Triangle Calculator

Triangle calculator

The triangle calculator is a versatile online triangle solver that allows you to quickly find all missing triangle measurements based on three known variables. By simply inputting the side lengths and angles of a triangle, this comprehensive tool instantly calculates the following properties:

• missing side lengths,
• missing triangle angles,
• area,
• perimeter,
• semiperimeter,
• heights to all sides of the triangle,
• medians to all sides of the triangle,
• inradius,
• circumradius.

Additionally, this triangle calculator provides the precise coordinates of the vertices, the centroid, the inscribed circle center (incenter), and the circumscribed circle center (circumcenter), assuming that the coordinates of vertex A are positioned at the origin [0, 0].

Directions for use

Using this online triangle calculator is incredibly straightforward. Simply enter any three known values into the input fields. These can be any combination of angles and side lengths. Note: At least one of the inputted values must be a side length; otherwise, the triangle will have infinite possible solutions (forming similar triangles).

Next, select your preferred units for the triangle angles—choose between degrees or radians. When using radians, use "pi" to represent π. For example, if your angle is $\frac{π}{3}$, simply enter "pi/3". Once you have inputted your known variables, click "Calculate." The solver will instantly generate all missing values from the list above alongside a schematic drawing of the triangle to help you easily visualize the results.

For those looking to understand the math behind the results, you can expand the "Show Calculation Steps" section. This provides a detailed breakdown of the solution algorithm and the specific geometric formulas used to find the final answer.

Limitations on the input values

For the triangle solver to work correctly, please keep the following geometric rules in mind:

First, at least one of the known values must be a side length.

When entering two angles and one side length, the sum of the provided angles must be strictly less than 180° or π.

When inputting three side lengths, the sum of any two sides must always be greater than the length of the remaining third side, in accordance with the triangle inequality theorem.

Calculation example

Imagine you are moving and want to borrow a moving truck from a friend. You need to load and unload heavy boxes, but the truck lacks a built-in ramp. You own a portable ramp, but you must ensure its dimensions match the truck's height before borrowing it.

Your ramp is not adjustable. You measure it and find that two of its sides are 1 m and 0.8 m long, and the angle opposite the 1 m side is exactly 85 degrees (as shown in the image). You also know the truck's tailgate height can be adjusted anywhere from 0.5 m to 1 m. The question is: will your ramp fit?

Given

• side b = 1;
• side c = 0.8;
• angle B = 85 degrees.

Solution

To determine whether your ramp fits, you need to solve the triangle and check if the length of side A falls within the truck's adjustable height range: 0.5 < a < 1.

By plugging the known values into our triangle side calculator, you will get the exact dimensions needed. While the tool computes all missing variables, we only need the missing side length for this practical example:

Answer

• Side a = 0.67376

• Side b = 1

• Side c = 0.8

• angle A = 42.16° = 42°9'35" = 0.73582 rad

• angle B = 85° = 1.48353 rad

• angle C = 52.84° = 52°50'25" = 0.92224 rad

The resulting ramp configuration looks like this:

As you can see, the required height is a ≈ 0.674 m. Since the truck's height is adjustable within the 0.5 < a < 1 range, the ramp fits perfectly! You can comfortably borrow the truck from your friend instead of renting one.

Triangle: definition and important formulas

In geometry, a triangle is a two-dimensional plane figure formed by the intersection of three straight, non-parallel lines. It can also be described as a basic polygon with three vertices and three edges. In everyday mathematics, the edges of a triangle are commonly referred to as its sides.

Conditions of existence of a triangle

For any triangle to exist, it must satisfy two fundamental rules: one concerning its sides, and another concerning its angles.

The rule for the sides is known as the triangle inequality theorem. It states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining third side. If the sum of the two shorter sides exactly equals the length of the third side, it forms a "degenerate" triangle.

A degenerate triangle is a theoretical case where all three vertices lie on the exact same straight line (effectively forming a flat line segment). Because this is a highly specialized case generally excluded from elementary geometry, our calculator does not consider it.

The rule for the angles states that the sum of the interior angles of any valid triangle must always equal exactly 180° (or π radians).

Triangle measurements

Let's explore the most essential triangle measurements alongside the core geometric formulas used to calculate them.

The perimeter of a triangle is the total distance around its outer edge, calculated by finding the sum of all three side lengths:

p = a + b + c

The semiperimeter is simply half of the triangle's perimeter:

$$s=\frac{p}{2}=\frac{a+b+c}{2}$$

The area of a triangle measures the total space enclosed within its three sides on a 2D plane. If you know the lengths of two sides and the measure of the angle between them (the included angle), you can calculate the area using this formula:

$$A=\frac{1}{2}a× b×\sin{C}$$

A triangle's height (or altitude) is a perpendicular line segment drawn from a vertex to the opposite side (or its extension). Since a triangle has three vertices, it inherently possesses three unique heights. The altitude drawn to side a is typically denoted as hₐ. Similarly, the other two heights are represented as $h_b$ and h꜀. The most straightforward way to find a triangle's height is by utilizing its area:

$$A=\frac{1}{2}× a× h_a=\frac{1}{2}× b× h_b=\frac{1}{2}× c× h_c$$

$$h_a=\frac{2A}{a}, h_b=\frac{2A}{b}, h_c=\frac{2A}{c}$$

A median of a triangle is a line segment connecting a vertex to the exact midpoint of the opposite side. Consequently, every triangle features three medians.

The median drawn to side a is denoted as mₐ. Likewise, the other two medians are represented as $m_b$ and m꜀. You can calculate the lengths of any median using the following formula:

$$m_a=\frac{1}{2}\sqrt{2b²+2c^2-a^2}$$

The inradius of a triangle is the radius of the largest possible circle inscribed perfectly inside the triangle, tangent to all three of its sides.

The length of the inradius r can be calculated using the area (A) and the semiperimeter (s):

$$r=\frac{A}{s}$$

The circumradius of a triangle is the radius of the circumscribed circle—a circle that passes perfectly through all three vertices of the triangle.

We can derive the length of the circumradius R by applying the Law of Sines:

$$2R=\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}$$

The Law of Sines is incredibly useful for finding missing side lengths or unknown angles. Another fundamental geometric theorem vital for solving triangles is the Law of Cosines:

$$a=\sqrt{b²+c^2-2bc\cos{A}}$$

$$b=\sqrt{a^2+c^2-2ac\cos{B}}$$

$$c=\sqrt{a^2+b²-2ab\cos{C}}$$

The formulas detailed above provide everything you need to calculate any triangle measurement manually. However, for maximum efficiency and precision, our online triangle calculator applies these exact formulas in the background to find all missing values in seconds!