometers came next at 0.72. Armpit (axillary) and forehead methods were notably less reliable, scoring 0.60 and 0.59 respectively.
In practical terms, this means an ear thermometer will catch a fever about 81 to 83% of the time, while a forehead thermometer catches it only about 54% of the time. Armpit readings miss nearly half of actual fevers. If accuracy matters, such as checking a child’s temperature when you suspect illness, an ear thermometer is your best bet among non-invasive options. Rectal thermometers remain the clinical gold standard for infants, though the study above didn’t include them in its comparison.
Measuring Slope and Roof Pitch in Degrees
In construction, slope is often described as “rise over run,” meaning how many inches a surface climbs vertically for every 12 inches of horizontal distance. A 6/12 pitch means the roof rises 6 inches for every 12 inches of horizontal run. To convert that ratio into degrees, you use the arctangent function: divide the rise by the run, then take the inverse tangent of the result.
For the 6/12 example: 6 ÷ 12 = 0.5, and the arctangent of 0.5 is about 26.6 degrees. Most smartphone calculator apps have an arctan or tan⁻¹ button that does this instantly. You can also use a digital inclinometer placed directly on the surface. Common roof pitches range from about 14 degrees (a low 3/12 slope) to 45 degrees (a steep 12/12 slope).
Degrees in Geographic Coordinates
Latitude and longitude use degrees to pinpoint any location on Earth. Latitude runs from 0° at the equator to 90° at the poles. Longitude runs from 0° at the Prime Meridian to 180° east or west. Each degree is subdivided into 60 minutes, and each minute into 60 seconds, written with symbols like 43°2’27” N, 77°14’30.60″ W.
One degree of latitude equals roughly 69 miles (111 kilometers) on the ground. One minute narrows that to about 1.15 miles, and one second to roughly 100 feet. This subdivision system gives GPS coordinates enough precision to locate a specific building. Many modern apps display coordinates in decimal degrees instead (like 43.0408° N), which is the same information in a different format. To convert, divide the minutes by 60 and the seconds by 3600, then add them to the whole degrees.
Converting Degrees to Radians
In trigonometry and higher math, angles are often expressed in radians rather than degrees. A full circle equals 360 degrees or 2π radians. To convert degrees to radians, multiply by π/180. So 90 degrees becomes 90 × π/180 = π/2 radians (about 1.571). To go the other direction, multiply radians by 180/π.
Radians are preferred in mathematics and physics because they’re based on the geometry of a circle itself, not an arbitrary division into 360 parts. If you’re working with sine, cosine, or tangent functions in a calculator or spreadsheet, check whether it’s set to degrees or radians mode. Getting this wrong is one of the most common sources of incorrect answers in trigonometry. Most scientific calculators have a “DEG/RAD” toggle, and spreadsheet functions like Excel’s SIN() expect radians by default.