Triangle Calculator
Use our free online triangle calculator to solve any triangle instantly. Enter any combination of sides and angles to find the missing values, calculate the area of a triangle, and identify the triangle type. This triangle solver works for right, acute, and obtuse triangles.
How to Use This Triangle Calculator
Using this online triangle calculator is simple:
- Enter any three known values, sides, angles, or a mix of both.
- Make sure at least one side value is included.
- Select degrees or radians as your angle unit.
- Click Calculate to instantly find all missing sides, angles, and the area.
What Is a Triangle Calculator?
A triangle calculator is an online tool that finds the unknown sides, angles, and area of a triangle based on the values you already know. You only need to enter three values, including at least one side, and the calculator determines the rest.
People use an online triangle calculator for geometry homework, construction measurements, engineering problems, and any situation that requires solving a triangle quickly and accurately.
This calculate triangle tool supports all triangle types and uses the Pythagorean theorem, the law of sines, and the law of cosines to find missing values.
Types of Triangles
Triangles are classified in two ways: by their sides and by their angles.
By Side Length
An equilateral triangle has all three sides equal in length. All three interior angles are also equal, each measuring 60°.
An isosceles triangle has two sides of equal length. The two angles opposite the equal sides are also equal.
A scalene triangle has all three sides of different lengths. All three interior angles are also different.
By Angle
A right triangle has one angle equal to exactly 90°. The side opposite the right angle is called the hypotenuse and is always the longest side.
An acute triangle has all three angles less than 90°.
An obtuse triangle has one angle greater than 90°.
Triangle Rules and Properties
Before calculating a triangle, it helps to understand the basic rules that all triangles follow.
The interior angles of any triangle always add up to 180°. If two angles are known, the third can always be found by subtracting the sum of the two known angles from 180°.
The sum of any two sides of a triangle must always be greater than the third side. This is known as the triangle inequality rule.
The longest side of a triangle is always opposite the largest angle, and the shortest side is always opposite the smallest angle.
Finding a Missing Side of a Triangle
Pythagorean Theorem (Right Triangles Only)
The Pythagorean theorem is used to find a missing side in a right triangle. If the two shorter sides are a and b, and the hypotenuse is c, the formula is:
Example: A right triangle has sides a = 3 and c = 5. Find side b.
- Step 1: Apply the formula: 3² + b² = 5²
- Step 2: Calculate: 9 + b² = 25
- Step 3: b² = 25 − 9 = 16
- Step 4: b = √16 = 4
Law of Sines
The law of sines is used to find a missing side or angle when you know two angles and one side (AAS or ASA), or two sides and an angle opposite one of them (SSA).
Formula:
Where a, b, c are the side lengths and A, B, C are the opposite angles.
Example: A triangle has side b = 2, angle B = 90°, and angle C = 45°. Find side c.
- Step 1: Set up the ratio: 2 / sin(90°) = c / sin(45°)
- Step 2: sin(90°) = 1, sin(45°) = 0.7071
- Step 3: c = 2 × 0.7071 / 1 = 1.414
Law of Cosines
The law of cosines is used when you know all three sides (SSS) or two sides and the included angle (SAS).
Formula:
Example: A triangle has sides a = 5, b = 7, and angle C = 60°. Find side c.
- Step 1: Apply the formula: c² = 5² + 7² − 2 × 5 × 7 × cos(60°)
- Step 2: cos(60°) = 0.5
- Step 3: c² = 25 + 49 − 70 × 0.5 = 74 − 35 = 39
- Step 4: c = √39 = 6.24
Finding Missing Angles
Using the Law of Cosines
When all three sides are known, each angle can be found using:
Example: A triangle has sides a = 6, b = 8, and c = 10. Find angle B.
- Step 1: Apply the formula: B = arccos((6² + 10² − 8²) / 2 × 6 × 10)
- Step 2: Calculate: B = arccos((36 + 100 − 64) / 120) = arccos(72 / 120)
- Step 3: B = arccos(0.6) = 53.13°
Using the Angle Sum Rule
When two angles are already known, the third is found by subtracting from 180°.
Example: A triangle has angles A = 45° and B = 75°. Find angle C.
- Step 1: C = 180° − A − B
- Step 2: C = 180° − 45° − 75°
- Step 3: C = 60°
Area of a Triangle
Base and Height Formula
The simplest way to calculate the area of a triangle is to use its base and height.
Example: A triangle has a base of 10 and a height of 6.
- Step 1: Area = (10 × 6) / 2
- Step 2: Area = 60 / 2 = 30 square units
Heron's Formula
Heron's formula calculates the area of a triangle when all three sides are known, without needing the height.
Where s is the semi-perimeter of the triangle.
Example: A triangle has sides a = 5, b = 6, c = 7.
- Step 1: s = (5 + 6 + 7) / 2 = 9
- Step 2: Area = √(9 × (9 − 5) × (9 − 6) × (9 − 7))
- Step 3: Area = √(9 × 4 × 3 × 2) = √216 = 14.70 square units
SAS Area Formula
When two sides and the included angle are known, the area is found using:
Example: A triangle has sides a = 8, b = 5, and an included angle C = 30°.
- Step 1: Area = 0.5 × 8 × 5 × sin(30°)
- Step 2: sin(30°) = 0.5
- Step 3: Area = 0.5 × 8 × 5 × 0.5 = 10 square units
FAQs
What is a triangle calculator?
A triangle calculator is an online tool that solves for the unknown sides, angles, and area of a triangle based on the values you enter.
How do I find a missing side of a triangle?
Use the Pythagorean theorem for right triangles, the law of sines when two angles and one side are known, or the law of cosines when two sides and the included angle are known.
How do I calculate the area of a triangle?
Use the formula Area = (base × height) / 2 if the base and height are known. Use Heron's formula if only the three side lengths are known. Use the SAS formula if two sides and the included angle are known.
What is the law of sines?
The law of sines states that the ratio of each side to the sine of its opposite angle is equal for all three sides of a triangle. It is used to solve triangles when two angles and one side are known.
What is the law of cosines?
The law of cosines relates the lengths of all three sides of a triangle to one of its angles. It is used to solve triangles when two sides and the included angle are known, or when all three sides are known.
What is Heron's formula?
Heron's formula calculates the area of a triangle using only the three side lengths. It first finds the semi-perimeter and then uses it to calculate the area.
What is a triangle angle calculator used for?
It is used to find one or more unknown angles in a triangle when some sides or angles are already known, using the law of sines, the law of cosines, or the angle sum rule.
How many values do I need to solve a triangle?
You need at least three values, and at least one of them must be a side length. Any valid combination of three sides and angles is enough to solve the full triangle.
What types of triangles can this calculator solve?
This calculator works for all triangle types: right, acute, obtuse, equilateral, isosceles, and scalene.
Is this triangle calculator free?
Yes. Our online triangle calculator is completely free with no sign-up required.