Triangle Area Formula and Other Triangle Formulas
By the Calculator.hair team · Updated · 8 min read
Short answer
The area of a triangle is A = ½ × base × height, so a base of 10 and a height of 6 give an area of 30. With only the three sides, use Heron's formula: A = √(s(s − a)(s − b)(s − c)), where s is half the perimeter. In a right triangle, the hypotenuse comes from the Pythagorean theorem: c² = a² + b².
Key takeaways
- A = ½ × base × height works for every triangle, as long as the height is perpendicular to the base.
- Heron's formula finds the area from three sides alone: sides 13, 14 and 15 give an area of 84.
- The Pythagorean theorem, a² + b² = c², applies only to right triangles, and c is always the side opposite the right angle.
- The law of cosines solves a triangle from two sides and the angle between them; the law of sines handles the cases where you know an angle and its opposite side.
- The 3-4-5, 45-45-90 and 30-60-90 triangles have fixed side ratios, so you can scale them without a calculator.
What is the formula for the area of a triangle?
The area of a triangle is A = ½ × b × h, where b is the length of any side chosen as the base and h is the perpendicular height from that base to the opposite corner. A triangle with a base of 10 and a height of 6 has an area of ½ × 10 × 6 = 30. In feet, that is 30 square feet, the same figure the triangle tool in our square footage calculator returns.
The ½ is there because two identical triangles fit together to make a parallelogram with area b × h. Any side can serve as the base, but the height must match it and meet it at a right angle. The slanted side is not the height unless the triangle has a right angle at the base. In an obtuse triangle, the height can fall outside the triangle, on an extension of the base.
If you do not know the height, pick the formula that fits what you do know. The triangle calculator takes any valid combination of three sides and angles and returns the area with every other measurement. For areas of other shapes, see area formulas.
| What you know | Area formula | Example | Area |
|---|---|---|---|
| Base and height | A = ½bh | b = 10, h = 6 | 30 |
| Three sides | A = √(s(s − a)(s − b)(s − c)) | 13, 14, 15 | 84 |
| Two sides and the angle between them | A = ½ab × sin C | 7, 10, C = 60° | 30.310889 |
| Legs of a right triangle | A = ½ab | 3, 4 | 6 |
| Side of an equilateral triangle | A = (√3 ÷ 4) × s² | s = 1 | 0.433013 |
How do you find the area of a triangle with three sides?
Use Heron's formula: find the semiperimeter s = (a + b + c) ÷ 2, then take A = √(s(s − a)(s − b)(s − c)). The semiperimeter is half the distance around the triangle.
Worked example with sides 13, 14 and 15. The semiperimeter is (13 + 14 + 15) ÷ 2 = 21. The three differences are 21 − 13 = 8, 21 − 14 = 7 and 21 − 15 = 6. Multiply: 21 × 8 × 7 × 6 = 7,056. The square root of 7,056 is 84, so the area is 84.
You can check that with the base-and-height formula. The height to the side of length 14 is 12, and ½ × 14 × 12 = 84. Our triangle calculator reports both the area and all three heights for these sides.
Heron's formula only applies when the sides can form a triangle. Each side must be shorter than the other two combined, so sides of 1, 2 and 3 flatten into a line with no area, and the calculator rejects them.
What is the Pythagorean theorem equation?
The Pythagorean theorem states that a² + b² = c² in any right triangle, where a and b are the two legs and c is the hypotenuse. The legs are the sides that form the right angle, and the hypotenuse is the longest side, opposite that angle.
To find the hypotenuse, add the squares of the legs and take the square root: c = √(a² + b²). With legs of 6 and 8, c = √(36 + 64) = √100 = 10. To find a missing leg, subtract instead: b = √(c² − a²). With a hypotenuse of 15 and a leg of 9, b = √(225 − 81) = √144 = 12. The Pythagorean theorem calculator shows this working for any two sides, including values entered as square roots.
The theorem also gives the straight-line distance between two points on a graph, because the horizontal and vertical gaps form the legs of a right triangle. The points (1, 1) and (4, 5) are 3 across and 4 up, so they sit 5 apart, which is what our distance calculator returns. The distance formula guide covers this in detail.
What are the common right triangles?
The common right triangles are the Pythagorean triples, such as 3-4-5, and the two special angle triangles, 45-45-90 and 30-60-90. A Pythagorean triple is a set of three whole numbers that satisfies a² + b² = c². Any multiple of a triple is also a triple, so 6-8-10 and 9-12-15 are scaled copies of 3-4-5 with the same angles.
The special angle triangles have fixed ratios that involve square roots. A 45-45-90 triangle has two equal legs and a hypotenuse √2 times as long. A 30-60-90 triangle has a hypotenuse twice the short leg and a long leg √3 times the short leg. So a 30-60-90 triangle with a short leg of 5 has a hypotenuse of 10 and a long leg of 8.660254.
