Special right triangles calculator 30 60 90

Question

In the day before computers when people actually had to draw angles, special tools called drawing triangles were used and the two most popular were the 30 60 90 and the 45 45 90 triangles.

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Right triangle
Special Right Triangles
What is the formula for 30 60 90 triangle Theorem?

These triangles have definite geometric relationships and it would be well worth your time to study the graphics on this page to learn about the ratios of their sides. If nothing else, it is worth noting that drawing the perpendicular bisector of an equilateral triangle (figure 1) produces a 30 60 90 triangle (figure 2) and bisecting a square along its diagonal (figure 3) yields a 45 45 90 triangle (figure 4).

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The graphics posted above show the 3 cases of a 30 60 90 triangle. If you know just 1 side of the triangle, the other 2 sides can be easily calculated.
For example, if you only know the short side (figure5), the medium side is found by multiplying this by the square root of 3 (about 1.732) and the hypotenuse is calculated by multiplying the short side by 2.
Looking at the middle section (figure 6), if you just know the "medium side", multiply this by (2 ÷ square root of 3) (about 1.155) to find the hypotenuse and multiply the medium side by (1 ÷ square root of 3) (about .5774) to calculate the short side.
The third graphic (figure 7) shows how to calculate the other sides if you only know the hypotenuse.

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This graphic shows how to calculate the sides of a 45 45 90 triangle.
If you just know the length of one "leg" of a 45 45 90 triangle (figure 8), multiply it by the square root of 2 (about 1.414) to obtain the hypotenuse length.
If you only know the hypotenuse (figure 9), multiply this by the reciprocal of the square root of 2 (about .707) to calculate the length of the leg.

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Figure 10 shows what occurs when we draw the bisector of the vertex angle of a 45 45 90 triangle.

Basically, when we bisect angle C, it divides triangle ABC into two 45 45 90 triangles and it bisects side AB.

For example, When Short side a = 2, then the long side b = 3.464102, Hypotenuse c = 4, Area = 3.46102, perimeter = 9.464102.

For example, When Perimeter = 12, then the short side a = 2.535898, long side b = 4.392305, Hypotenuse c = 5.071797, Area = 5.569219.

Please provide 2 values below to calculate the other values of a right triangle. If radians are selected as the angle unit, it can take values such as pi/3, pi/4, etc.

a = ∠α =b = ∠β =c = h =A =areaP =perimeter

RelatedTriangle Calculator | Pythagorean Theorem Calculator

Right triangle

A right triangle is a type of triangle that has one angle that measures 90°. Right triangles, and the relationships between their sides and angles, are the basis of trigonometry.

In a right triangle, the side that is opposite of the 90° angle is the longest side of the triangle, and is called the hypotenuse. The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides. Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90°) for side c, as shown below. In this calculator, the Greek symbols α (alpha) and β (beta) are used for the unknown angle measures. h refers to the altitude of the triangle, which is the length from the vertex of the right angle of the triangle to the hypotenuse of the triangle. The altitude divides the original triangle into two smaller, similar triangles that are also similar to the original triangle.

If all three sides of a right triangle have lengths that are integers, it is known as a Pythagorean triangle. In a triangle of this type, the lengths of the three sides are collectively known as a Pythagorean triple. Examples include: 3, 4, 5; 5, 12, 13; 8, 15, 17, etc.

Area and perimeter of a right triangle are calculated in the same way as any other triangle. The perimeter is the sum of the three sides of the triangle and the area can be determined using the following equation:

Special Right Triangles

30°-60°-90° triangle:

The 30°-60°-90° refers to the angle measurements in degrees of this type of special right triangle. In this type of right triangle, the sides corresponding to the angles 30°-60°-90° follow a ratio of 1:√3:2. Thus, in this type of triangle, if the length of one side and the side's corresponding angle is known, the length of the other sides can be determined using the above ratio. For example, given that the side corresponding to the 60° angle is 5, let a be the length of the side corresponding to the 30° angle, b be the length of the 60° side, and c be the length of the 90° side.:

Angles: 30°: 60°: 90°

Ratio of sides: 1:√3:2

Side lengths: a:5:c

Then using the known ratios of the sides of this special type of triangle:

As can be seen from the above, knowing just one side of a 30°-60°-90° triangle enables you to determine the length of any of the other sides relatively easily. This type of triangle can be used to evaluate trigonometric functions for multiples of π/6.

45°-45°-90° triangle:

The 45°-45°-90° triangle, also referred to as an isosceles right triangle, since it has two sides of equal lengths, is a right triangle in which the sides corresponding to the angles, 45°-45°-90°, follow a ratio of 1:1:√2. Like the 30°-60°-90° triangle, knowing one side length allows you to determine the lengths of the other sides of a 45°-45°-90° triangle.