Triangle Calculator Guide: Sides, Angles, Area and Formulas

Types of Triangles

Triangles are classified by their sides and angles. Understanding the type determines which formula to use.

By Sides

• Equilateral: all three sides equal, all angles 60°
• Isosceles: two sides equal, two base angles equal
• Scalene: no sides equal, no angles equal

By Angles

• Acute: all angles less than 90°
• Right: one angle exactly 90°
• Obtuse: one angle greater than 90°

According to the Mathematical Association of America, the triangle is the simplest polygon and forms the basis of trigonometry, structural engineering, and computer graphics. Every polygon can be decomposed into triangles — a process called triangulation used in everything from GPS to 3D rendering.

The Pythagorean Theorem (Right Triangles)

For any right triangle with legs a and b and hypotenuse c:

a² + b² = c²

Worked Example

A right triangle has legs of 5 cm and 12 cm. Find the hypotenuse.

c² = 5² + 12²
c² = 25 + 144 = 169
c = √169 = 13 cm

This 5-12-13 is one of the well-known Pythagorean triples. Others include 3-4-5, 8-15-17, and 7-24-25. According to Euclid's Elements (circa 300 BCE), there are infinitely many such triples, and the theorem has over 400 known proofs — more than any other theorem in mathematics.

Triangle Area Formulas

There are several ways to calculate the area of a triangle depending on what information you have.

Heron's Formula Example

A triangle has sides a = 7, b = 8, c = 9. Find its area.

s = (7 + 8 + 9) / 2 = 12
A = √[12 × (12–7) × (12–8) × (12–9)]
A = √[12 × 5 × 4 × 3]
A = √720
A = 26.83 square units

Heron's formula dates to the 1st century CE and remains one of the most practical formulas in surveying. The U.S. Bureau of Land Management uses triangulation-based area calculations for property boundary surveys across 245 million acres of public land.

The Law of Sines

For any triangle with sides a, b, c and opposite angles A, B, C:

a / sin(A) = b / sin(B) = c / sin(C)

Use the law of sines when you know:

• Two angles and one side (AAS or ASA)
• Two sides and a non-included angle (SSA — the ambiguous case)

Worked Example (AAS)

Given: angle A = 40°, angle B = 70°, side a = 10. Find side b.

10 / sin(40°) = b / sin(70°)
10 / 0.6428 = b / 0.9397
15.557 = b / 0.9397
b = 15.557 × 0.9397
b = 14.62

The Law of Cosines

For any triangle:

c² = a² + b² – 2ab × cos(C)

Use the law of cosines when you know:

• Two sides and the included angle (SAS)
• All three sides (SSS) and need to find an angle

Worked Example (SAS)

Given: a = 8, b = 11, angle C = 37°. Find side c.

c² = 8² + 11² – 2(8)(11) × cos(37°)
c² = 64 + 121 – 176 × 0.7986
c² = 185 – 140.55
c² = 44.45
c = 6.67

The law of cosines is essential in GPS navigation. According to the IEEE, trilateration algorithms used in smartphone GPS rely on solving triangles formed between satellites and the receiver, processing these calculations billions of times daily across 6.8 billion mobile devices worldwide.

Special Right Triangles

Two special right triangles appear so frequently in math and engineering that their ratios are worth memorizing:

A 30-60-90 triangle with a short side of 5 has a hypotenuse of 10 and a long side of 5√3 ≈ 8.66. According to the National Council of Teachers of Mathematics, these two triangles account for the majority of triangle problems on standardized tests including the SAT and ACT.

Frequently Asked Questions

How do I find the area of a triangle?

The most common formula is Area = ½ × base × height. If you know all three sides but not the height, use Heron's formula: Area = √[s(s–a)(s–b)(s–c)] where s = (a+b+c)/2. For two sides and the included angle, use Area = ½ × a × b × sin(C).

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It only works for right triangles (those with a 90-degree angle). For example, a triangle with sides 3, 4, and 5 satisfies 9 + 16 = 25.

How do I use the law of cosines?

The law of cosines is c² = a² + b² – 2ab × cos(C), where C is the angle opposite side c. Use it when you know two sides and the included angle (SAS) or all three sides (SSS) and need to find an angle. It generalizes the Pythagorean theorem — when C = 90°, cos(C) = 0 and it reduces to a² + b² = c².

Can a triangle have two right angles?

No. The interior angles of any triangle must sum to exactly 180 degrees. Two right angles alone would total 180 degrees, leaving 0 degrees for the third angle, which is impossible. A triangle can have at most one right angle (90°) or one obtuse angle (greater than 90°).

What is the law of sines used for?

The law of sines states a/sin(A) = b/sin(B) = c/sin(C). Use it when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA, the ambiguous case). It lets you find unknown sides or angles by setting up proportions.