Triangle Formula, Area, Perimeter, Solved Examples
A triangle formula is an enclosed geometric figure defined by three angles, three sides, and three vertices. It is denoted as △PQR when its vertices are labeled as P, Q, and R. Triangles are frequently encountered in everyday life, from familiar objects like road signs and sandwiches to their prominence in various contexts.
Sawat Sayyed11 Sept, 2023
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Triangle Formula: Triangle formulas encompass essential equations employed to determine unknown dimensions and characteristics of triangles. These formulas predominantly pertain to the triangle's area and perimeter. Let's delve into a comprehensive understanding of these fundamental triangle formulas.
Triangle
A triangle is an enclosed geometric figure defined by three angles, three sides, and three vertices. It is denoted as △ PQR when its vertices are labeled as P, Q, and R. Triangles are frequently encountered in everyday life, from familiar objects like road signs and sandwiches to their prominence in various contexts. This page will provide an in-depth exploration of the triangle's shape, its definition, its constituent elements, the various types of triangles, and the inherent properties that characterize them.Also Check - Solid Shapes Formula
Parts of A Triangle
A triangle is composed of distinct elements. It is defined by its possession of three angles, three sides, and three vertices. The provided illustration of triangle PQR visually exemplifies these components, depicting the sides, vertices, and internal angles of the triangle. In the depicted triangle:
The angles are identified as
∠
PQR,
∠
QRP, and
∠
RPQ.
The sides are denoted as side PQ, side QR, and side RP.
The vertices are named P, Q, and R.
A noteworthy observation is that the sum of all angles within the triangle invariably equals 180°.
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Properties of Triangles Formula
Various geometric shapes exhibit distinct characteristics related to their sides and angles, aiding in their identification. The significant attributes specific to triangles are outlined below.- A triangle comprises three sides, three vertices, and three interior angles.
- The angle sum property of a triangle dictates that the total of its interior angles always equals 180°. For instance, in triangle PQR shown above, angle P + angle Q + angle R = 180°.
- The Triangle Inequality Theorem asserts that the combined length of any two sides of a triangle is invariably greater than the length of the third side.
- In accordance with the Pythagorean Theorem, in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: Hypotenuse² = Base² + Altitude².
- The longest side of a triangle corresponds to the one opposite the largest angle.
- The Exterior Angle Theorem of a triangle states that the exterior angle formed at a vertex is equivalent to the sum of the interior angles that are not adjacent to it.
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What are Triangle Formulas?
Two crucial formulas in the realm of triangles are those for calculating the triangle's area and perimeter. Distinct triangle formulas come into play for various types of triangles. Let's explore a comprehensive overview of the fundamental formulas associated with triangles.All Triangle Formulas
The collection of triangle formulas primarily encompasses those concerning the area and perimeter of triangles. In the subsequent sections, we will delve into an exploration of these formulas.Area of Triangle Formulas
The concept of a triangle's area pertains to the region enclosed by the triangle. It is precisely half of the product attained by multiplying the length of its base by its corresponding altitude (height). This measurement is consistently expressed in square units, reflecting its inherently two-dimensional nature. The triangle ABC depicted below provides a visual representation of the base and height, which are integral components employed in the computation of a triangle's area. The area of a triangle is calculated by taking half the product of its base and height. This can be expressed as Area = ½ × base × height.
- While this formula applies to any triangle, the use of Heron's formula becomes necessary for scalene triangles. Heron's formula for the area of a scalene triangle is given as:
Area = √(s * (s - a) * (s - b) * (s - c)),
- where 's' represents the semi-perimeter of the triangle, calculated as half of its perimeter: s = (a + b + c)/2.
- For equilateral triangles, the area can be determined using the formula: Area = (√3/4) * a², with 'a' being the side length.
- In the case of an isosceles triangle, the formula for its area is: Area = ½ * Base * Height, where the height is computed as √(a² - b²/4). In this context, 'a' represents the equal side length, and 'b' is the base.
- The Pythagorean theorem proves helpful for determining an unknown side of a right-angled triangle if the other two sides are known. Mathematically, it is stated as h² = p² + b², where 'h' is the hypotenuse (the longest side), 'p' is the perpendicular side, and 'b' signifies the base. Once all the sides are established, the triangle's area can be calculated using the previously mentioned formulas.
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Perimeter of Triangle Formulas
The sum of all sides of a triangle constitutes its perimeter, and this can be mathematically expressed as follows: Perimeter of a triangle (P) = a + b + c, where 'a', 'b', and 'c' represent the lengths of the triangle's three sides. For equilateral triangles, the formula for perimeter is simplified to: Perimeter of equilateral triangle = a + a + a = 3a. Here, 'a' denotes the side length, and it's important to note that in an equilateral triangle, all three sides are of equal length. When dealing with isosceles triangles, the perimeter formula takes the form: Perimeter of isosceles triangle = s + s + b = 2s + b. In this context, 's' stands for the length of the two equal sides, while 'b' represents the base length of the isosceles triangle.
Triangle Formula FAQs
What is meant by the Area of the triangle?
The area of a triangle refers to the region encompassed by the three sides of the triangle. Computation of this area involves the use of diverse formulas contingent upon the triangle's specific type, and the result is typically expressed in square units such as cm², inches², and the like.
Which are the fundamental formulas for triangles?
The pair of fundamental triangle formulas encompass the calculation of a triangle's area and its perimeter. These essential mathematical expressions are as follows:
Area of a triangle:
Formula: A = ½ * base * height
The perimeter of a triangle:
Formula: P = a + b + c
Define Triangle.
A triangle is an enclosed geometric figure defined by three angles, three sides, and three vertices. It is denoted as △PQR when its vertices are labeled as P, Q, and R.
Explain the meaning of the Triangle formula.
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