Triangle Calculator

Please enter three values at least one of which must be a side into the six fields below, then click the Calculate button. If radians are selected as the angle unit, you may enter values such as π/2, π/4, and so on.

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A triangle is a fundamental geometric shape known as a polygon, characterized by having exactly three vertices. A vertex is defined as a point at which two or more lines, curves, or edges intersect; in the specific case of a triangle, each vertex is formed by the meeting of two straight line segments. These three vertices are connected pairwise by three straight edges, creating a closed, three-sided figure.

Triangles are commonly identified and named according to their vertices. For instance, a triangle whose vertices are labeled a, b, and c is written as Δabc. This notation helps clearly distinguish one triangle from another, especially in mathematical diagrams and problem-solving contexts.

In geometry, triangles are further classified based on the properties of their sides and their internal angles. When classification is based on side lengths, several distinct types emerge. A triangle in which all three sides are of equal length is known as an equilateral triangle; in addition, all of its interior angles are equal. If a triangle has exactly two sides of the same length, it is called an isosceles triangle, which also means it has two angles of equal measure. Finally, when all three sides of a triangle are of different lengths, the triangle is referred to as scalene. In this case, none of the angles are equal. These different types of triangles are often illustrated visually to highlight their unique properties and differences.

Tick marks placed along the edges of a triangle are a widely used form of notation in geometry to indicate the relative lengths of the sides. When two or more sides display the same number of tick marks, it signifies that those sides are equal in length. Similarly, internal angles of a triangle are often marked using curved arcs drawn at the vertices. Angles that share the same number of concentric arcs are understood to be equal in measure, while differing numbers of arcs indicate angles of different sizes. This visual system of marks allows the properties of a triangle to be quickly identified without requiring numerical measurements.

As illustrated by the triangles above, there is a direct relationship between the lengths of a triangle’s sides and the measures of its internal angles. Because of this relationship, an equilateral triangle not only has three sides of equal length but also three internal angles that are equal in measure. In contrast, triangles with unequal sides will generally have angles of differing sizes. It is important to note, however, that diagrams shown in calculators or illustrative tools are not always drawn to scale. Although a triangle may appear equilateral and include angle markings that suggest equality, it is not necessarily equilateral unless the actual values confirm it. These diagrams are meant to serve as visual representations only. Once specific side lengths or angle measures are entered, the calculator will adjust its output to accurately reflect the true shape and proportions of the triangle.

Triangles can also be classified according to their internal angles, and this classification falls into two main groups: right triangles and oblique triangles. A right triangle is defined as a triangle that contains one angle measuring exactly 90 degrees. This right angle is typically indicated by a small square drawn at the corresponding vertex. The side opposite the right angle is the longest side of the triangle and is known as the hypotenuse. Any triangle that does not contain a right angle is referred to as an oblique triangle. Oblique triangles can be further divided into obtuse and acute types. In an obtuse triangle, one of the internal angles is greater than 90 degrees, while in an acute triangle, all three internal angles are less than 90 degrees. These distinctions, as shown in the accompanying diagrams, help provide a clear and systematic way to understand and categorize different types of triangles.

Triangle facts, theorems, and laws

A triangle cannot have more than one vertex with an internal angle that is greater than or equal to 90 degrees. If a shape were to contain two such angles, the remaining angle would be forced to be zero or negative, which is not possible in a triangle. As a result, a triangle may have at most one right angle or one obtuse angle; otherwise, it would no longer meet the basic geometric definition of a triangle.

One of the most important properties of triangles is that the sum of their interior angles is always equal to 180 degrees. This rule applies to all triangles, regardless of their shape or size. In addition, each exterior angle of a triangle has a special relationship with its interior angles. Specifically, the measure of an exterior angle is equal to the sum of the two interior angles that are not adjacent to it. Another convenient way to determine the measure of an exterior angle is to subtract the measure of the interior angle at the same vertex from 180 degrees. These angle relationships are widely used in solving geometric problems involving unknown angles.

Triangles also obey a fundamental rule regarding side lengths known as the triangle inequality theorem. According to this principle, the sum of the lengths of any two sides of a triangle must always be greater than the length of the remaining side. If this condition is not satisfied, a triangle cannot be formed. This rule helps determine whether a given set of three lengths can represent a valid triangle and plays an essential role in both geometry and trigonometry.

