Triangle Calculator
Three sides are enough. Heron's formula gets the area without a height, which matters because measuring three sides of a real triangle is easy and getting a perpendicular height is not.
Area
6.0000
- Area
- 6.0000
- Perimeter
- 12.0000
- Type
- Scalene, right-angled
- Angle A
- 36.87°
- Angle B
- 53.13°
- Angle C
- 90.00°
- The area comes from Heron's formula, which needs only the three sides. That matters in practice: measuring three sides of a real triangle is easy, and getting a perpendicular height off one is not.
- The angles come from the law of cosines and always add to 180 degrees. If they do not, the sides were mistyped.
About triangle calculations
The familiar area formula — half base times height — needs a height, and a height is awkward to obtain. It is the perpendicular distance from a vertex to the opposite side, which on a real triangle means constructing a right angle you cannot see. Heron's formula needs only the three side lengths, which a tape measure gives directly.
For the 3-4-5 triangle it gives an area of 6, a perimeter of 12, and angles of 36.87, 53.13 and 90 degrees. That right angle is not a coincidence — 3-4-5 is the smallest set of whole numbers satisfying Pythagoras, which is why it is used to square up foundations and corners.
Not every three numbers make a triangle. Any two sides added together must exceed the third, or the shape cannot close — sides of 1, 2 and 10 leave the short pair unable to reach. This is the triangle inequality, and the calculator says so rather than returning a meaningless area.
The angles come from the law of cosines and always sum to 180 degrees. That sum is a useful check on your own measurements: if three measured sides give angles that add to 180, the arithmetic worked, and if the shape in front of you disagrees, one of the measurements is wrong.
What it works out
- Area by Heron's formula, from three sides
- All three angles
- Perimeter and the triangle type
- A check that the sides can form a triangle at all
The formula
Heron: s = (a+b+c) ÷ 2, then Area = √(s(s−a)(s−b)(s−c))
The semi-perimeter s is half the way round. For 3, 4 and 5 that is 6.
The area is the square root of 6 × 3 × 2 × 1, which is the square root of 36 — exactly 6. Heron's formula rarely lands on a whole number that neatly; the 3-4-5 triangle is unusually well behaved.
The angles come from the law of cosines, which is Pythagoras generalised to triangles that are not right-angled. Rearranged for an angle it gives cos A = (b² + c² − a²) ÷ 2bc, applied once per corner.
For 3-4-5 that produces 36.87, 53.13 and exactly 90 degrees. The right angle appears because 3² + 4² = 5², and the law of cosines reduces to Pythagoras whenever that holds.
The triangle inequality is checked first. Sides of 1, 2 and 10 make no triangle, and Heron's formula would return the square root of a negative number if it were applied anyway.
- s
- The semi-perimeter — half of a + b + c.
- a, b, c
- The three sides. Any two must add to more than the third.
- Law of cosines
- Gives each angle from the three sides. Reduces to Pythagoras at 90 degrees.
A worked example
A triangle with sides of 3, 4 and 5.
That works out to 6.0000 .
- Area
- 6.0000
- Perimeter
- 12.0000
- Type
- Scalene, right-angled
- Angle A
- 36.87°
- Angle B
- 53.13°
- Angle C
- 90.00°
Questions
How do I find the area of a triangle without the height?
Heron's formula. Take half the perimeter, subtract each side from it in turn, multiply all four together and take the square root. Only the three sides are needed.
What is Heron's formula?
Area = √(s(s−a)(s−b)(s−c)) where s is half the perimeter. It dates from the first century and is the standard way to get an area from side lengths alone.
Why is 3-4-5 a right-angled triangle?
Because 9 + 16 = 25, satisfying Pythagoras. It is the smallest set of whole numbers that does, which is why builders use it to square up corners with nothing but a tape measure.
Can any three lengths make a triangle?
No. Any two must add to more than the third — the triangle inequality. Sides of 1, 2 and 10 cannot close, because the short pair can never reach across the long one.
Do the angles always add to 180 degrees?
In flat geometry, yes, always. It is a useful check: if your measured sides give angles summing to 180 but the shape in front of you looks different, a measurement is wrong.
What makes a triangle scalene, isosceles or equilateral?
How many sides are equal — none, two, or all three. That is separate from whether it is acute, right-angled or obtuse, which is about the largest angle, so both are reported.