Triangle Calculator
Calculate triangle area and perimeter instantly. Supports base/height method and side lengths.
Formula
Area = ½ × base × height | Perimeter = a + b + c | Heron: Area = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
Area is half the product of the base and the perpendicular height measured to that base. Perimeter is simply the sum of the three side lengths. When you know all three sides but no height, Heron's formula gives the area directly from the semi-perimeter s = (a+b+c)/2.
Worked Examples
Example 1: Area from base and height
Problem:A triangular sail has a base of 10 m and a perpendicular height of 5 m. Find its area.
Solution:Area = 0.5 x base x height = 0.5 x 10 x 5 = 25 square metres. The height must be measured at a right angle to the base, not along a slanted side.
Result:Area = 25 m²
Example 2: Perimeter of a 3-4-5 triangle
Problem:Find the perimeter of a triangle with sides 3, 4 and 5 units.
Solution:P = a + b + c = 3 + 4 + 5 = 12 units. Check the triangle inequality: 3 + 4 = 7 > 5, so these sides do form a triangle. Because 3² + 4² = 9 + 16 = 25 = 5², it is also a right triangle.
Result:Perimeter = 12 units
Example 3: Area from three sides (Heron's formula)
Problem:A plot has sides 13 m, 14 m and 15 m but no measured height. Find its area.
Solution:Semi-perimeter s = (13 + 14 + 15) / 2 = 21 s - a = 8, s - b = 7, s - c = 6 Area = sqrt(21 x 8 x 7 x 6) = sqrt(7056) = 84 square metres. Cross-check with base 14: height = 2 x 84 / 14 = 12 m, and 0.5 x 14 x 12 = 84.
Result:Area = 84 m² (height to the 14 m side = 12 m)
Frequently Asked Questions
Do I need the height to calculate area?
Not always. If you have the base and height, use 0.5×b×h. If you have 3 sides, use Heron's Formula. If you have 2 sides and an angle, use 0.5×a×b×sin(C).
What is the sum of angles in a triangle?
The sum of the interior angles of any triangle in a flat plane is always 180 degrees (or π radians).
What is a Right Triangle?
A triangle containing one 90-degree angle. The side opposite the right angle is called the hypotenuse, and the Pythagorean theorem (a² + b² = c²) applies.
What is an Isosceles Triangle?
A triangle with at least two sides of equal length. The angles opposite the equal sides are also equal.
What is a Scalene Triangle?
A triangle where all three sides have different lengths, and all three angles are different.
What is an Equilateral Triangle?
A triangle where all three sides are equal. All angles are also equal (60 degrees each).
Can a triangle have two right angles?
No. Since the sum of angles must be 180°, two 90° angles would leave 0° for the third angle, which is impossible in Euclidean geometry.
What is the Triangle Inequality Theorem?
It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Otherwise, the sides won't connect.
How are triangles used in construction?
Triangles are the strongest shape because they cannot be deformed without changing the length of a side. This is why trusses and bridges are built using triangular patterns.
What are special right triangles?
Special right triangles have fixed side ratios that make calculations exact without a calculator. In a 45-45-90 triangle (an isosceles right triangle), the two legs are equal and the hypotenuse is leg × √2. If each leg is 1, the hypotenuse is √2 ≈ 1.414. In a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2, where the shortest side is opposite the 30° angle and the hypotenuse is twice the shortest side. These triangles appear constantly in engineering, architecture, and physics — for instance, a roof pitch of 45° forms a 45-45-90 triangle, and an equilateral triangle bisected diagonally creates two 30-60-90 triangles. Knowing these ratios eliminates the need for trigonometric tables in common scenarios.
Background & Theory
Triangle Area:
The most common formula is Area = ½ × base × height.
- Base: Any side of the triangle.
- Height: The perpendicular distance from the base to the opposite vertex.
Triangle Perimeter:
The sum of the lengths of all three sides.
- P = a + b + c
Heron's Formula:
Calculates area from 3 sides (a, b, c):
- s = (a+b+c)/2 (semi-perimeter)
- Area = √(s(s-a)(s-b)(s-c))