Calculators

Geometry

Plane

Right Triangle Shape

A = angle A
a = side a
B = angle B
b = side b
C = angle C
c = side c

K = area
P = perimeter

See Diagram Below:
ha = altitude of a
hb = altitude of b
hc = altitude of c

*Length units are for your reference only since the value of the resulting lengths will always be the same no matter what the units are.

A right triangle is a special case of a triangle where one angle is equal to 90 degrees. In the case of a right triangle the length of the sides have a predictable relationship where a² + b² = c². This formula is known as the Pythagorean Theorem.

In our calculations for a right triangle we only consider 2 known sides to calculate the other 7 unknowns. For example, if we know a and b we can calculate c using the Pythagorean Theorem. c = √(a² + b²). Once we know sides a, b, and c we can calculate the perimeter = P, the semiperimeter = s, the area = K, and the altitudes: ha, hb, and hc. Let us know if you have any other suggestions!

Pythagorean Triples

Any set of three positive integers that satisfies the Pythagorean equation a² + b² = c² is known as a Pythagorean triple. In these right triangles the length of the two sides and the hypotenuse are positive integers where a < c and b < c.

Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). Showing the math with these triples in (a, b, c) format:

Pythagorean Triple (3, 4, 5)

• a² + b² = c²
• 3² + 4² = 5²
• 9 + 16 = 25

Pythagorean Triple (5, 12, 13)

• a² + b² = c²
• 5² + 12² = 13²
• 25 + 144 = 169

You might find it interesting to know that you can create sets of Pythagorean triples using formulas. Use one equation for odd numbers and another equation for even numbers.

Using an odd number n, the triple is:

• (a, b, c) = (n, (n² - 1) ÷ 2, (n² + 1) ÷ 2)
• If a = 9
• b = (n² - 1) ÷ 2 = (9² - 1) ÷ 2 = (81 - 1) ÷ 2 = 80 ÷ 2 = 40
• c = (n² + 1) ÷ 2 = (9² + 1) ÷ 2 = (81 + 1) ÷ 2 = 82 ÷ 2 = 41
• The Pythagorean triples are (9, 40, 41)

Using an even number n, the triple is:

• (n, ((n ÷ 2)² - 1), (n ÷ 2)² + 1))
• If a = 8
• b = (n ÷ 2)² - 1 = (8 ÷ 2)² + 1 = 4² - 1 = 16 - 1 = 15
• c = (n ÷ 2)² + 1 = (8 ÷ 2)² + 1 = 4² + 1 = 16 + 1 = 17
• The Pythagorean triples are (8, 15, 17)

Formulas and Calculations for a right triangle:

• Pythagorean Theorem for Right Triangle: a² + b² = c²
• Perimeter of Right Triangle: P = a + b + c
• Semiperimeter of Right Triangle: s = (a + b + c) / 2
• Area of Right Triangle: K = (a * b) / 2
• Altitude a of Right Triangle: ha = b
• Altitude b of Right Triangle: hb = a
• Altitude c of Right Triangle: hc = (a * b) / c

1. Given sides a and b find side c and the perimeter, semiperimeter, area and altitudes

• a and b are known; find c, P, s, K, ha, hb, and hc
• c = √(a² + b²)
• P = a + b + c
• s = (a + b + c) / 2
• K = (a * b) / 2
• ha = b
• hb = a
• hc = (a * b) / c

2. Given sides a and c find side b and the perimeter, semiperimeter, area and altitudes

• a and c are known; find b, P, s, K, ha, hb, and hc
• b = √(c² - a²)
• P = a + b + c
• s = (a + b + c) / 2
• K = (a * b) / 2
• ha = b
• hb = a
• hc = (a * b) / c

For more information on right triangles see:

Weisstein, Eric W. "Right Triangle." From MathWorld--A Wolfram Web Resource. Right Triangle.

Weisstein, Eric W. "Altitude." From MathWorld--A Wolfram Web Resource. Altitude.