Triangles – The Strongest Shape in Geometry
Triangles are the simplest polygons and one of the strongest shapes in nature and engineering. Every other polygon can be divided into triangles, making them the foundation of geometry.
What Is a Triangle?
A triangle is a closed shape with three straight sides and three angles. The sum of the interior angles of any triangle is always 180°. This is one of the most important facts in all of geometry.
Types of Triangles by Sides
| Type | Sides | Angles |
|---|---|---|
| Equilateral | All three sides equal | All angles = 60° |
| Isosceles | Two sides equal | Base angles equal |
| Scalene | No sides equal | No angles equal |
Types of Triangles by Angles
| Type | Defining Angle |
|---|---|
| Acute triangle | All three angles less than 90° |
| Right triangle | One angle exactly 90° |
| Obtuse triangle | One angle greater than 90° |
The Angle Sum Property
Third angle = 180 − 50 − 70 = 60°.
x + 2x + 3x = 180. 6x = 180. x = 30. Angles: 30°, 60°, 90°. This is a right triangle.
Pythagoras' Theorem
In a right-angled triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides.
a² + b² = c², where c is the hypotenuse (the side opposite the right angle).
c² = 3² + 4² = 9 + 16 = 25. c = √25 = 5.
b² = 13² − 5² = 169 − 25 = 144. b = √144 = 12.
Perimeter and Area of a Triangle
| Measurement | Formula |
|---|---|
| Perimeter | Sum of all three sides: P = a + b + c |
| Area | Half base times height: A = ½ × b × h |
Key Takeaways
- All triangles have interior angles summing to 180°.
- Equilateral: all equal. Isosceles: two equal. Scalene: none equal.
- Pythagoras: a² + b² = c² applies only to right triangles.
- Area = ½ × base × height.
Practice Questions
- A triangle has angles 40° and 95°. Find the third angle and classify the triangle by angles.
- An isosceles triangle has a vertex angle of 40°. Find the base angles.
- Find the hypotenuse of a right triangle with legs 8 and 15.
- A right triangle has hypotenuse 17 and one leg 8. Find the other leg.
- Find the area of a triangle with base 10 cm and height 7 cm.