Right Triangle Calculator

Use the Right Triangle Calculator to accurately calculate and analyze a right triangle. Determine side lengths, angles, height, area, and perimeter quickly.

Right Triangle Calculator

Side A:

Side B:

Side C (hypotenuse):

∠α:

∠β:

How to Use the Right Triangle Calculator

1. Input Values:
- Side A, Side B, or Side C: Enter the length of any two sides of the right triangle. The third side will be automatically calculated using the Pythagorean theorem.
- ∠α (Alpha) or ∠β (Beta): Enter the value of either angle in degrees. The other angle will be calculated as 90 degrees minus the specified angle.
- Height (h): Optionally, you can provide the height of the triangle perpendicular to the base.
- Area: If you know the area of the triangle, you can enter its value.
- Perimeter: Similarly, if you know the perimeter of the triangle, you can input it.

Note: At least two valid inputs are required for the calculations to work.

2. Click the "Calculate" button to perform the calculations.

3. View the Results:
- Side A, Side B, and Side C: These values represent the lengths of the triangle sides.
- ∠α (Alpha) and ∠β (Beta): The calculated angles in degrees.
- Height (h): If not provided, it will be calculated based on the other inputs.
- Area: The area of the triangle based on the given inputs.
- Perimeter: The total length of all three sides.
- Inradius: The radius of the inscribed circle within the triangle.
- Circumradius: The radius of the circumscribed circle around the triangle.

Note: If any input was not provided, the calculator will automatically calculate it based on the other available values.

Example:
Suppose you know Side A = 4 units and ∠β = 30 degrees.

1. Enter the value "4" for Side A and "30" for ∠β.

2. Click "Calculate."

3. The results will be displayed:
- Side A: 4.00 units
- Side B: 1.93 units
- Side C (hypotenuse): 4.62 units
- ∠α: 60 degrees
- ∠β: 30 degrees
- Height (h): 3.46 units
- Area: 6.93 square units
- Perimeter: 10.54 units
- Inradius: 0.86 units
- Circumradius: 2.31 units

The calculator will also provide a visual representation of the right triangle with labeled sides and angles.

Use the Right Triangle Calculator to explore various configurations and properties of right triangles quickly and accurately.

What is a Right Triangle?

A right triangle is a fundamental geometric shape that consists of three sides and three angles. It is called a "right" triangle because it contains one right angle, which measures exactly 90 degrees (90°). This right angle is formed when one of the sides is perpendicular to the other side, creating a perfect L-shape.

Let's break down the key concepts of a right triangle:

1. Sides:
- Hypotenuse (Side C): The longest side of the right triangle and is always opposite the right angle.
- Legs (Side A and Side B): The two shorter sides that form the right angle. These sides are adjacent to the right angle.

2. Angles:
- Right Angle (∠C): The angle that measures 90 degrees and is formed between the hypotenuse and one of the legs.
- Acute Angles (∠A and ∠B): The two smaller angles of the right triangle that are less than 90 degrees.

3. Pythagorean Theorem:
The Pythagorean theorem is a fundamental formula that relates the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b). Mathematically, it can be written as:
c^2 = a^2 + b^2

For example, let's consider a right triangle with side lengths of a = 3 units and b = 4 units.

Using the Pythagorean theorem:
c^2 = a^2 + b^2
c^2 = 3^2 + 4^2
c^2 = 9 + 16
c^2 = 25

Taking the square root of both sides, we find:
c = √25
c = 5 units

So, in this example, the hypotenuse (Side C) of the right triangle measures 5 units.

Right triangles have several important properties and relationships. Here are a few:

- The sum of the interior angles of any triangle is always 180 degrees. In a right triangle, since one angle is fixed at 90 degrees, the sum of the other two angles is always 90 degrees.
- The lengths of the legs and the hypotenuse are related by the Pythagorean theorem.
- The area of a right triangle can be calculated using the formula: Area = (1/2) * base * height, where the base and height are the lengths of the legs.

