Right Angle Triangle Calculator

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Right Angle Triangle Calculator
Sides a and b (Legs)Side a and Hypotenuse cSide a and Angle αHypotenuse c and Angle α
Results:
function updateInputs(){var mode=document.getElementById('calc_mode').value;var l1=document.getElementById('label1');var l2=document.getElementById('label2');document.getElementById('calculatorAnswer').style.display='none';if(mode=='sides_ab'){l1.innerHTML='Side a:';l2.innerHTML='Side b:';}else if(mode=='side_a_hyp'){l1.innerHTML='Side a:';l2.innerHTML='Hypotenuse c:';}else if(mode=='side_a_angle'){l1.innerHTML='Side a:';l2.innerHTML='Angle α (deg):';}else if(mode=='hyp_angle'){l1.innerHTML='Hypotenuse c:';l2.innerHTML='Angle α (deg):';}}function calculateTriangle(){var mode=document.getElementById('calc_mode').value;var v1=parseFloat(document.getElementById('input1').value);var v2=parseFloat(document.getElementById('input2').value);var showSteps=document.getElementById('steps').checked;var ans=document.getElementById('answer');var container=document.getElementById('calculatorAnswer');if(isNaN(v1)||isNaN(v2)){alert('Please enter valid numeric values');return;}var a,b,c,alpha,beta,area,perimeter,stepsHtml=";if(mode=='sides_ab'){a=v1;b=v2;c=Math.sqrt(a*a+b*b);alpha=Math.atan(a/b)*(180/Math.PI);beta=90-alpha;if(showSteps){stepsHtml+='c = √(a² + b²) = √('+a+'² + '+b+'²) = '+c.toFixed(4)+'
';stepsHtml+='α = arctan(a/b) = '+alpha.toFixed(4)+'°
';}}else if(mode=='side_a_hyp'){a=v1;c=v2;if(a>=c){alert('Hypotenuse must be longer than side a');return;}b=Math.sqrt(c*c-a*a);alpha=Math.asin(a/c)*(180/Math.PI);beta=90-alpha;if(showSteps){stepsHtml+='b = √(c² – a²) = √('+c+'² – '+a+'²) = '+b.toFixed(4)+'
';}}else if(mode=='side_a_angle'){a=v1;alpha=v2;if(alpha=90){alert('Angle must be between 0 and 90');return;}var rad=alpha*(Math.PI/180);c=a/Math.sin(rad);b=Math.sqrt(c*c-a*a);beta=90-alpha;if(showSteps){stepsHtml+='c = a / sin(α) = '+a+' / sin('+alpha+'°) = '+c.toFixed(4)+'
';}}else if(mode=='hyp_angle'){c=v1;alpha=v2;if(alpha=90){alert('Angle must be between 0 and 90');return;}var rad=alpha*(Math.PI/180);a=c*Math.sin(rad);b=c*Math.cos(rad);beta=90-alpha;if(showSteps){stepsHtml+='a = c * sin(α) = '+c+' * sin('+alpha+'°) = '+a.toFixed(4)+'
';}}area=0.5*a*b;perimeter=a+b+c;var output='Side a: '+a.toFixed(4)+'
';output+='Side b: '+b.toFixed(4)+'
';output+='Hypotenuse c: '+c.toFixed(4)+'
';output+='Angle α: '+alpha.toFixed(4)+'°
';output+='Angle β: '+beta.toFixed(4)+'°
';output+='Area: '+area.toFixed(4)+'
';output+='Perimeter: '+perimeter.toFixed(4)+'
';if(showSteps){output+='
'+stepsHtml+'
';}ans.innerHTML=output;container.style.display='block';}

Using the Right Angle Triangle Calculator

The right angle triangle calculator is an essential tool for students, engineers, and woodworkers. It simplifies the process of solving for missing dimensions in a triangle where one angle is exactly 90 degrees. By providing just two known values—such as two sides or one side and an acute angle—you can instantly determine all other properties, including the hypotenuse, area, and perimeter.

To use this calculator, simply select your known variables from the dropdown menu, enter the values, and click "Calculate."

Side a & Side b (Legs)
These are the two sides that meet at the 90-degree angle. They are often called the "opposite" and "adjacent" sides depending on which angle you are looking from.
Hypotenuse (c)
The longest side of the triangle, located directly opposite the right angle.
Angle α (Alpha)
One of the two acute angles. In a right triangle, α + β always equals 90 degrees.

How the Calculations Work

Right triangle math relies on two primary pillars: the Pythagorean Theorem and Trigonometry (SOH CAH TOA). Here are the fundamental formulas used by our right angle triangle calculator:

Pythagorean Theorem: a² + b² = c²

Area = ½ × base × height

Trig Ratios: sin(α) = a/c, cos(α) = b/c, tan(α) = a/b

  • Solving for Hypotenuse: If you know sides a and b, c = √(a² + b²).
  • Solving for an Angle: Use inverse trig functions, such as α = arctan(a/b).
  • Perimeter: The sum of all three sides (a + b + c).
  • Area: Since the legs are perpendicular, the area is simply half the product of the legs.

Practical Calculation Example

Example: A construction worker needs to build a ramp. The height of the deck is 3 feet (Side a), and the horizontal distance is 4 feet (Side b). What is the length of the ramp (Hypotenuse)?

Step-by-step solution:

  1. Identify knowns: a = 3, b = 4.
  2. Apply Pythagorean Theorem: c = √(3² + 4²)
  3. Calculate: c = √(9 + 16) = √25
  4. Result: c = 5 feet.
  5. Find Area: Area = (3 × 4) / 2 = 6 square feet.

Common Questions

What is a right angle triangle?

A right angle triangle is a triangle where one of the interior angles is exactly 90 degrees. This creates a unique relationship between the sides that allows for the use of the Pythagorean theorem.

Can I use this calculator for any triangle?

No, this specific tool is designed for triangles with a 90-degree angle. For scalene, isosceles, or equilateral triangles without a right angle, you would need to use the Law of Sines or the Law of Cosines.

How do you find the angle if you only know the sides?

You can use the inverse trigonometric functions. For example, if you know the opposite side (a) and the adjacent side (b), the angle α is found using arctan(a/b). Most scientific calculators and our right angle triangle calculator perform this automatically.

What is the sum of angles in a right triangle?

Like all triangles, the interior angles sum to 180 degrees. Since one angle is 90 degrees, the other two acute angles must sum to 90 degrees (they are complementary).

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