Area of Triangle Definitions and Examples

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Area of Triangle Definitions and Examples

Introduction

Triangle definitions and examples are essential for mathematics. In this blog post, we will explore three different area triangle definitions and examples. By doing so, you will be able to better understand how these concepts work and how they can benefit your math skills .

What is the Area of a Triangle?

The area of a triangle is one-half times the base times height.

Formula: A = 1/2 × b × h.

Definition of a Triangle

Triangle is a three-sided figure with two angles that measure 90 degrees. The vertex angle is the angle between the base and the apex of the triangle, while the other two angles are called the side angles. A right triangle has a right vertex angle and a sum of its side angles that equals 180 degrees. In case of a triangle with an acute angle, like 90 degrees, one of its angles is multiplied by 3 to find its measure.

The Area of a Triangle

A triangle is a three-sided geometric figure with two equal sides and a third side that is shorter than the other two. A triangle has six points of intersection, which are called vertices. The three vertices that are equal in size form the triangle’s base. The point at the midpoint of the base is called the vertex of the Triangle.

The three vertices on one side of the triangle are called interior angles. The angle opposite any interior angle on one side is called an exterior angle. An exterior angle on one side is also considered to be an interior angle if it shares a common vertex with another exterior angle on that same side.

The sum of the angles in a triangle is 180 degrees. This property allows you to find any two angles in a Triangle by adding their corresponding angles and then dividing by 2 because 180° = 360°

Area of Triangle Using Heron’s Formula

Triangle Area using Heron’s Formula

The area of a triangle is given by the following formula:

A = base_a^2 + height_a^3

If the triangle is right angled, then the value of “base_a” is 0 and all other values are 1. If the triangle is not right angled, then “base_a” can be any real number and “height_a” will always be positive.

Area of Triangle With 2 Sides and Included Angle (SAS)

Formula: Area= (a x b x sin c)/2, where a, b are the two sides and c is the angle between them.

Another definition is the sum of the lengths of the three sides, also known as the length measure or linear measure. Another way to think about it is to imagine cutting each side off at a point halfway between the two ends and counting how many inches (or centimeters) that cuts make. This is also called the base length or sometimes just base.

The third definition is called the circumradius and it’s just what you’d expect-the radius around one of the triangle’s corners. Finally, there’s a special type of triangle called an equilateral triangle, which has all its angles at 90 degrees. All these definitions are important when working with Triangles in problems and calculations, so it’s worth getting used to them!

Area of a Right-Angled Triangle

Area of a right-angled triangle is the sum of the three sides. The base is always the longest side, and the height is the shortest side.

The area of a triangle can be found by multiplying the base length by the height. For example, if you have a triangle with a base of 10 inches and a height of 12 inches, then its area would be 120 square inches.

Types of Triangles

There are many types of triangles, some more common than others. Here are a few:

Isosceles Triangle

An isosceles triangle has two equal sides and the third side is the same length as the other two.

Triangle in a Right Vertical Plane

In geometry, a triangle in a right vertical plane is one that has its vertices centered at the top, bottom, and left edges of the paper. A right triangle in this context has one angle that measures 90 degrees.

Conclusion

In this article, we discussed the three most common types of triangles and their definitions. After reading this, you should have a better understanding of what a triangle is and how to identify it in various situations. I hope that you enjoy learning about these important shapes!

Area of Triangle

Result

A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

Definition

Defining inequalities

y>=0 and y (a^2 + c^2) + x sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))<=c sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) + b^2 y and a^2 y + x sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))>=y (b^2 + c^2)

Lamina properties

(c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)

a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b

(data not available)

sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)

x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))

Mechanical properties

J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)

J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)

J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))

J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)

r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c)
r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))

Distance properties

a | b | c

p = a + b + c

r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))

R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

max(a, b, c)

χ = 1

s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c)

A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

Alternate form

1/4 sqrt((-a - b - c) (a - b - c) (a + b - c) (a - b + c))

Alternate forms assuming a, b, and c are positive

1/4 sqrt(a + b - c) sqrt(a - b + c) sqrt(-a + b + c) sqrt(a + b + c) (-1)^(⌊-(arg(a + b - c) + arg(a - b + c) + arg(-a + b + c) - π)/(2 π)⌋)

1/4 sqrt(a + b - c) sqrt(a - b + c) sqrt(-a + b + c) sqrt(a + b + c) exp(i π floor(-arg(a + b - c)/(2 π) - arg(a - b + c)/(2 π) - arg(-a + b + c)/(2 π) + 1/2))

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