Right triangles have various applications in mathematics, physics, and engineering. They are particularly useful for solving problems involving distances, angles of elevation and depression, and trigonometric functions.

Special Right Triangles

In addition to the general properties of right triangles, there are two special types of right triangles that have distinct characteristics and can be easily recognized based on their side lengths and angles. These special right triangles are the 45-45-90 triangle and the 30-60-90 triangle.

1. 45-45-90 Triangle:
In a 45-45-90 triangle, the two legs are congruent (equal in length) and the angles are 45 degrees, 45 degrees, and 90 degrees. This triangle is also known as an isosceles right triangle.

Key properties of a 45-45-90 triangle:
- The two legs are equal in length.
- The length of the hypotenuse is equal to √2 times the length of either leg.
- The angles are 45 degrees, 45 degrees, and 90 degrees.

For example, let's consider a 45-45-90 triangle with a leg length of 5 units.

Properties:
- Leg A = Leg B = 5 units
- Hypotenuse (Side C) = 5√2 units
- ∠A = ∠B = 45 degrees
- ∠C = 90 degrees

The 45-45-90 triangle is commonly encountered in various geometric and trigonometric applications due to its simple and predictable properties.

2. 30-60-90 Triangle:
In a 30-60-90 triangle, the angles are 30 degrees, 60 degrees, and 90 degrees. The ratios of the side lengths in this triangle have a specific relationship.

Key properties of a 30-60-90 triangle:
- The side lengths are in a ratio of 1 : √3 : 2.
- The shortest side (opposite the 30-degree angle) is half the length of the hypotenuse.
- The medium side (opposite the 60-degree angle) is √3 times the length of the shortest side.
- The longest side (the hypotenuse) is twice the length of the shortest side.

For example, let's consider a 30-60-90 triangle with a shortest side length of 4 units.

Properties:
- Shortest Side (Side A) = 4 units
- Medium Side (Side B) = 4√3 units
- Hypotenuse (Side C) = 8 units
- ∠A = 30 degrees
- ∠B = 60 degrees
- ∠C = 90 degrees

The 30-60-90 triangle is useful in trigonometry, especially when dealing with special angles and trigonometric functions such as sine, cosine, and tangent.

Right Triangle Angle Relationships

In a right triangle, the angles have specific relationships that are important to understand. These relationships provide insights into the properties and calculations involving right triangles.

1. Right Angle (∠C):
The right angle (∠C) in a right triangle measures exactly 90 degrees. It is formed by the intersection of the two legs and is always the largest angle in the triangle. The right angle is denoted by a small square at the vertex.

2. Acute Angles (∠A and ∠B):
The acute angles (∠A and ∠B) in a right triangle are the two smaller angles that are less than 90 degrees. These angles are formed between one of the legs and the hypotenuse. The sum of the measures of the acute angles is always 90 degrees.

3. Complementary Angles:
The acute angles (∠A and ∠B) in a right triangle are complementary angles, meaning their sum is always 90 degrees. This relationship holds true in all right triangles.

4. Relationship between Angles and Side Lengths:
The measures of the angles in a right triangle are closely related to the ratios of the side lengths. The trigonometric functions (sine, cosine, and tangent) define these relationships. For example:
- The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
- The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

These trigonometric functions enable us to calculate missing side lengths or angle measures in right triangles.

Understanding the angle relationships in a right triangle is crucial for solving problems involving angles, side lengths, and trigonometric functions. These relationships form the basis of trigonometry and have wide-ranging applications in fields such as physics, engineering, and navigation. By leveraging these relationships, we can make accurate measurements, perform calculations, and analyze various real-life situations involving right triangles.

Pythagorean Theorem and Right Triangles

The Pythagorean theorem is a fundamental concept in mathematics, specifically in the context of right triangles. It establishes a relationship between the lengths of the sides of a right triangle and plays a crucial role in solving problems involving right triangles. Let's explore the Pythagorean theorem and its significance in more detail:

1. The Pythagorean Theorem Statement:
The Pythagorean theorem states that in a 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. Mathematically, it can be expressed as:

c^2 = a^2 + b^2

Here, c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides, which are the legs of the right triangle.

