Triangle area calculator
Sides of 13, 14 and 15 give 84 square units in the triangle area calculator. Heron's formula gets there from the sides alone, and the height to the side of 14 comes out as exactly 12.
Area
- Area Heron's formula from the three sides: s = 21, area = sqrt(s(s-a)(s-b)(s-c)).
- 84
- Sides
- a = 13, b = 14, c = 15
- Angles Summing to 180 degrees, as every plane triangle must.
- A = 53.13deg, B = 59.49deg, C = 67.38deg
- Area
- 84
- Perimeter
- 42
- Type
- scalene, acute
- Heights The perpendicular from each side to the opposite vertex. Area is half the base times the height for any of the three.
- to a: 12.9231, to b: 12, to c: 11.2
- Circumradius The radius of the circle through all three vertices.
- 8.125
- Inradius The radius of the largest circle that fits inside.
- 4
| a | b | c | Multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6-8-10, 9-12-15, 30-40-50 |
| 5 | 12 | 13 | 10-24-26, 15-36-39 |
| 8 | 15 | 17 | 16-30-34 |
| 7 | 24 | 25 | 14-48-50 |
| 20 | 21 | 29 | 40-42-58 |
| 9 | 40 | 41 | 18-80-82 |
- The Pythagorean theorem holds only for right triangles. For any other triangle the law of cosines generalizes it: c squared equals a squared plus b squared minus 2ab cos(C), which reduces to Pythagoras when C is 90 degrees and cos(C) is zero.
- Any two sides must add to more than the third, or the shape does not close. That triangle inequality is checked here before anything else is computed.
How the Triangle Calculator works
The basic formula for the area of a triangle is A = ½ × base × height, where the height is the perpendicular distance from the base to the opposite corner. Two copies of any triangle fit together into a parallelogram with the same base and height, and the triangle is half of it. The trouble is that the height is rarely what you have. Usually it is the three sides, and that is what this calculator takes.
From three sides the area is Heron's formula. Work out s, half the perimeter, then A = √(s(s - a)(s - b)(s - c)). For sides 13, 14 and 15 the perimeter is 42, so s = 21, and the area is √(21 × 8 × 7 × 6) = √7056 = 84. No angle and no height are needed, which is the point of it: a surveyor who can measure the three edges of a plot can get its area without ever finding a right angle.
The page then gives every height, which closes the loop with ½ × base × height. Take the side of 14 as the base and the height to it is 12, and ½ × 14 × 12 = 84. Take 13 as the base and the height is 168 / 13, about 12.92; take 15 and it is 11.2. Each product is the same 84, because any side can be the base as long as the height is measured to it. The common mistake with the basic formula is using a slanted side as the height. In this triangle that would mean ½ × 14 × 13 = 91 instead of 84. That is too big, as it always is, because a slanted side is longer than the perpendicular.
From two sides and the angle between them, the area is A = ½ab sin C. That is the same formula, since b sin C is the height to side a. In this triangle the angle between the sides of 13 and 14 is about 67.38°, whose sine is exactly 12/13, and ½ × 13 × 14 × 12/13 = 84 again. Choose the two-sides-and-an-angle mode for that case. For a right triangle the two legs are base and height, so the area is half their product: legs of 5 and 12 give 30.
The answer is in square units of whatever the sides were measured in: sides in meters, area in square meters. Doubling every side quadruples the area, so a 26-28-30 triangle has an area of 336.
Not every three numbers make a triangle. Any two sides must add up to more than the third, and the page checks that before anything else. Sides of 2, 3 and 10 do not close, and Heron's formula would be taking the square root of a negative number. If two sides add up to exactly the third, the "triangle" is flat and its area is zero.
For the area of other shapes, including a triangle from its base and height directly, the area calculator has a triangle shape. 13-14-15 is a favorite textbook example because its sides are consecutive whole numbers and its area and one of its heights come out whole too.
Triangle Calculator FAQ
How do I find the area of a triangle with three sides?
Use Heron's formula: with s as half the perimeter, A = √(s(s - a)(s - b)(s - c)). For sides 13, 14 and 15, s = 21 and the area is √7056 = 84.
What is the formula for the area of a triangle?
A = ½ × base × height, where the height is perpendicular to the base. A base of 14 with a height of 12 gives 84.
How do I find the area from two sides and an angle?
A = ½ab sin C, where C is the angle between sides a and b. Sides of 13 and 14 with an angle of about 67.38° between them give 84.
Why do I get a different answer with the slanted side?
Because the height must be perpendicular to the base. A slanted side is always longer than the perpendicular, so using it overstates the area.
More from the Triangle Calculator
- Right triangle calculator
- Law of cosines calculator
- Law of sines calculator
- Equilateral triangle calculator
- Isosceles triangle solver
- Pythagorean theorem calculator