Triangle Calculator

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Triangle Calculator

Welcome to the Triangle Calculator! To get started, please provide at least one side and two other values for the following fields, then click “Calculate.” If you’re using radians for angles, you can enter values such as pi/2, pi/4, etc.

Triangle Overview

A triangle is a polygon with three sides and three vertices, which are points where two or more lines or edges meet. These three vertices are connected by three edges, or line segments. Triangles are typically labeled based on their vertices, for example, a triangle with vertices A, B, and C is denoted as ΔABC.

Triangles can be categorized based on the lengths of their sides and their internal angles:

Equilateral Triangle: All three sides are equal in length, and all angles are 60°.
Isosceles Triangle: Two sides are of equal length, and the angles opposite these sides are equal.
Scalene Triangle: All three sides are of different lengths, and all internal angles are different.

Types of Triangles Based on Angles

Right Triangle: One angle is 90°, and the longest side, opposite the right angle, is called the hypotenuse.
Oblique Triangle: These are non-right triangles, which can either be:
- Acute Triangle: All angles are less than 90°.
- Obtuse Triangle: One angle is greater than 90°.

Important Triangle Theorems and Laws

Interior Angles: The sum of the interior angles of any triangle is always 180°.
Exterior Angles: The exterior angle is equal to the sum of the two interior angles that are not adjacent to it.
Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Key Triangle Theorems

Pythagorean Theorem (for Right Triangles): In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):
a2+b2=c2a^2 + b^2 = c^2
For example, if a=3a = 3 and c=5c = 5, then:
32+b2=52⇒9+b2=25⇒b2=16⇒b=43^2 + b^2 = 5^2 \quad \Rightarrow \quad 9 + b^2 = 25 \quad \Rightarrow \quad b^2 = 16 \quad \Rightarrow \quad b = 4
Law of Sines: The ratio of a side of a triangle to the sine of its opposite angle is constant:
asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
For example, if b=2b = 2, B=90°B = 90°, and C=45°C = 45°, you can solve for cc using:
2sin⁡(90°)=csin⁡(45°)⇒c=2×22=2\frac{2}{\sin(90°)} = \frac{c}{\sin(45°)} \quad \Rightarrow \quad c = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}
Law of Cosines: To calculate angles when all sides are known:
A=arccos⁡(b2+c2−a22bc)A = \arccos \left( \frac{b^2 + c^2 – a^2}{2bc} \right)
Example: For a=8a = 8, b=6b = 6, and c=10c = 10, you can find BB using:
B=arccos⁡(82+102−622×8×10)=arccos⁡(0.8)≈36.87°B = \arccos \left( \frac{8^2 + 10^2 – 6^2}{2 \times 8 \times 10} \right) = \arccos(0.8) \approx 36.87°

Area of a Triangle

There are several methods to calculate the area of a triangle, depending on the available information:

Base and Height:
Area=12×Base×Height\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}
Example: If the base is 5 and the height is 6, then:
Area=12×5×6=15\text{Area} = \frac{1}{2} \times 5 \times 6 = 15
Two Sides and Included Angle (using the sine of the included angle):
Area=12×a×b×sin⁡(C)\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)
Example: For a=9a = 9, b=7b = 7, and C=30°C = 30°:
Area=12×9×7×sin⁡(30°)=15.75\text{Area} = \frac{1}{2} \times 9 \times 7 \times \sin(30°) = 15.75
Heron’s Formula (when all sides are known):
s=a+b+c2(semi-perimeter)s = \frac{a + b + c}{2} \quad \text{(semi-perimeter)} Area=s(s−a)(s−b)(s−c)\text{Area} = \sqrt{s(s – a)(s – b)(s – c)}
Example: For a=3a = 3, b=4b = 4, and c=5c = 5:
s=3+4+52=6s = \frac{3 + 4 + 5}{2} = 6 Area=6(6−3)(6−4)(6−5)=6\text{Area} = \sqrt{6(6 – 3)(6 – 4)(6 – 5)} = 6

Median, Inradius, and Circumradius

Median: A median is a line segment connecting a vertex to the midpoint of the opposite side. The medians of a triangle intersect at the centroid.
Inradius: The inradius is the radius of the largest circle that can fit inside the triangle, touching all three sides. It can be calculated using the area AA and semi-perimeter ss:
Inradius=As\text{Inradius} = \frac{A}{s}
Circumradius: The circumradius is the radius of the circle that passes through all three vertices of the triangle. It can be calculated using the formula:
Circumradius=a2sin⁡(A)\text{Circumradius} = \frac{a}{2 \sin(A)}
where aa is a side, and AA is the angle opposite that side.

Conclusion

Triangles are fundamental shapes in geometry, and their properties are closely tied to their side lengths and angles. Whether calculating angles, side lengths, or area, understanding these key theorems and formulas helps solve a wide range of geometric problems.