What are the types of triangles by sides?+
There are three types by sides. An equilateral triangle has all three sides equal; as a direct consequence, all three interior angles are exactly 60° and the shape is perfectly symmetric about all three altitudes. An isosceles triangle has exactly two sides equal (the legs); the two base angles - the angles opposite the equal sides - are always equal to each other. A scalene triangle has all three sides of different lengths, which means all three angles are different as well. Every triangle falls into exactly one of these three categories.
What are the types of triangles by angles?+
There are three types by angles. An acute triangle has all three interior angles strictly less than 90°. A right triangle has exactly one angle equal to 90°; the side opposite this right angle is called the hypotenuse and is always the longest side. An obtuse triangle has exactly one angle greater than 90°. A triangle cannot have more than one right or obtuse angle - if two angles were each 90° or more, their sum alone would equal or exceed 180°, leaving no positive measure for the third angle.
Can a triangle be both isosceles and right?+
Yes - the 45-45-90 triangle is both isosceles (two equal legs) and right (one 90° angle). Its sides are always in the ratio 1 : 1 : √2. The two legs are equal and the two base angles are each exactly 45°. If the legs have length s, the hypotenuse has length s√2. For example, legs of 5 and 5 give a hypotenuse of approximately 7.071. This is one of the two "special right triangles" taught in geometry alongside the 30-60-90 triangle.
Can a triangle be both isosceles and obtuse?+
Yes - for example, sides 5-5-8. The two equal sides are 5, and the base is 8. The apex angle (the angle between the two equal legs, opposite the base) is obtuse. Using the Law of Cosines: cos(C) = (25+25−64)/(2×5×5) = −14/50 = −0.28, so C = arccos(−0.28) ≈ 106.3°. The two base angles are each approximately 36.9°, which is less than 45°. Whenever the base of an isosceles triangle is longer than its legs (times √2), the apex angle is obtuse.
How do you classify a triangle from its side lengths?+
First, verify the triangle inequality: each side must be less than the sum of the other two. Then classify by sides: equilateral if a=b=c, isosceles if any two sides are equal, scalene otherwise. For the angle classification, identify the longest side (call it c). If a²+b² > c², all angles are less than 90° and the triangle is acute. If a²+b² = c² (within a small tolerance for floating-point), one angle is exactly 90° and it is a right triangle. If a²+b² < c², the angle opposite c exceeds 90° and the triangle is obtuse. This is the quick algebraic test; using the Law of Cosines gives you the exact angles.
What is a scalene triangle?+
A scalene triangle has all three sides of different lengths and therefore all three interior angles of different measures. There are no lines of symmetry and no equal angles. Most real-world triangles encountered in engineering, surveying, and everyday construction are scalene. The classic 3-4-5 right triangle is scalene (3 ≠ 4 ≠ 5), as is the 5-12-13 Pythagorean triple. A scalene triangle can be acute, right, or obtuse depending on its specific side lengths - "scalene" describes only the side relationship, not the angle type.
What determines if a triangle is acute?+
A triangle is acute if and only if all three interior angles are strictly less than 90°. The equivalent algebraic test using side lengths: a triangle with sides a, b, c is acute if and only if a²+b²>c², a²+c²>b², and b²+c²>a² - that is, the square of each side is less than the sum of squares of the other two. In practice you only need to check against the longest side, since the other two conditions are automatically satisfied for shorter sides. For example, sides 5-7-8: longest side 8, check 25+49=74>64 ✔ - acute. But sides 5-7-9: check 25+49=74<81 - not acute.
What is the largest possible angle in a triangle?+
Theoretically, the largest possible angle in a triangle approaches 180° (but never reaches it). As one angle approaches 180°, the triangle degenerates into a straight line segment - the other two angles approach 0°. In practice, any triangle with an angle between 90° and just under 180° is classified as obtuse. The largest realistic obtuse angle is constrained by the triangle inequality: for an isosceles triangle with equal sides s and base b, the apex angle C satisfies cos(C) = 1 − b²/(2s²), so as b approaches 2s (the limit where the triangle inequality is still satisfied), cos(C) approaches −1 and C approaches 180°. A triangle with exactly one 180° angle would collapse to a line and is called a degenerate triangle.
What is the triangle inequality and why does it matter?+
The triangle inequality theorem states that in any valid triangle, the sum of any two side lengths must be strictly greater than the third side. You must check all three combinations: a+b>c, a+c>b, and b+c>a. If any one of these fails, the three "sides" cannot close into a triangle - the ends simply do not meet. For example, sides 2, 3, 10 are invalid because 2+3=5 < 10. This theorem arises from the straight-line principle in geometry: the shortest path between two points is a straight line. Taking a detour via a third point (the triangle path) must always be longer than the direct route. This calculator validates the triangle inequality before any computation and shows an error if it is violated.
How are the six triangle types combined?+
With three side types and three angle types, there are in theory nine combinations, but only six actually occur. The six valid combinations are: equilateral acute (all sides equal - always acute, never right or obtuse), isosceles acute, isosceles right (the 45-45-90), isosceles obtuse, scalene acute, scalene right (e.g. 3-4-5), and scalene obtuse. The three impossible combinations are equilateral right, equilateral obtuse, and any triangle with two right or two obtuse angles. An equilateral triangle cannot be right or obtuse because all its angles are forced to 60°. This calculator identifies the exact combination from your three side lengths.