Contents
The following article is an introduction to triangles and their properties, in addition to types of triangles based on sides and angles, and important basic definitions such as altitude, bisector, median, incircle, and circumcircle. Furthermore, it includes fundamental formulas for calculating a triangle’s area and perimeter. There are also links to more detailed articles for each definition and concept. Happy reading.
Definition and Properties of a Triangle: A triangle is a geometric shape composed of three straight sides that meet at three vertices. These sides and vertices define three interior angles. The triangle is considered one of the simplest closed shapes in geometry and is a crucial foundation for many mathematical concepts and theories.
One of the properties of a triangle is that the sum of the lengths of any two sides must be greater than the third side. Also, if the length of one side is greater than another, then the angle opposite the longer side is greater than the angle opposite the other side.
Triangle Sides and Angles
A triangle consists of three sides and three angles.
The sum of the interior angles in any triangle is 180 degrees. See the article Triangle Side and Angle Calculation Exercises which includes 16 different exercises for all cases.
Thus, if we know the measure of two angles in a triangle, we can certainly determine the third angle, as it equals 180 - (sum of the measures of the other two angles).
Example: If the measures of angles A and B in a triangle are 30 and 20, then the third angle will be 180 - (30+20) = 150
Exercise: What is the sum of a triangle's angles?
Generally, and by convention, we denote angles with capital letters (A, B, C) and sides with lowercase Latin letters (a, b, c). For example, side 'a' is opposite angle A, side 'b' is opposite angle B, and so on.
Types of Triangles According to Angle Measure
Let's start by classifying angles as follows:
- Acute angle is an angle whose measure is less than 90 degrees.
- Obtuse angle is an angle whose measure is greater than 90 degrees.
- Right angle is an angle whose measure is 90 degrees.
According to this classification, triangles are divided into 3 types based on angle measure:
- Right-angled triangle: Contains one angle measuring 90 degrees. The side opposite the right angle is called the hypotenuse, which is the longest side in a right-angled triangle.
Example: A triangle with sides 3 cm, 4 cm, 5 cm is a right-angled triangle, which you can verify using the Triangle Calculator. - Acute-angled triangle: All its angles are acute (less than 90°).
- Obtuse-angled triangle: One of its angles is obtuse (greater than 90°).
Pythagorean Theorem in Right Triangles
The Pythagorean theorem for a right triangle states that the square of the **hypotenuse's length = the sum of the squares of the lengths of the two legs**. This is one of the important theorems in trigonometry that all students should memorize.
Pythagoras deduced this theorem approximately 2000 years BC.
Question: We have a right-angled triangle at A, with legs measuring 10 cm and 24 cm. What is the length of the hypotenuse according to the Pythagorean theorem?
Types of Triangles According to Side Lengths
Triangles can be classified based on their side lengths:
- Isosceles triangle: Has two sides of equal length. The angles adjacent to these two sides are also equal.
- Equilateral triangle: All its sides are equal in length, and its angles are equal (each 60 degrees).
Example: A triangle with all sides 5 cm and another triangle with side lengths 7 and another 100. All these triangles have equal angle measures, with each angle being 60 degrees.
- Scalene triangle: All its sides are different in length. One of its properties is that its angle measures are also different. Example: A triangle with sides 2 cm, 3 cm, 4 cm.
From the above, we conclude that if there is a right isosceles triangle, the measure of the two equal angles is 45 degrees, because the last angle is 90 degrees.
Area of a Triangle
The area of a triangle is the amount of surface enclosed within its sides. It is usually calculated using the base and height, which is the general formula for calculating the area of a triangle.
Area of a triangle = ½ × Base × Height
Example: A triangle with a base of 8 cm and a height of 6 cm has an area of: (1/2) × 8 × 6 = 24 square cm.
Question: What is the area of a triangle with a base = 3cm and a height = 3cm?
Perimeter of a Triangle
The perimeter of a triangle is the sum of the lengths of its three sides. Perimeter = Side 1 + Side 2 + Side 3. See the article Triangle Perimeter Formula, which contains all necessary formulas for calculating a triangle's perimeter.
Example: A triangle with sides 5 cm, 7 cm, and 9 cm. Its perimeter is 5 + 7 + 9 = 21 cm.
Question: What is half the perimeter of the following triangle?
Altitudes
An altitude is a line segment drawn from one vertex of a triangle perpendicular to the opposite side (or its extension). Every triangle has three altitudes, all of which intersect at a single point known as the orthocenter. Each triangle has three altitudes.
In a right-angled triangle, the two legs are altitudes of the triangle. The third altitude is the perpendicular line from the right angle to the hypotenuse.
Medians
A median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side. A triangle has 3 medians that intersect at a point called the centroid of the triangle, or the point of concurrency of the medians.
One of the properties of the point of concurrency of the medians is that it divides each median into two parts, one of which is twice the length of the other. The part further from the vertex is twice as long as the part closer to it.
Angle Bisectors
An angle bisector of a triangle is a line segment connecting any angle in the triangle to its opposite side, dividing that angle into two equal angles. The three angle bisectors intersect at a point called the incenter of the triangle.
Question: Medians are line segments that divide the interior angles of a triangle into two equal angles.
The Incircle of a Triangle (Internally Tangent Circle)
The incircle is the unique circle that is tangent to all three sides of a triangle internally. The center of the incircle of a triangle is the intersection point of the internal angle bisectors of the triangle. This point is equidistant from the sides of the triangle (i.e., if a perpendicular is drawn from this point to each of the triangle's sides, these three perpendiculars will be equal and each will be equal to the radius of the incircle).
The incircle of a triangle is the largest circle that can be drawn inside the triangle.
The radius of the incircle can be calculated using the relationship:
Example: A triangle has an area of 12 square cm and a semi-perimeter of 8 cm. What is the radius of its incircle?
Applying the formula, it is 12 / 8 = 1.5 cm.
The Circumcircle of a Triangle (Circle Passing Through Vertices)
The circumcircle is the circle that passes through the vertices of the triangle. Its radius (R) is the distance from the center of this circle (the intersection point of the perpendicular bisectors of the triangle's sides) to any of the triangle's vertices. The radius of the circumcircle can be calculated using the relationship:
Where a, b, and c are the lengths of the triangle's sides and A is its area.
Exercise: Find the radius of the circumcircle for a triangle with sides 3 cm, 4 cm, 5 cm, and an area of 6 square cm.
Solution: The radius of the circumcircle is calculated according to the relationship: (3 × 4 × 5) / (4 × 6) = 2.5 cm.