The table lists each triangle at its smallest size, as our right triangle calculator computes it. Angles are the two acute angles, rounded to two decimal places where they are not exact.
| Triangle | Legs | Hypotenuse | Acute angles | Area |
|---|---|---|---|---|
| 3-4-5 | 3, 4 | 5 | 36.87°, 53.13° | 6 |
| 5-12-13 | 5, 12 | 13 | 22.62°, 67.38° | 30 |
| 8-15-17 | 8, 15 | 17 | 28.07°, 61.93° | 60 |
| 45-45-90 | 1, 1 | √2 ≈ 1.414214 | 45°, 45° | 0.5 |
| 30-60-90 | 1, √3 ≈ 1.732051 | 2 | 30°, 60° | 0.866025 |
What are the types of triangles?
Triangles are classified by their sides as equilateral, isosceles or scalene, and by their largest angle as acute, right or obtuse. Every triangle gets one label from each group, and the three angles always add up to 180°.
Once you have the sides, you can find the angle type without measuring anything. Call the longest side c and compare c² with a² + b². If they are equal, the triangle is right. If c² is smaller, it is acute. If c² is larger, it is obtuse. For sides 5, 6 and 8, a² + b² = 25 + 36 = 61, and c² = 64, so the triangle is obtuse, with its largest angle at 92.87°.
| Type | Definition | Example sides | Largest angle |
|---|---|---|---|
| Equilateral | All three sides equal, all angles 60° | 1, 1, 1 | 60° |
| Isosceles | Two sides equal, and the two angles opposite them equal | 5, 5, 6 | 73.74° |
| Scalene | No two sides equal | 13, 14, 15 | 67.38° |
| Acute | All angles under 90° | 13, 14, 15 | 67.38° |
| Right | One angle of exactly 90° | 3, 4, 5 | 90° |
| Obtuse | One angle over 90° | 5, 6, 8 | 92.87° |
When do you use the law of cosines?
Use the law of cosines when you know two sides and the angle between them, or all three sides. The formula is c² = a² + b² − 2ab × cos C, where C is the angle opposite side c. When C is 90°, cos C is 0, the last term disappears, and the formula becomes the Pythagorean theorem.
Worked example: a = 7, b = 10 and C = 60°. Since cos 60° = 0.5, c² = 49 + 100 − (2 × 7 × 10 × 0.5) = 149 − 70 = 79, so c = √79 = 8.888194. The triangle calculator agrees and gives the other two angles as A = 43.003912° and B = 76.996088°, which add with 60° to 180°.
To find an angle from three sides, rearrange it: cos C = (a² + b² − c²) ÷ (2ab), then apply the inverse cosine. Make sure your calculator is in degree mode if your angles are in degrees.
When do you use the law of sines?
Use the law of sines when you know an angle and the side opposite it, plus one more side or angle. The formula is a ÷ sin A = b ÷ sin B = c ÷ sin C, which says each side divided by the sine of its opposite angle gives the same ratio.
Worked example: a = 10, A = 30° and B = 45°. The third angle is C = 180° − 30° − 45° = 105°. The shared ratio is 10 ÷ sin 30° = 20. So b = 20 × sin 45° = 14.142136 and c = 20 × sin 105° = 19.318517. That ratio of 20 is also twice the radius of the circle through the triangle's three corners, and the triangle calculator reports that radius as 10.
Take care when you know two sides and an angle that is not between them. That information can fit no triangle, one triangle or two different triangles, which is known as the ambiguous case. Check whether a second solution exists before you trust the first one.
- Three sides: law of cosines for an angle, or Heron's formula for the area.
- Two sides and the angle between them: law of cosines.
- Two angles and any side: law of sines.
- Two sides and an angle not between them: law of sines, then check the ambiguous case.
- Right triangle with two sides known: Pythagorean theorem.
Calculators in this guide
Frequently asked questions
How do you find the hypotenuse of a right triangle?
Square both legs, add them and take the square root. Legs of 3 and 4 give √(9 + 16) = √25 = 5.
Does the Pythagorean theorem work on every triangle?
No, only on right triangles. For other triangles, use the law of cosines, which adds a correction term based on the angle.
How do you find the height of a triangle from its area?
Rearrange the area formula to h = 2A ÷ b. A triangle with an area of 84 and a base of 14 has a height of 2 × 84 ÷ 14 = 12.
What is the formula for the perimeter of a triangle?
Add the three sides: P = a + b + c. Sides of 13, 14 and 15 have a perimeter of 42 and a semiperimeter of 21.
Can a triangle have two right angles?
No. The angles of a triangle add up to 180°, so two 90° angles would leave nothing for the third.