2. Application of the Pythagorean Theorem:
The Pythagorean theorem allows us to calculate the length of any side of a right triangle when the lengths of the other two sides are known. We can use it to find missing side lengths, determine if a triangle is a right triangle, or check if a set of side lengths forms a valid right triangle.

3. Pythagorean Triples:
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the Pythagorean theorem. In other words, (a, b, c) form the side lengths of a right triangle. Some well-known Pythagorean triples include (3, 4, 5), (5, 12, 13), and (8, 15, 17). These triples have been studied extensively throughout history due to their significance in geometry and number theory.

4. Applications of the Pythagorean Theorem:
The Pythagorean theorem finds applications in various fields, including architecture, engineering, physics, and navigation. It allows us to calculate distances, determine heights, design structures, and solve real-life problems involving right triangles. For example, it is used in measuring diagonal distances, determining cable lengths, calculating forces in mechanical systems, and creating accurate maps.

Understanding the Pythagorean theorem and its application to right triangles is essential in solving geometric problems and practical situations. It provides a fundamental tool for calculations involving right triangles and serves as a basis for more advanced concepts in geometry, trigonometry, and mathematics as a whole.

Right Triangle Problems with Solutions

1. Problem: Find the length of the hypotenuse in a right triangle with side lengths of 3 units and 4 units.

Solution:
Using the Pythagorean theorem, we can calculate the length of the hypotenuse (c):

c^2 = a^2 + b^2
c^2 = 3^2 + 4^2
c^2 = 9 + 16
c^2 = 25
c = √25
c = 5 units

Therefore, the length of the hypotenuse is 5 units.

2. Problem: Determine the length of the missing leg in a right triangle with a hypotenuse of 10 units and one leg measuring 6 units.

Solution:
Using the Pythagorean theorem, we can find the length of the missing leg (b):

a^2 + b^2 = c^2
6^2 + b^2 = 10^2
36 + b^2 = 100
b^2 = 100 - 36
b^2 = 64
b = √64
b = 8 units

Therefore, the length of the missing leg is 8 units.

3. Problem: Calculate the area of a right triangle with a base of 12 units and a height of 5 units.

Solution:
The area of a right triangle can be calculated using the formula A = (1/2) * base * height.

A = (1/2) * 12 * 5
A = 30 square units

Therefore, the area of the right triangle is 30 square units.

4. Problem: Determine the measure of angle ∠A in a right triangle with side lengths of 5 units and 13 units.

Solution:
To find the measure of angle ∠A, we can use the inverse trigonometric function (arcsine).

∠A = arcsin(opposite/hypotenuse)
∠A = arcsin(5/13)
∠A ≈ 23.58 degrees

Therefore, the measure of angle ∠A is approximately 23.58 degrees.

5. Problem: Given the angles ∠A = 30 degrees and ∠B = 60 degrees in a right triangle, calculate the length of the hypotenuse if the shortest leg measures 4 units.

Solution:
To find the length of the hypotenuse, we can use the trigonometric function cosine.

Hypotenuse = adjacent / cos(angle)
Hypotenuse = 4 / cos(30 degrees)
Hypotenuse = 4 / √3/2
Hypotenuse = 8/√3 units

Therefore, the length of the hypotenuse is 8/√3 units.

Solving right triangle problems involves applying the Pythagorean theorem, trigonometric functions, and basic geometric principles. These examples demonstrate how to find missing side lengths, calculate areas, and determine angle measures in right triangles. By understanding these techniques, you can successfully solve various problems related to right triangles.

Right Triangle FAQs

1. What is a right triangle?
A right triangle is a type of triangle that has one angle measuring 90 degrees, known as the right angle. The other two angles in a right triangle are acute angles, meaning they are less than 90 degrees. The side opposite the right angle is called the hypotenuse, while the other two sides are known as the legs.

