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Opposite to angle A
Opposite to angle B
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Everything You Need to Know About Triangles
Triangles are the simplest polygon, and also the most important. Every polygon, no matter how many sides it has, can be broken down into triangles. A pentagon is made of three triangles. A hexagon is made of four. This means that if you understand triangles, you have the foundation for understanding all shapes in geometry.
Our calculator works with any triangle. Enter three sides, two sides and an angle, or one side and two angles, and it computes everything else: the remaining sides, all three angles, the area, the perimeter, and the triangle type.
Triangle types by sides: Equilateral (all three sides equal, all angles 60 degrees). Isosceles (two sides equal, two angles equal). Scalene (all sides different, all angles different). By angles: Acute (all angles less than 90 degrees). Right (one angle exactly 90 degrees). Obtuse (one angle greater than 90 degrees).
The Rules Every Triangle Follows
Angle sum rule: The three interior angles of any triangle always add up to exactly 180 degrees. This is true for every triangle that has ever existed or ever will exist. If you know two angles, you can always find the third by subtracting their sum from 180. For example, if two angles are 45 and 70 degrees, the third is 180 minus 115 = 65 degrees.
Triangle inequality rule: The sum of any two sides must be greater than the third side. This means 3, 4, and 10 cannot form a triangle because 3 + 4 = 7, which is less than 10. But 3, 4, and 5 work because 3 + 4 = 7 (greater than 5), 3 + 5 = 8 (greater than 4), and 4 + 5 = 9 (greater than 3). If three lengths fail this test, they simply cannot form a triangle.
Longest side, largest angle: The longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle. This is a useful shortcut for estimating angles when you know the side lengths, or checking your work after a calculation.
How to Find the Area of a Triangle
Base times height method: The simplest formula is area = 1/2 x base x height. If a triangle has a base of 10 inches and a height of 6 inches, the area is 10 x 6 / 2 = 30 square inches. The height is the perpendicular distance from the base to the opposite corner, not the length of a side. For some triangles, you may need to draw the height inside or outside the triangle.
Heron's formula: When you know all three sides but not the height, Heron's formula saves the day. First, find the semi-perimeter: s = (a + b + c) / 2. Then area = the square root of s x (s-a) x (s-b) x (s-c). For sides 7, 8, and 9: s = 12, area = the square root of 12 x 5 x 4 x 3 = the square root of 720 = about 26.8 square units. This formula is over 2,000 years old and still one of the most useful tools in geometry.
Using trigonometry: If you know two sides and the angle between them, area = 1/2 x a x b x sin(C). For sides 10 and 8 with a 30-degree angle between them: area = 1/2 x 10 x 8 x 0.5 = 20 square units. Our calculator uses whichever method gives the most accurate result based on the information you provide.
Why Triangles Are the Strongest Shape
Structural strength: Triangles cannot be pushed out of shape without changing the length of one of their sides. A square can be pushed into a diamond shape, but a triangle stays rigid. This is why bridges, roof trusses, cranes, and the Eiffel Tower are all built using triangular frameworks. Engineers call this property rigidity, and it makes triangles the go-to shape for anything that needs to be strong and stable.
Truss bridges: A typical truss bridge is made entirely of interconnected triangles. Each triangle distributes forces evenly across all three sides, preventing any single piece from bearing too much weight. The Golden Gate Bridge, though it looks curved, is held up by a steel framework of thousands of triangles.
Everyday examples: Tripods (three legs, very stable), musical instrument stands, camera tripods, and even the way you position your feet when standing are all based on the stability of triangles. Three points always define a flat plane, which is why three-legged stools never wobble but four-legged chairs sometimes do.
Special Triangles Worth Knowing
Equilateral triangle: All sides equal, all angles 60 degrees. These appear in honeycombs, tile patterns, and traffic signs. They are perfectly symmetrical, which makes them useful in design and engineering. The area formula simplifies to (side squared x the square root of 3) / 4.
45-45-90 right triangle: An isosceles right triangle where the two legs are equal and the angles are 45, 45, and 90 degrees. The hypotenuse is always the leg length times the square root of 2 (about 1.414). If each leg is 5, the hypotenuse is about 7.07.
30-60-90 right triangle: A special right triangle where the sides are in the ratio 1 to the square root of 3 to 2. If the shortest side is 5, the longer leg is about 8.66 and the hypotenuse is 10. These triangles appear constantly in math competitions and engineering problems because their ratios are fixed and predictable.
Frequently Asked Questions
Multiply the base by the height, then divide by 2. The formula is A = 1/2 x base x height. So if a triangle has a base of 10 inches and a height of 6 inches, the area is 10 x 6 / 2 = 30 square inches. The height is the perpendicular distance from the base to the opposite corner, not the length of a side.
Equilateral triangles have all three sides equal and all three angles are 60 degrees. Isosceles triangles have two sides equal and the angles opposite those sides are also equal. Scalene triangles have all three sides different and all three angles different. Our calculator identifies the type automatically based on your measurements.
No. The three angles inside any triangle always add up to exactly 180 degrees. If a triangle had two 90-degree angles, that would already be 180 degrees with no room for a third angle, which is impossible. The largest angle a triangle can have is just under 180 degrees, and the smallest is just above 0 degrees.
The perimeter is the total distance around the outside of the triangle, which is simply the sum of all three sides. If the sides are 3, 4, and 5 inches, the perimeter is 3 + 4 + 5 = 12 inches. For equilateral triangles with side length s, the perimeter is just 3s since all sides are equal.
Heron's formula lets you find the area of a triangle when you know all three side lengths but not the height. First, calculate the semi-perimeter (half the perimeter): s = (a + b + c) / 2. Then use the formula: Area = the square root of s x (s-a) x (s-b) x (s-c). This is over 2,000 years old and is still one of the most useful triangle formulas.
Triangles are the strongest shape in engineering because any force applied to a triangle is distributed evenly through all three sides. Unlike a square, which can be pushed out of shape into a diamond, a triangle cannot be deformed without changing the length of one of its sides. This is why bridges, roof trusses, and the Eiffel Tower are built with triangular frameworks.
The triangle inequality rule says that the sum of any two sides must be greater than the third side. So for sides a, b, and c: a + b must be greater than c, a + c must be greater than b, and b + c must be greater than a. For example, sides of 3, 4, and 10 cannot form a triangle because 3 + 4 = 7, which is less than 10.
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