Angle calculator: add, subtract, multiply and divide degrees, minutes and seconds
Do math with angles the way they are written in the field book: degrees, minutes and seconds. The calculator adds, subtracts, multiplies and divides, shows the result in every format and explains each step.
Type the angles with the symbols ° ′ ″ (or 25:30:15) and use + − × ÷ and parentheses. A number without a symbol is a multiplier or divisor; added to an angle, it counts as degrees.
Paste the angles from a field book column or from Excel: 92°15′20″, 92 15 20 or 92.2556. Shows the sum, the mean and the angular misclosure if they are the angles of a polygon.
Result
Type an expression above.
How to do it by hand
Adding: add the degrees, minutes and seconds separately. Every 60″ carries 1 minute, and every 60′ carries 1 degree. E.g.: 25°30′15″ + 10°45′50″ = 35°75′65″ → 35°76′05″ → 36°16′05″.
Subtracting: if the top seconds (or minutes) are smaller, borrow 1′ = 60″ (or 1° = 60′) from the column on the left. E.g.: 50°10′20″ − 20°30′40″ → 49°69′80″ − 20°30′40″ = 29°39′40″.
Multiplying and dividing: multiply each part and carry every 60 to the left. To divide, divide the degrees, turn the remainder into minutes (× 60) and add it to the minutes, then do the same with the seconds.
Why not just use decimals? Because 25°30′ is 25.5°, not 25.30°: the base is 60, not 10. Adding 25.30 + 10.45 gives the wrong answer.
FAQ
How do you add degrees, minutes and seconds?
Add each part separately (degrees with degrees, minutes with minutes, seconds with seconds). If the seconds reach 60, subtract 60 and add 1 minute; if the minutes reach 60, subtract 60 and add 1 degree. E.g.: 25°30′15″ + 10°45′50″ = 36°16′05″.
How do you subtract angles when you run short of seconds or minutes?
Borrow from the column on the left: 1 degree becomes 60 minutes and 1 minute becomes 60 seconds. E.g.: 50°10′20″ − 20°30′40″: borrow 1′ (20″ becomes 80″) and 1° (9′ becomes 69′), then 49°69′80″ − 20°30′40″ = 29°39′40″.
How do you multiply or divide an angle by a number?
Multiply the degrees, minutes and seconds by the number and carry every 60 to the column on the left. To divide, divide the degrees, turn the remainder into minutes (× 60), add it to the minutes and divide; then do the same with the seconds. E.g.: 100°20′30″ ÷ 3 = 33°26′50″.
What is the sum of the interior angles of a polygon?
(n − 2) × 180°, where n is the number of sides. A quadrilateral adds up to 360°, a pentagon to 540° and a hexagon to 720°. The exterior angles add up to (n + 2) × 180°. The difference between the measured and the theoretical sum is the angular misclosure of the traverse.
How do you do this on a scientific calculator?
Use the degrees-minutes-seconds key (°′″ or DMS): type 25 °′″ 30 °′″ 15 °′″ + 10 °′″ 45 °′″ 50 °′″ = and the calculator shows 36°16′5″. Without that key, convert to decimal degrees, do the math and convert back.