Area of a Triangle Calculator

Area of a triangle

The simplest rule is half of a side times the height that stands on it. The height is the perpendicular distance from the opposite corner to the line that contains that side; it is not the length of another side unless the triangle has a right angle there.

A = ½ × base × heightAny of the three sides can serve as the base, as long as the height is measured to that same side.

When the height is not known, the other measurements you do have decide which formula to use. The calculator covers all seven common situations, listed in the next section. Throughout this page the sides are called a, b and c, and the angles α, β and γ sit opposite them: α faces a, β faces b and γ faces c.

How to use the calculator

1. Pick the case that matches your data in Solve triangle by. The form then shows only the fields that case needs.
2. Choose the unit of the answer in Metric.
3. Fill in the measurements. Every length has its own unit selector, so different units can be mixed; angles are in degrees.
4. Press Calculate area. The result is given in squared units with the unit written out, for example 22.591692 dm² (square decimeters).

Changing the Metric list only changes the unit of the answer. Set the unit next to each measurement to what you actually measured. The result is rounded to at most six decimal places.

Rules for solving a triangle

Each name lists what is known: S is a side, A an angle, H a height (or, in HL, the hypotenuse) and L a leg. For the angle cases the letters follow the order around the triangle, which shows whether an angle sits between two sides.

The formulas behind each case are as follows.

SSS (Heron): s = (a + b + c) ÷ 2, A = √( s (s − a)(s − b)(s − c) )

SAS: A = ½ × b × c × sin α

ASA: A = c² ÷ ( 2 × (cot α + cot β) ) = c² sin α sin β ÷ ( 2 sin(α + β) )

AAS: β = 180° − α − γ, a = c sin α ÷ sin γ, A = ½ × a × c × sin β

SSA: sin γ = c sin β ÷ b, then α = 180° − β − γ and A = ½ × b × c × sin α

HL: A = ½ × b × √(c² − b²)b is the leg and c is the hypotenuse; the second leg comes from the Pythagorean theorem.

The SSA case deserves care. Because the known angle is not between the two known sides, the side opposite it can sometimes swing to two different positions. If both are valid triangles the calculator prints two rows, Area and Area (2-nd possible). If the side opposite the angle is too short to reach, no triangle exists and you see "Such a triangle cannot exist!".

Worked examples

SH: a roof gable. The base of a gable is 12 m and its height is 7 m. A = ½ × 12 × 7 = 42 m².

SSS: sides 13, 14 and 15. The semi-perimeter is s = (13 + 14 + 15) ÷ 2 = 21. Then A = √(21 × 8 × 7 × 6) = √7056 = 84. The unit is whatever you used for the sides, squared.

SAS: two sides and the angle between them. Take b = 8, c = 11 and α = 35°. A = ½ × 8 × 11 × sin 35° = 44 × 0.573576 ≈ 25.2374.

ASA: two angles and the side between them. Let c = 10, α = 40° and β = 65°. The third angle is γ = 75°. A = 10² sin 40° sin 65° ÷ (2 sin 75°) ≈ 30.1557.

AAS: the same triangle, described differently. Keep c = 10 and α = 40°, but give the angle opposite c instead: γ = 75°. Then β = 65°, a = 10 × sin 40° ÷ sin 75° ≈ 6.6546 and A = ½ × 6.6546 × 10 × sin 65° ≈ 30.1557. The result agrees with the ASA example, as it must.

SSA: the ambiguous case. Let b = 6, c = 8 and β = 35°. Then sin γ = 8 × sin 35° ÷ 6 ≈ 0.7648, so γ is either about 49.89° or about 130.11°. These give α ≈ 95.11° and α ≈ 14.89°, hence two triangles with areas of about 23.9045 and 6.1657. Both appear in the results.

HL: a right triangle. The hypotenuse is 13 and one leg is 5. The other leg is √(169 − 25) = 12, so A = ½ × 5 × 12 = 30.

What the calculator does with unusual input

• Impossible triangles. In SSA or HL mode, measurements that cannot form a triangle produce the message "Such a triangle cannot exist!". In SSS mode, sides that fail the triangle inequality (for example 1, 2 and 10) return nan, which means "not a number".
• Missing angles. ASA and AAS need both angles to be greater than zero, and SSA needs a non-zero side b. Otherwise an explanatory message appears.
• Angle totals. The calculator does not check that the angles add up to less than 180°. Entering 100° and 80° for ASA, or 100° and 100° for AAS, returns an enormous or negative number that has no geometric meaning.
• Blank fields. In the SH, SAS and HL modes an empty length or angle counts as zero, so the area is 0. In SSS mode a missing side usually gives nan.

Practical applications

Triangles are the building block of area measurement. A surveyor who can measure only the three boundary lengths of a plot uses Heron's formula. A carpenter sizing a gable end uses base and height. Sail makers, truss designers and landscape planners combine two sides and an angle, because angles are easy to read with a protractor or a total station. In computer graphics every surface is cut into triangles, and the SAS expression is the cross-product area of two edge vectors.

For the other classic flat shapes, see the rectangle, square, trapezoid, parallelogram and circle calculators. The sine and cosine functions used above are available in the sine calculator, the cosine calculator and the trigonometry calculator.

Common mistakes

• Using a slanted side as the height. The height must meet the base at a right angle.
• Using an angle that is not between the sides in SAS. If the angle is opposite one of the sides, you are in the SSA case instead.
• Entering radians. All angles are degrees. An angle of 1.2 is just a bit more than one degree.
• Forgetting the half. Base times height is the area of a parallelogram; the triangle is half of it.
• Assuming SSA has one answer. Check whether a second row appears.