All About Triangles
A triangle is one of the most fundamental shapes in geometry. Three sides, three angles, three vertices — and a surprising number of rules and relationships that make triangles show up everywhere from architecture to trigonometry.
The Basics
A triangle is a polygon with exactly three sides and three angles. The points where the sides meet are called vertices (singular: vertex).
Every triangle has:
- Three sides (line segments)
- Three interior angles
- Three vertices
The three angles of a triangle always add up to the same total — more on that in a moment.
The Angle Sum Rule
No matter how big or small a triangle is, no matter its shape:
$$\text{angle}_1 + \text{angle}_2 + \text{angle}_3 = 180°$$
The three interior angles of any triangle always add up to exactly 180 degrees. This is one of the most useful facts in all of geometry.
Example: A triangle has angles of 45° and 70°. What is the third angle?
$$180° - 45° - 70° = 65°$$
The third angle is 65°.
Example: A triangle has two equal angles of 55° each. What is the third?
$$180° - 55° - 55° = 70°$$
Types by Sides
Triangles are often classified by whether their sides are equal.
Equilateral triangle — all three sides are the same length. Because the sides are equal, all three angles are also equal: each one is 60°. A perfectly symmetric triangle.
Isosceles triangle — exactly two sides are equal. The two angles opposite those equal sides (called the base angles) are also equal to each other. If you know one base angle, you know the other.
Scalene triangle — all three sides are different lengths. All three angles are also different. Most triangles you draw at random will be scalene.
Types by Angles
Triangles are also classified by their largest angle.
Acute triangle — all three angles are less than 90°. The triangle looks "sharp."
Right triangle — one angle is exactly 90°. The side opposite the right angle is the longest side, called the hypotenuse. The other two sides are called legs. Right triangles are especially important because of the Pythagorean theorem.
Obtuse triangle — one angle is greater than 90°. A triangle can only have one obtuse angle (since two would already exceed 180°).
Note that these categories can overlap with the side-based types. For example, you can have an acute scalene triangle, or a right isosceles triangle (with angles of 45°, 45°, and 90°).
Area of a Triangle
The area of any triangle is:
$$A = \frac{1}{2} \times \text{base} \times \text{height}$$
The base is any side of the triangle. The height is the perpendicular distance from that base to the opposite vertex — straight up, not along a slanted side.
Example: A triangle has a base of 10 cm and a height of 6 cm.
$$A = \frac{1}{2} \times 10 \times 6 = 30 \text{ cm}^2$$
Example: A right triangle has legs of 5 and 8.
For a right triangle, the two legs are perpendicular to each other, so one leg is the base and the other is the height:
$$A = \frac{1}{2} \times 5 \times 8 = 20$$
The Triangle Inequality
Not just any three lengths can form a triangle. There's a rule: the sum of any two sides must be greater than the third side.
$$a + b > c, \quad a + c > b, \quad b + c > a$$
Example: Can sides of length 3, 4, and 8 form a triangle?
Check: \(3 + 4 = 7\), which is not greater than 8. So no — these lengths cannot form a triangle. The two short sides can't reach each other.
Example: Can sides of length 5, 7, and 9 form a triangle?
- \(5 + 7 = 12 > 9\) ✓
- \(5 + 9 = 14 > 7\) ✓
- \(7 + 9 = 16 > 5\) ✓
Yes, these lengths form a valid triangle.
Special Right Triangles
Two right triangle shapes appear so often in math and physics that they're worth memorizing.
The 30-60-90 triangle has angles of 30°, 60°, and 90°. If the shortest side (opposite the 30° angle) has length \(x\), then the other sides are always \(x\sqrt{3}\) and \(2x\). See the full lesson on 30-60-90 triangles.
The 45-45-90 triangle (also called an isosceles right triangle) has angles of 45°, 45°, and 90°. If the two legs each have length \(x\), the hypotenuse is always \(x\sqrt{2}\).
These ratios are fixed regardless of the size of the triangle, which makes them powerful shortcuts.
For any other triangle — given any combination of three sides and angles — the Triangle Calculator walks through the work using the Pythagorean theorem, Law of Sines, or Law of Cosines as appropriate.
Practice Problems
A triangle has angles of 38° and 75°. What is the third angle?
Show answer\(180° - 38° - 75° = 67°\)
A triangle has three equal angles. What is each angle?
Show answer\(180° \div 3 = 60°\). This is an equilateral triangle.
A triangle has a base of 14 and a height of 5. What is its area?
Show answer\(A = \frac{1}{2} \times 14 \times 5 = 35\)
Can a triangle have sides of length 2, 5, and 9?
Show answerNo. \(2 + 5 = 7\), which is less than 9. The triangle inequality fails.
A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
Show answerUsing the Pythagorean theorem: \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\).
An isosceles triangle has two base angles of 48° each. What is the vertex angle?
Show answer\(180° - 48° - 48° = 84°\)