Triangle Calculator
How do you calculate the area of a triangle?
Three different formulas find the area depending on what you know: Related: use the Area Calculator to go further, or the Circumference Calculator for a different angle.
| What You Know | Formula | Example |
|---|---|---|
| Base and height | Area = ½ × b × h | Base 8m, height 5m: Area = 20 m² |
| Three sides (Heron's formula) | s=(a+b+c)/2; Area=√(s(s-a)(s-b)(s-c)) | Sides 5,6,7: s=9; Area=√(9×4×3×2)=√216=14.7 m² |
| Two sides + included angle | Area = ½ × a × b × sin(C) | Sides 6m, 8m, angle 30°: Area=½×6×8×0.5=12 m² |
| Right triangle (two legs) | Area = ½ × a × b | Legs 3m and 4m: Area = ½×3×4 = 6 m² |
What are the triangle types and their properties?
Classification by sides
| Type | Sides | Angles | Key Property |
|---|---|---|---|
| Equilateral | All three equal (a=b=c) | All 60° | Most symmetrical triangle — also equiangular |
| Isosceles | Two sides equal | Base angles equal | The equal sides are called "legs"; unequal side is the "base" |
| Scalene | No sides equal | No angles equal | Most general form — a right triangle is usually scalene |
Classification by angles
| Type | Angle Condition | Pythagorean Rule | Example |
|---|---|---|---|
| Acute | All angles < 90° | a²+b² > c² for all side pairs | Equilateral (60-60-60) |
| Right | One angle = 90° exactly | a²+b² = c² | 3-4-5, 5-12-13, 8-15-17 |
| Obtuse | One angle > 90° | a²+b² < c² | Sides 2-3-4: 4+9=13 < 16 ✓ obtuse |
Law of Sines and Law of Cosines — when to use each
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Use when you know: two angles and any side (AAS or ASA), OR two sides and the angle opposite one of them (SSA — the ambiguous case). Example: A=30°, B=70°, a=10. Find b: b/sin(70°) = 10/sin(30°). b = 10 × sin(70°)/sin(30°) = 10 × 0.9397/0.5 = 18.79.
Law of Cosines: c² = a² + b² − 2ab×cos(C)
Use when you know: three sides (SSS — to find angles), OR two sides and the included angle (SAS — to find the third side). Example: sides 5, 7, and angle between them 60°. Third side² = 25 + 49 − 2(5)(7)cos(60°) = 74 − 35 = 39. Third side = √39 = 6.24.
Pythagorean triples: right triangle side combinations
| Legs (a, b) | Hypotenuse (c) | Scale Multiples |
|---|---|---|
| 3, 4 | 5 | 6-8-10; 9-12-15; 30-40-50 |
| 5, 12 | 13 | 10-24-26; 15-36-39 |
| 8, 15 | 17 | 16-30-34 |
| 7, 24 | 25 | 14-48-50 |
| 20, 21 | 29 | 40-42-58 |
| 9, 40 | 41 | 18-80-82 |
Frequently asked questions
How do you calculate the area of a triangle without the height?
Use Heron's formula when you know all three sides but not the height. s = (a+b+c) ÷ 2 (semi-perimeter). Area = √[s(s−a)(s−b)(s−c)]. For sides 5, 6, 7: s=9. Area = √(9×4×3×2) = √216 = 14.70. You can also find the height once you know the area: h = 2 × Area ÷ base.
What is the sum of angles in a triangle?
The interior angles of every triangle — regardless of type or size — sum to exactly 180°. This is a fundamental theorem of Euclidean geometry. The missing angle = 180° minus the sum of the other two. For angles 47° and 83°: third angle = 180° − 47° − 83° = 50°.
Can a triangle have two right angles?
No. If a triangle had two 90° angles, the three angles would sum to 180° + the third angle, which is impossible (they must total exactly 180°). A triangle can have at most one right angle and at most one obtuse angle.
How do you check if three sides can form a valid triangle?
Triangle inequality theorem: any side must be less than the sum of the other two sides. For sides a, b, c: a+b > c, a+c > b, and b+c > a must all be true. Sides 3, 4, 8: 3+4=7 which is NOT greater than 8 — these cannot form a triangle. Sides 3, 4, 6: 3+4=7 > 6 ✓, 3+6=9 > 4 ✓, 4+6=10 > 3 ✓ — valid triangle.
What is the largest area triangle with a fixed perimeter?
The equilateral triangle has the largest area for any given perimeter. This is a consequence of the isoperimetric inequality. With perimeter 30: equilateral triangle (sides 10, 10, 10) has area = (√3/4) × 100 = 43.3. A 5-12-13 right triangle with perimeter 30: area = ½ × 5 × 12 = 30 — 30% smaller despite the same perimeter.
Sources & References
Sources: Euclid, Elements Books I and IV — triangle theorems; Law of Sines and Cosines — standard trigonometry (Ptolemy, 2nd century AD, formalized in modern form by 15th–16th century European mathematicians); Heron of Alexandria, Metrica (c. 60 AD) — Heron's area formula; NIST Digital Library of Mathematical Functions — trigonometric identities.
← Back to Math Calculators · See also: Area Calculator · Circumference Calculator · Quadratic Formula Calculator
Types of triangles and their properties
| Type | Definition | Properties |
|---|---|---|
| Equilateral | All 3 sides equal | All angles = 60°; 3 lines of symmetry |
| Isosceles | 2 sides equal | 2 equal base angles; 1 line of symmetry |
| Scalene | All sides different | All angles different; no lines of symmetry |
| Right-angled | One 90° angle | Hypotenuse² = side₁² + side₂² (Pythagorean theorem) |
| Acute | All angles < 90° | All altitudes inside triangle |
| Obtuse | One angle > 90° | One altitude outside triangle |
Triangle formulas reference: area, perimeter, and angles
Perimeter: P = a + b + c (sum of all three sides).
Area by base and height: A = ½ × base × height. Height must be perpendicular to the base.
Area by Heron's formula (3 sides known): s = (a+b+c)/2; Area = √[s(s−a)(s−b)(s−c)]. For a triangle with sides 7, 8, 9: s = 12; Area = √[12×5×4×3] = √720 ≈ 26.83.
Area by two sides and included angle: A = ½ × a × b × sin(C).
Law of cosines (finding a side): c² = a² + b² − 2ab·cos(C).
Law of sines: a/sin(A) = b/sin(B) = c/sin(C) = 2R (where R = circumradius).
Real-world applications of triangle calculations
Architecture: roof pitch calculations use right triangles — a 6:12 pitch means 6 inches rise per 12 inches horizontal run, with a hypotenuse of √(6²+12²) = 13.4 inches per foot of run. Navigation: surveyors use triangulation to measure distances across rivers or between landmarks. Engineering: truss bridges rely on triangular structures because triangles are the only polygon that cannot be deformed without changing side lengths. Related: Area Calculator · Volume Calculator