A triangle is a 3-sided shape with three vertices. A right triangle is a triangle with one right angle, that is, one vertex is equal to 90°. Given that the three vertices of a triangle must sum to 180°, by definition, the other two angles will be each be less than 90°.
In a right-angled triangle, the two shorter sides will meet at a right angle, and the side opposite the right angle is known as the hypotenuse. The reason this distinction is important is that the calculations applied to this type of triangle are simpler than for more complex 3-sided shapes, which may require trigonometry.
A right triangle is a simple case, which speaks directly to the Pythagorean theorem without the use of complex equations.
We use the Pythagorean Theorem to solve for various values relating to the right-angled triangle.
This theorem shows that if you have a right triangle, the length of the hypotenuse is the square root of the sum of the square of the sides.
The hypotenuse is the side that lies opposite of the right angle, and will always be the longest side.
If we know the length of two sides, we can always calculate the length of the third side of the triangle using this theorem as long as the triangle is a right-angled triangle. Take a look at the diagram below.
C - Hypotenuse B A
The Theorem of Pythagoras is written as follows:
Using this theorem, you can calculate the various parameters of a right-angled triangle.
The formula for the length of the Hypotenuse C, is written as follows:
In this example, we have the two shorter sides, and we sue the theorem to solve for the length of the hypotenuse:
C - Hypotenuse B A
In this example, B = 3, and A = 4. Thus, the equation to solve for the length of the Hypotenuse is written as follows:
From this, we can calculate that the length of the Hypotenuse is 5.
The example above is known as a Pythagorean triple; a set of three positive integers related according to the theorem. Only right triangles can yield Pythagorean triples, though not all right triangles will.
These relationships have fascinating implications for set theory, as well as associations with the Fibonacci series, and opens a real of fascinating fundamental mathematics!
One of the greatest mathematical mysteries of the century, which remains unsolved to this day, was posed by Pierre Fermat.
In Fermat’s Last Theorem, he states that there could not be a set of three integer numbers that would satisfy the Pythagorean Theorem; aⁿ + bⁿ = cⁿ for n bigger than 2.
This conjecture has not been proven mathematically, and it's considered one of the most important mathematical problems of the century.
For a right-angled triangle where the two shorter sides are equal in length, we say that A = B, therefore we can write that:
H, in this case, stands for the Hypotenuse, which is the Diagonal of the square we are trying to calculate.
Thus, the diagonal is the square root of twice the square of one of the sides, as follows:
And so
The formula for the area of a triangle is half of the length of the base multiplied by the height.
If you imagine the area of a rectangle as being the width, multiplied by the length, and then consider a diagonal intersection of the rectangle resulting in a triangle then it becomes clear why the calculation involves half the base, multiplied by the height.
Hypotenuse Height Base
In the case of the diagram above, we calculate the area as follows:
If we know the lengths of the shorter sides, then this calculation is an easy one, we insert the length of the base (side B) and the height (side A) into the above equation to find the answer. If you do have only one of these lengths, and the length of the hypotenuse, then you can calculate the length of the missing side using the Pythagorean theorem. More simply, you can use the hypotenuse as the base, but you will still need to use the theorem to calculate the height of the triangle to get the area. Why not use this handy calculator instead!
We have several calculators related to the measurement of triangles, their perimeter, and areas, and how these related more broadly to scientific questions!
• We use right triangles as the basis for all calculus as we use this formula in calculating gradients, or slopes. Visit our handy calculator for working out gradients or slopes of lines.
• Sundials use the measurement of shadows and right-angle geometry to tell time. This concept is based on the same foundational mathematical concepts.
