Triangle

For other uses, see Triangle (disambiguation).

A black triangle

A triangle is a shape, or a part of two dimensional space. It has three straight sides and three vertices. The three angles of a triangle always add up to 180° (180 degrees). This can be proved when you put 3 of the same triangle together to make a trapezoid. It is the polygon with the least possible number of sides. A triangle with vertices A, B, C is written as $\triangle ABC$. The study of geometry related to triangles is called trigonometry. Modern computers usually use triangles to make more complex drawings.

Types of triangles

This is an isosceles right triangle, because it has a right angle and two of its sides have the same length.

Triangles can be grouped according to how many of their sides are equal:

• if all the three sides of a triangle have the same length, then it is an equilateral triangle.
• if a triangle has two sides with the same length, then it is an isosceles triangle.
• if all the three sides of a triangle have different lengths, then we have a scalene triangle.

Triangles can also be grouped by their angles:

• if a triangle has a right angle, that is, if one of the angles of that triangle measures 90° (90 degrees), then it is a right triangle. The opposite side to the right angle is the hypotenuse.
• if a triangle has an obtuse angle, that is, if one of the angles of that triangle is larger than 90°, then it is an obtuse triangle.
• if a triangle has only acute angles, that is, if all the angles of that triangle are less than 90°, then it is an acute triangle.

Equilateral triangle Isosceles triangle Scalene triangle Right Angled triangle Obtuse Angled triangle Acute Angled triangle

Area

b is base and h is height

The area of a triangle is equal to base times height times one half.

$area={\frac {base\times height}{2}}$

If you have a triangle with a base of 4 and a height of 8, and a triangle with a base of 8 and a height of 4, due to the commutative property of multiplication, both triangles have the same area. Knowing this, you can divide any shape cake equally along the center by dividing the perimeter by the number of people, making sure everyone gets the same amount of toppings.

To prove this, each of the 3 quadrilateral pieces can be divided further into triangles.

After this, the 6 triangles can be measured to find that all 3 of the original pieces have the same area.

Center

A triangle with circumcenter (blue,) orthocenter (red,) incenter (yellow,) and centroid (green)

Different people have different ways of finding the center of the triangle. Some people say the center of the biggest circle that fits inside the triangle is the center of the triangle, which is the same point as the intersection of the angle bisectors. This is called the incenter. Some people say that when you draw lines from the corners to the middles of the opposite sides, that's the center. This is called the centroid. Some people say that the center is the intersection of the altitudes of each side that touch the corners. This is called the orthocenter.

Related pages

• Law of sines
• Pythagorean theorem
• Sierpinski Triangle
• Reuleaux triangle