Triangle Area Calculator
Find a triangle’s area from base and height, from three sides (Heron’s formula), or from two sides and the included angle (SAS). Each run shows the formula, the substitution, and whether the sides form a triangle.
½ × 10 × 6 = 30
- A triangle’s area is half the base times the matching height.
- A = ½ × 10 × 6 = 30.
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How to Use the Triangle Area Calculator
Pick the knowns you have, enter the values, and select Calculate. Base and height uses A = ½bh. Three sides uses Heron’s formula and checks the triangle inequality. SAS uses two sides and the angle between them.
- Choose Base & height, Three sides, or SAS.
- Enter positive lengths. For SAS, enter the included angle in degrees, strictly between 0° and 180°.
- Read the area, the substitution, and — for three sides — the perimeter, semi-perimeter, and kind.
This page finds area of a triangle. A circle’s area and circumference belong on the Circle Calculator. Converting square units belongs on the Area Converter. A missing SAS side or SSS angle belongs on the Law of Cosines Calculator. A triangular prism uses this area as B, then V = Bh on the Triangular Prism Calculator. A triangular-base pyramid uses this area as B, then V = (1/3)Bh on the Pyramid Calculator.
The Three Triangle Area Formulas
Any side can be the base. The height is the perpendicular distance from the opposite vertex to the line of that base, including a height that lands outside an obtuse triangle. This is the definition prealgebra courses start with.
When three sides are known, Heron’s formula does not need a drawn height. Add the sides, take half of that sum as the semi-perimeter s, then take the square root of s(s − a)(s − b)(s − c). It is algebraically the same area you would get from ½ab sin C after recovering an angle, so a 3-4-5 triangle still returns 6.
SAS means two sides and the included angle — the angle between those two sides. C must be in (0°, 180°). An included angle of 0° or 180° is a flat line. An obtuse included angle still has a positive sine, so the area is still positive.
Triangle Area Calculator Examples
Base 10 and height 6
A = ½ × 10 × 6 = 30 square units. Units are the square of whatever you typed.
A 3-4-5 triangle by Heron
s = (3 + 4 + 5) / 2 = 6. A = √[6(3)(2)(1)] = √36 = 6. Same answer as ½ × 3 × 4 because 3-4-5 is right. Kind is scalene.
SAS: sides 5 and 7, included 60°
A = ½ × 5 × 7 × sin 60° = 17.5 × (√3 / 2) ≈ 15.155. The third side is not an input, so perimeter stays blank.
Isosceles 5, 5, 6
s = 8. A = √[8 × 3 × 3 × 2] = √144 = 12. Two equal sides classify it as isosceles.
Degenerate 1, 2, 3
1 + 2 = 3, so the points are collinear. s = 3 and s − 3 = 0, so A = 0. The calculator reports a degenerate (flat) triangle instead of inventing a height.
Impossible 1, 1, 3
1 + 1 is less than 3. No triangle exists. The product inside Heron’s square root is negative, and the calculator stops with the triangle inequality.
Triangle Inequality and Kinds
Three positive lengths form a non-degenerate triangle if and only if each side is shorter than the sum of the other two. Equality is the degenerate, area-zero case. A strict failure cannot form a triangle at all.
When you enter three sides, the tool also labels equilateral (all sides equal), isosceles (exactly two equal), scalene (all different), or degenerate. It does not classify acute / right / obtuse from sides in the metrics row. Use the Pythagorean Theorem Calculator when you specifically need a² + b² versus c².
Common Triangle Area Mistakes
Using a slanted side as the height
In A = ½bh the height must be perpendicular to the chosen base. A side that is not the altitude is not h.
Feeding SAS a non-included angle
The angle must sit between the two given sides. An angle that is not between them is ASA or AAS, and ½ab sin C does not apply until you find the included angle.
Entering 0° or 180° as the included angle
Those endpoints are a flat line. The calculator rejects them. Use a value strictly between 0 and 180.
Reporting the semi-perimeter as the area
s is half the perimeter. The area is the square root that follows, not s itself.
Mixing this page with a unit converter
The result is already in square units of the inputs. To convert m² to ft² or acres, use the Area Converter.
Triangle Area Calculator FAQs
What is the formula for the area of a triangle?
A = ½bh when you have a base and the perpendicular height. Heron’s formula when you have three sides. A = ½ab sin C when you have two sides and the included angle.
How do you use Heron’s formula?
s = (a + b + c) / 2, then A = √[s(s − a)(s − b)(s − c)]. The sides must satisfy the triangle inequality.
Can three sides have area zero?
Yes. When a + b = c the triangle is degenerate (flat) and Heron returns 0.
What is SAS area?
½ab sin C for the angle C between sides a and b. C must be between 0° and 180°, exclusive.
Does this convert square units?
No. Use the Area Converter for m², ft², and acres.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.