2. What is the Pythagorean theorem?
The Pythagorean theorem is a mathematical principle that relates the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The Pythagorean theorem is expressed as c^2 = a^2 + b^2, where c represents the hypotenuse, and a and b represent the legs of the right triangle.

3. How do I use the Pythagorean theorem to find a missing side length?
To use the Pythagorean theorem to find a missing side length in a right triangle, you need to know the lengths of the other two sides. You can rearrange the formula c^2 = a^2 + b^2 to solve for the missing side. For example, if you know the lengths of the legs (a and b), you can calculate the length of the hypotenuse (c). Alternatively, if you know the hypotenuse and one leg, you can find the length of the missing leg.

4. What are special right triangles?
Special right triangles are right triangles with specific angle measures and side ratios that make calculations easier. The two most common types of special right triangles are the 45-45-90 triangle and the 30-60-90 triangle. In a 45-45-90 triangle, the two acute angles are both 45 degrees, and the sides have a ratio of 1:1:√2. In a 30-60-90 triangle, the acute angles are 30 degrees and 60 degrees, and the sides have a ratio of 1:√3:2.

5. How do I calculate the area of a right triangle?
The area of a right triangle can be calculated using the formula A = (1/2) * base * height. The base and height of the right triangle should be perpendicular to each other, with the base being one of the legs or the hypotenuse. Multiply the base and height, and then divide the result by 2 to obtain the area.

6. What are the trigonometric functions used in right triangles?
The trigonometric functions commonly used in right triangles are sine, cosine, and tangent. These functions relate the ratios of the sides of a right triangle to the angles. The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

7. How do I find the angles in a right triangle?
In a right triangle, one angle is always 90 degrees. To find the other two angles, you can use trigonometric functions. For example, if you know the lengths of the two legs, you can use the sine function to find one of the acute angles. If you know the lengths of the hypotenuse and one leg, you can use the cosine function to find one of the acute angles. The sum of the angles in any triangle is always 180 degrees.

8. What is the relationship between the sides in a 45-45-90 triangle?
In a 45-45-90 triangle, the two acute angles are both 45 degrees. The sides have a specific ratio: the length of each leg is equal, and the length of the hypotenuse is √2 times the length of either leg. For example, if one leg measures 5 units, the other leg will also measure 5 units, and the hypotenuse will be 5√2 units.

9. How can I determine if three given side lengths form a right triangle?
To determine if three given side lengths form a right triangle, you can use the Pythagorean theorem. Square the lengths of the two shorter sides and add them together. Then, square the length of the longest side (hypotenuse). If the sum of the squares of the shorter sides is equal to the square of the longest side, the triangle is a right triangle.

10. Can a right triangle have two equal legs?
Yes, a right triangle can have two equal legs. This type of right triangle is called an isosceles right triangle. In an isosceles right triangle, the two legs are congruent, meaning they have the same length. The non-right angles in an isosceles right triangle are always 45 degrees each.

11. How do I calculate the inradius and circumradius of a right triangle?
The inradius of a right triangle is the radius of the inscribed circle, which touches all three sides of the triangle. It can be calculated using the formula r = (a + b - c) / 2, where a and b are the lengths of the legs, and c is the length of the hypotenuse. The circumradius is the radius of the circumscribed circle, which passes through all three vertices of the triangle. It can be calculated using the formula R = c / 2, where c is the length of the hypotenuse.

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Right Triangle Calculator

Right Triangle Calculator

​How to Use the Right Triangle Calculator

​What is a Right Triangle?

Special Right Triangles

​Right Triangle Angle Relationships

​Pythagorean Theorem and Right Triangles

​Right Triangle Problems with Solutions

​Right Triangle FAQs

How to Use the Right Triangle Calculator

What is a Right Triangle?

Right Triangle Angle Relationships

Pythagorean Theorem and Right Triangles

Right Triangle Problems with Solutions

Right Triangle FAQs