Triangles are one of the most fundamental shapes in geometry, consisting of three sides, three vertices, and three angles. They play a vital role in mathematics, engineering, and everyday problem-solving. This guide breaks down essential triangle formulas to help you calculate area, perimeter, semi-perimeter, and more.
The area of a triangle can be calculated using its base and height:
A : Area P : Perimeter b: Base h: Height
\[A= { b * h \over 2 } \]
This formula is useful for right-angled and other standard triangles when the base and height are known.
The perimeter of a triangle is the sum of the lengths of its three sides:
\[ P = a + b + c \]
Where \(a, b, c\) are the sides of the triangle
The semi-perimeter is half the perimeter and is used in Heron’s formula for area:
\[s= { a + b + c \over 2 } {= {p \over 2} }\]
When the lengths of all three sides are known, the area of the triangle can be calculated using Heron’s formula:
\[A= { \sqrt{s(s-a)(s-b)(s-c)} }\]
where \(s\) is a semi-perimeter and \(a, b, c\) are the sides of the triangle
The sum of the interior angles of any triangle is always:
\[\alpha + \beta + \gamma = 180^\circ \]
where \(\alpha, \beta, \gamma\) represent the three interior angles
In geometry, a median of a triangle is a line segment that joins a vertex of the triangle to the midpoint of the opposite side. Medians are fundamental in dividing a triangle into smaller regions of equal area and are widely used in calculations involving centroid and triangle properties.
In the context of geometry, \(CN = AN\) typically refers to the property of a median of a triangle. Here's what it means:
\[ CN = AN \]
A median of a triangle is a line segment joining a vertex of the triangle to the midpoint of the opposite side. For example, in a triangle \(ABC\) the median from vertex \(C\) to the midpoint \(N\) of side \(AB\) is called \(CN\).
To calculate the length of a median in a triangle, we use the following formula:
\[{m = \frac{1}{2} \sqrt{2a^2 + 2c^2 - b^2} }\]
Where:
In geometry, an angle bisector is a line or ray that divides an angle into two equal parts. It is an essential concept in triangle geometry, used in various calculations and constructions, such as finding proportional relationships between sides and determining specific segment lengths. Let’s explore the key formulas and properties of angle bisectors.
The angle bisector divides the opposite side of a triangle into two segments that are proportional to the other two sides of the triangle.
Mathematically:
\[\frac{CI}{CB} = \frac{AI}{AB}\]
Where:
This property helps in solving problems involving triangle segment ratios.
The length of the angle bisector from a vertex to the opposite side can be calculated using the formula:
\[g = \frac{2}{a+c} \sqrt{ac \cdot s(s-b)}\]
where:
\[s = \frac{a+b+c}{2}\]