Triangle Calculator
Types of Triangles
Triangles are the simplest polygon and the foundation of geometry. They are classified in two independent ways:
The triangle inequality theorem states that any two sides must sum to more than the third side: a + b > c, a + c > b, and b + c > a. If this is not satisfied, no triangle exists.
Sum of Angles and Interior/Exterior Angles
The interior angles of any triangle always sum to 180°. This is the most fundamental triangle fact and the starting point for nearly every triangle problem. It can be proved by drawing a line parallel to one side through the opposite vertex and using alternate angles.
An exterior angle is formed by extending one side of the triangle. The exterior angle equals the sum of the two non-adjacent interior angles (the remote interior angles). This is the exterior angle theorem.
Area Formulas
where s = (a + b + c) / 2 (semi-perimeter)
The Sine Rule
The sine rule relates every side to the angle opposite it:
Use the sine rule when you know:
- AAS (two angles and any side) — find the third angle (= 180° − A − B), then use the rule to find missing sides.
- ASA (two angles and the side between them) — same process.
- SSA (two sides and a non-included angle) — beware of the ambiguous case: there may be 0, 1, or 2 valid triangles. Always check whether the supplementary angle (180° − angle found) also produces a valid triangle.
The Cosine Rule
Use the cosine rule when you know:
- SAS (two sides and the included angle) — find the third side.
- SSS (all three sides) — find any angle.
The cosine rule is a generalisation of Pythagoras’ theorem. When A = 90°, cos A = 0, and the formula reduces to a² = b² + c².
Special Triangles: 30-60-90 and 45-45-90
These special right triangles have side ratios that produce exact values:
- 30°-60°-90° triangle: Sides in ratio 1 : √3 : 2. If the short leg = x, then long leg = x√3 and hypotenuse = 2x. This gives the exact values sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3.
- 45°-45°-90° triangle: Sides in ratio 1 : 1 : √2. If legs = x, hypotenuse = x√2. This gives sin 45° = cos 45° = √2/2, tan 45° = 1.
These triangles appear constantly in GCSE exact value questions and A-Level proofs.
Heron’s Formula
Hero (or Heron) of Alexandria (c. 10–70 AD) developed a formula to find the area of a triangle using only its three side lengths, without needing the height:
This is useful when you know all three sides but not the height. It’s particularly valuable in coordinate geometry when finding the area of a polygon.
Worked Examples
Example 1: Using the sine rule
In triangle ABC: a = 10 cm, angle A = 35°, angle B = 65°. Find side b.
First: angle C = 180 − 35 − 65 = 80°
Sine rule: b / sin B = a / sin A
b = a × sin B / sin A = 10 × sin 65° / sin 35° = 10 × 0.9063 / 0.5736 = 15.80 cm
Example 2: Using the cosine rule (SAS)
Sides b = 7 cm, c = 9 cm, included angle A = 48°. Find side a.
a² = b² + c² − 2bc cos A
a² = 49 + 81 − 2 × 7 × 9 × cos 48°
a² = 130 − 126 × 0.6691 = 130 − 84.31 = 45.69
a = √45.69 ≈ 6.76 cm
Example 3: Heron’s formula
Triangle with sides 8 cm, 11 cm, 13 cm. Find the area.
s = (8 + 11 + 13)/2 = 16
Area = √(16 × (16−8) × (16−11) × (16−13))
Area = √(16 × 8 × 5 × 3) = √1920 = 43.82 cm²
Similar and Congruent Triangles
Congruent triangles are identical in shape and size. The congruence conditions are:
- SSS: All three pairs of sides are equal.
- SAS: Two sides and the included angle are equal.
- ASA / AAS: Two angles and a side are equal (in corresponding positions).
- RHS: Right angle, hypotenuse, and one other side are equal (right triangles only).
Similar triangles have the same shape but different sizes. Their corresponding angles are equal and their sides are in the same ratio (the scale factor). If scale factor = k, then areas are in ratio k² and volumes (for 3D) in ratio k³. Similarity is tested at GCSE and is the basis of trigonometry itself.
Triangles in GCSE (AQA and Edexcel)
At GCSE, you are expected to:
- Find missing sides and angles in right triangles using Pythagoras and SOHCAHTOA.
- Calculate areas using ½bh and ½ab sin C.
- Apply the sine rule and cosine rule to non-right-angled triangles (Higher tier).
- Classify triangles and verify congruence or similarity.
- Use the triangle inequality to determine if a triangle is possible.
Remember: when finding an angle using the sine rule, there may be two possible answers (the ambiguous case). Always check both: the supplementary angle (180° minus the calculated value) may also be valid if it keeps all angles positive and summing to 180°.
Coordinate Geometry with Triangles
Given three vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):
- Side lengths: Use the distance formula: AB = √((x₂−x₁)² + (y₂−y₁)²)
- Area (shoelace formula): Area = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
- Centroid (midpoint of medians): ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
- Circumcentre: The point equidistant from all three vertices. Found by solving two perpendicular bisector equations simultaneously.
Frequently Asked Questions
What are the types of triangles?
By sides: equilateral (all equal, all 60°), isosceles (two sides equal), scalene (all different). By angles: acute (all <90°), right-angled (one 90°), obtuse (one >90°). A triangle can belong to categories from both classifications, e.g. an isosceles right triangle.
How do you find the area of a triangle?
Three main methods: (1) ½ × base × perpendicular height — when height is known. (2) Heron’s formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 — when all three sides are known. (3) ½ × a × b × sin C — when two sides and the included angle are known.
What is the sine rule?
The sine rule: a/sin A = b/sin B = c/sin C. Use it when you know two angles and one side (AAS/ASA) to find missing sides, or two sides and a non-included angle (SSA) to find an angle. Beware the ambiguous case with SSA: there may be two valid triangles.
What is the cosine rule?
The cosine rule: a² = b² + c² − 2bc cos A. Use it with SAS (two sides and included angle → find third side) or SSS (all three sides → find any angle via cos A = (b²+c²−a²)/(2bc)). It generalises Pythagoras: when A=90°, it becomes a²=b²+c².
How do you find a missing angle in a triangle?
If two angles are known: third angle = 180° − A − B. For a right triangle with a known side ratio: use arctan, arcsin, or arccos (SOHCAHTOA). For any triangle with sides known: use the cosine rule rearranged: cos A = (b²+c²−a²)/(2bc). With two sides and a non-included angle: use the sine rule.
What is Heron’s formula?
Heron’s formula finds triangle area from three sides. Compute s = (a+b+c)/2. Then Area = √(s(s−a)(s−b)(s−c)). For sides 5, 6, 7: s=9, Area = √(9×4×3×2) = √216 ≈ 14.70 square units. Named after Hero of Alexandria (c. 10–70 AD).