The Pythagorean theorem is a key result that applies specifically to right triangles. It states that, in any right triangle, the square of the length of the hypotenuse the side opposite the right angle is equal to the sum of the squares of the lengths of the other two sides. This relationship allows for the calculation of unknown side lengths when two sides are known. Moreover, any triangle whose side lengths satisfy this condition must be a right triangle. There are also well known special right triangles, such as the 30°–60°–90°, 45°–45°–90°, and 3–4–5 triangles, which have fixed side ratios that simplify calculations. If the legs of a right triangle are labeled a and b, and the hypotenuse is labeled c, the Pythagorean theorem is written as:

a² + b² = c²

For example, if a = 3 and c = 5, the value of b can be found as follows:

3² + b² = 5²
9 + b² = 25
b² = 16
b = 4

Another important relationship in triangle geometry is described by the law of sines. This law states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. The law of sines is especially useful for finding unknown sides or angles when sufficient information is given. If the sides of a triangle are labeled a, b, and c, and their opposite angles are labeled A, B, and C, respectively, the law of sines can be written as:

a / sin(A) = b / sin(B) = c / sin(C)

Using this relationship, if b, B, and C are known, the value of c can be calculated by comparing b / sin(B) with c / sin(C). It is important to note that, under certain conditions, the same set of measurements can produce two different triangle configurations, a situation known as the ambiguous case.

For example, given b = 2, B = 90°, and C = 45°, the value of c can be found as follows:

2 / sin(90°) = c / sin(45°)
c = 2 × (sin(45°) / sin(90°))
c = 2 × (√2 / 2) = √2

Finally, when the lengths of all three sides of a triangle are known, each internal angle can be calculated using inverse trigonometric functions. These formulas are derived from the law of cosines and are especially useful when solving triangles with no right angles. If a, b, and c are the known side lengths, the angles A, B, and C can be found using the following equations:

A = arccos((b² + c² − a²) / (2bc))
B = arccos((a² + c² − b²) / (2ac))
C = arccos((a² + b² − c²) / (2ab))

For instance, if a = 8, b = 6, and c = 10, angle B can be calculated as:

B = arccos((8² + 10² − 6²) / (2 × 8 × 10))
B = arccos(0.8) ≈ 36.87°

These formulas and principles together provide a comprehensive framework for analyzing and solving problems involving triangles.

Area of a Triangle

There are several different formulas that can be used to calculate the area of a triangle, and the choice of equation depends on what information about the triangle is available. Because triangles can be described in many ways using side lengths, angles, or a combination of both different area formulas are useful in different situations. Among these, the most widely known and frequently used method for finding the area of a triangle is the formula that involves the base and the height.

In this approach, the area of a triangle is calculated using the expression
Area = ½ × b × h,
where b represents the base of the triangle and h represents its height. The term “base” does not refer to a specific side; rather, it can be any side of the triangle that is conveniently chosen. Once a base is selected, the corresponding height is defined as the length of the line segment drawn from the vertex opposite the base straight down to the base itself, forming a right angle. This perpendicular distance is what gives the height its meaning in the formula.

It is important to note that the height does not always lie entirely within the triangle. In some cases, particularly with obtuse triangles, the perpendicular line from the opposite vertex may intersect the extension of the base rather than the base segment itself. Regardless of where this perpendicular meets the base or its extension, the distance still represents the height associated with that base.

This base-and-height formula is especially useful because of its simplicity and visual clarity, making it ideal for introductory geometry and practical applications. However, when the height is difficult to determine or not directly known, other formulas such as those involving trigonometry or only side lengths may be more convenient. Even so, the base-and-height method remains a foundational concept for understanding how the area of a triangle is defined and calculated.

When the lengths of two sides of a triangle and the measure of the angle included between those two sides are known, the area of the triangle can be calculated using a trigonometric formula. This method is especially useful when the height of the triangle is not easily determined but angle information is available. The variables in the formula correspond to the sides and angles of the triangle as shown in the calculator diagram referenced above.

The area of a triangle can be expressed using any pair of sides and the sine of the angle between them, as shown in the following equivalent formulas:

Area = ½ ab sin(C)
= ½ bc sin(A)
= ½ ac sin(B)

For example, if side a = 9, side b = 7, and the included angle C = 30°, the area can be calculated as follows:

Area = ½ × 7 × 9 × sin(30°)
Area = ½ × 63 × 0.5
Area = 15.75

Another widely used method for finding the area of a triangle is Heron’s formula. Unlike the trigonometric approach, Heron’s formula does not require knowledge of any angles, nor does it require selecting a base or determining a height. Instead, it relies solely on the lengths of all three sides of the triangle. This makes it particularly useful when only side length measurements are available.

Heron’s formula is written as:

Area = √[s(s − a)(s − b)(s − c)]

where s represents the semiperimeter of the triangle and is calculated using:

s = (a + b + c) / 2

For example, if the sides of the triangle are a = 3, b = 4, and c = 5, the semiperimeter is:

s = (3 + 4 + 5) / 2 = 6

Substituting these values into Heron’s formula gives:

Area = √[6(6 − 3)(6 − 4)(6 − 5)]
Area = √(6 × 3 × 2 × 1)
Area = √36 = 6

Both of these methods provide reliable ways to calculate the area of a triangle, and the choice between them depends on whether side lengths, angle measures, or both are known.

Median, inradius, and circumradius

Median

The median of a triangle is a line segment that begins at one vertex of the triangle and extends to the midpoint of the side directly opposite that vertex. Since a triangle has three vertices, it also has three medians one corresponding to each vertex. Each median divides the opposite side into two equal segments, helping to balance the triangle geometrically.

