Triangle Calculator
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Triangle Area (Base & Height)
Triangle Calculation Results
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A triangle is a polygon with 3 sides and 3 angles. Sum of angles = 180°. Area = (Base × Height) / 2. Perimeter = Side A + Side B + Side C.
Complete Triangle Calculator Guide
Triangles are fundamental geometric shapes with three sides and three angles. This guide explains how to calculate triangle properties and their real-world applications.
Triangle Types
| Type | Definition | Properties |
|---|---|---|
| Equilateral | All 3 sides equal | All angles = 60°, Perfectly symmetric |
| Isosceles | 2 equal sides | 2 equal angles, Symmetric |
| Scalene | All sides different | All angles different, No symmetry |
| Right Triangle | One 90° angle | a² + b² = c², Pythagorean theorem applies |
| Acute | All angles < 90° | All angles sharp, Sum = 180° |
| Obtuse | One angle > 90° | One wide angle, Sum = 180° |
Triangle Formulas
| Property | Formula | Variables |
|---|---|---|
| Area (Base & Height) | A = (b × h) / 2 | b = base, h = height |
| Area (Heron's Formula) | A = √[s(s-a)(s-b)(s-c)] | s = semi-perimeter, a,b,c = sides |
| Perimeter | P = a + b + c | a, b, c = side lengths |
| Height | h = (2 × Area) / base | Area = triangle area, base = base side |
| Semi-perimeter | s = (a + b + c) / 2 | a, b, c = sides, used in Heron's formula |
Real-World Applications
- Architecture: Roof trusses, support beams, structural framework
- Engineering: Bridge design, load distribution, force analysis
- Land Surveying: Triangulation to measure distances and areas
- Navigation: Determining position using triangulation
- Construction: Cutting materials, framing, roofing
- Art & Design: Composition, symmetry, visual balance
- Physics: Vector calculations, force triangles
Understanding Angles in Triangles
- Sum of angles: Always 180° in any triangle
- Right angle: Exactly 90° (right triangle)
- Acute angle: Less than 90° (acute triangle has 3 of these)
- Obtuse angle: Greater than 90° (only 1 in obtuse triangle)
- External angle: Equal to sum of 2 non-adjacent internal angles
Practical Examples
Base = 10 cm, Height = 8 cm
Area = (10 × 8) / 2 = 40 cm²
This is a right triangle with base and height perpendicular.
Sides: A = 5 cm, B = 6 cm, C = 7 cm
Perimeter = 5 + 6 + 7 = 18 cm
This is a scalene triangle (all sides different).
Area = 40 cm², Base = 10 cm
Height = (2 × 40) / 10 = 8 cm
Same triangle as Example 1.
Heron's Formula (Advanced)
When you have all 3 sides but no height:
- Semi-perimeter: s = (a + b + c) / 2
- Area = √[s(s-a)(s-b)(s-c)]
- Example: Sides 5, 6, 7 cm
- s = (5 + 6 + 7) / 2 = 9
- Area = √[9(9-5)(9-6)(9-7)] = √[9 × 4 × 3 × 2] = √216 ≈ 14.7 cm²
Triangle Inequality Theorem
For 3 sides to form a valid triangle:
- a + b > c
- b + c > a
- a + c > b
- Sum of any 2 sides must be greater than the 3rd side
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Frequently Asked Questions
1. What is the sum of angles in a triangle?
The sum of all three angles in any triangle is always 180 degrees.
2. How do I calculate triangle area without height?
Use Heron's Formula: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2. This works when you have all 3 sides.
3. What's the difference between isosceles and equilateral?
Equilateral has all 3 sides equal. Isosceles has 2 equal sides. All equilateral are isosceles, but not vice versa.
4. Can a triangle have two right angles?
No. If 2 angles are 90°, the third would be 0°, which is impossible. A triangle can have at most one right angle.
5. What is the Pythagorean theorem?
For right triangles: a² + b² = c² where c is the hypotenuse (longest side). Used to find missing side lengths.
6. How do I find the height of a triangle?
If you know area and base: h = (2 × Area) / base. Or use Heron's formula to find area first, then calculate height.
7. What is semi-perimeter used for?
Semi-perimeter (s) is used in Heron's Formula to calculate area when you have all 3 sides but no height.
8. Can you have a triangle with sides 3, 4, 8?
No. It violates triangle inequality: 3 + 4 = 7, which is not greater than 8. Sum of 2 sides must exceed the 3rd.
9. What's the smallest triangle area possible?
As dimensions approach 0, area approaches 0. Minimum area is limited by your precision and unit system.
10. How do angles relate to side lengths?
Larger angles are opposite longer sides. In a right triangle, the right angle (90°) is opposite the hypotenuse (longest side).
11. What is an equilateral triangle's area formula?
For equilateral with side a: Area = (√3 / 4) × a². All angles are 60°, all sides equal.
12. How do I verify if a triangle is right-angled?
Check if a² + b² = c². If true for the longest side c, it's a right triangle. Also check if one angle equals 90°.