This Right-Angled Triangle Calculator helps you find the missing side lengths or angles in a right triangle using the fundamental trigonometric principles (SOH CAH TOA) and the Pythagorean theorem. Enter any two known values to solve for the remaining unknown variables in degrees.
Right Triangle Calculator in Degrees
Calculator in Degrees Formula
The calculation relies on three primary relationships (SOH CAH TOA) and the Pythagorean Theorem:
Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
Pythagorean Theorem: $\text{Opposite}^2 + \text{Adjacent}^2 = \text{Hypotenuse}^2$
Variables Explained
- Angle $\theta$ (Degrees): The non-right angle you are solving for or using as a basis for calculation. Must be between 0 and 90.
- Opposite Side (O): The length of the side across from angle $\theta$.
- Adjacent Side (A): The length of the side next to angle $\theta$, which is not the hypotenuse.
- Hypotenuse (H): The longest side, opposite the $90^\circ$ angle.
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What is Calculator in Degrees?
A “Calculator in Degrees” typically refers to any mathematical tool designed to perform trigonometric functions where the angles are measured in degrees, rather than radians. Degrees are the most common unit of angular measurement used in practical applications like construction, surveying, navigation, and primary engineering.
The core function involves solving right-angled triangles—triangles that include one $90^\circ$ angle. By leveraging the fixed relationships between the angles and the ratios of the side lengths (Sine, Cosine, and Tangent), you can determine all unknown elements of the triangle if you know at least one side and one other angle, or any two side lengths.
Ensuring your calculator uses degrees is crucial, as the results for functions like $\sin(45)$ are vastly different depending on whether the input $45$ is interpreted as $45$ degrees or $45$ radians. Our calculator is specifically configured to interpret all angle inputs as degrees for real-world usability.
How to Calculate Degrees (Example)
Suppose you know the Opposite side (O = 10) and the Adjacent side (A = 7.5). Here is how you find the angle $\theta$ and the Hypotenuse (H):
- Find the Angle ($\theta$): Use the Tangent relationship: $\tan(\theta) = O/A$. $$\theta = \arctan(10 / 7.5) = \arctan(1.333)$$ $$\theta \approx 53.13 \text{ degrees}$$
- Find the Hypotenuse (H): Use the Pythagorean Theorem: $O^2 + A^2 = H^2$. $$H = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25}$$ $$H = 12.5 \text{ units}$$
- Verify with Sine/Cosine: You can check your work: $\sin(53.13^\circ) \approx 0.8$. $O/H = 10/12.5 = 0.8$. The results are consistent.
Frequently Asked Questions (FAQ)
- What is the difference between degrees and radians?
- Degrees divide a circle into 360 parts. Radians define the angle based on the arc length of a circle’s radius. $180 \text{ degrees} = \pi \text{ radians}$. Trigonometric calculators often default to radians, so explicitly working in degrees is essential for many practical fields.
- Can I solve a triangle if only one side is known?
- No. For a right-angled triangle, you must know at least two components: either two sides, or one side and one acute angle ($\theta$ or $90^\circ – \theta$).
- What does SOH CAH TOA mean?
- It is a mnemonic to remember the fundamental trigonometric ratios: Sine is Opposite over Hypotenuse; Cosine is Adjacent over Hypotenuse; Tangent is Opposite over Adjacent.
- Is the hypotenuse always the longest side?
- Yes, in a right-angled triangle, the hypotenuse (the side opposite the $90^\circ$ angle) is always the longest side.