All three medians of a triangle intersect at a single point known as the centroid. The centroid represents the average, or mean, position of all the points within the triangle and is often described as the triangle’s center of mass. This point has the special property of always lying inside the triangle, regardless of the triangle’s shape or size. Additionally, the centroid divides each median into a fixed ratio, with the portion from the vertex to the centroid being twice as long as the portion from the centroid to the midpoint of the opposite side. The accompanying figure below illustrates the medians of a triangle and shows how they intersect at the centroid for clarity.

he three medians of a triangle are denoted by the line segments $m_a$ma​, $m_b$mb​, and $m_c$mc​, each corresponding to a specific vertex of the triangle. Every median connects a vertex to the midpoint of the side opposite that vertex. The length of each of these medians can be determined using the following formulas:

Let \(a\), \(b\), and \(c\) be the triangle’s side lengths. For example, with \(a = 2\), \(b = 3\), and \(c = 4\), the median \(m_a\) is calculated as:

\( m_a = \sqrt{\frac{2b^2 + 2c^2 – a^2}{4}} \)

\( m_b = \sqrt{\frac{2a^2 + 2c^2 – b^2}{4}} \)

\( m_c = \sqrt{\frac{2a^2 + 2b^2 – c^2}{4}} \)

Inradius

The inradius is the radius of the largest circle that can fit inside a polygon in this case, a triangle. It is perpendicular to each side of the triangle. To find the inradius, first determine the incenter by constructing two angle bisectors. The inradius is then the perpendicular distance from the incenter to any side of the triangle, since the incenter is equidistant from all sides.

For this calculator, the inradius of a triangle is determined using its area and semiperimeter. The formulas used are:$$\text{inradius} = \frac{\text{area}}{s}$$inradius=sarea​ $$s = \frac{a + b + c}{2}$$s=2a+b+c​

where $a$a, $b$b, and $c$c represent the lengths of the triangle’s sides.

Circumradius

The circumradius of a triangle is defined as the radius of a unique circle that passes through all three vertices of the triangle. This special circle is known as the circumcircle, and it exists for every type of triangle, whether it is acute, obtuse, or right-angled. The center of the circumcircle is called the circumcenter, which is a significant point in triangle geometry. The circumcenter is found as the point of intersection of the perpendicular bisectors of each side of the triangle. In other words, if you draw a line perpendicular to the midpoint of each side of the triangle, these three lines will converge at a single point the circumcenter. The distance from the circumcenter to any of the triangle’s vertices is equal and is what we refer to as the circumradius.

It is important to note that the location of the circumcenter varies depending on the type of triangle. For acute triangles, the circumcenter lies inside the triangle; for right triangles, it is located at the midpoint of the hypotenuse; and for obtuse triangles, it is positioned outside the triangle. Despite this variation in location, every triangle has a circumcenter and a corresponding circumradius, making the circumcircle a fundamental and universally applicable concept in triangle geometry. Understanding the circumradius and circumcenter is essential not only in pure geometry but also in applications such as construction, navigation, and even in certain areas of physics where triangular configurations appear.

In this calculator, the circumradius is determined using the formula:$$\text{circumradius} = \frac{a}{2 \sin(A)}$$circumradius=2sin(A)a​

where $a$a represents a side of the triangle, and $A$A is the angle opposite that side.

While the formula uses side $a$a and angle $A$A here, it can be applied to any side of the triangle along with its opposite angle.

Frequently Asked Questions.

1 : What is a Triangle Calculator?

A Triangle Calculator is an online tool or software that helps you find missing sides, angles, perimeter, or area of a triangle. You just enter the known values, and it calculates the unknown ones using formulas like the Pythagorean theorem or trigonometric rules.

2 : What is the area of ∆?

The area of a triangle (∆) is the space inside it. You can calculate it using:

• Base × Height ÷ 2
  Example: Area = (base × height) ÷ 2
• Heron’s formula (if all sides are known):
  $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$Area=s(s−a)(s−b)(s−c)​
  where $a, b, c$a,b,c are the sides, and $s = \frac{a+b+c}{2}$s=2a+b+c​ is the semi-perimeter.

3 : What is △?

△ is the symbol for a triangle. It represents a three-sided polygon with three angles and three sides.

4 : How to calculate angles in a triangle?

You can calculate angles in a triangle using:

• Sum of angles rule: The sum of all angles in a triangle is always 180°.
  Example: If two angles are 50° and 60°, the third angle = 180° − (50° + 60°) = 70°.
• Trigonometry: If you know the sides, use formulas like sine, cosine, or tangent laws.

6 : What is a triangle for kids?

A triangle is a shape with three straight sides and three corners (angles). It can look like a slice of pizza or a pyramid. Some triangles are tall and skinny, some are short and wide, and some have all sides the same!