Triangle Area Calculator

Find the area of any triangle using base & height, three sides (Heron's formula), or two sides and an angle (SAS).

△ What is the Area of a Triangle?

The area of a triangle is the measure of the two-dimensional region enclosed within its three sides. It is expressed in square units (cm², m², in², etc.) and represents how much flat surface the triangle covers. Unlike the perimeter which measures the boundary, area measures the interior space.

Triangles are the simplest polygon and the foundational shape in all of geometry. They appear in architecture, engineering, art, navigation, and computer graphics. The ability to calculate triangle area accurately is essential in fields ranging from land surveying (where irregularly shaped plots are divided into triangles) to 3D rendering (where every curved surface is approximated by a mesh of triangles).

There are three main ways to calculate a triangle's area depending on what measurements you have available. The base and height method (Area = ½ × b × h) is the most straightforward and works whenever you can measure the base and the perpendicular height. Heron's formula requires only the three side lengths and is invaluable when height is difficult to measure directly. The SAS formula (Area = ½ × p × q × sin C) applies when you know two sides and the angle between them, which is common in surveying and trigonometry problems.

The perimeter of a triangle is the total length of its three sides added together: P = a + b + c. Triangle classification is based on side lengths: equilateral (all equal), isosceles (two equal), or scalene (none equal). This calculator determines the type automatically from the side inputs so you always know what kind of triangle you are working with.

Formulas and Derivation

Method 1 — Base and Height:

A = ½ × b × h

b = Base length (any side of the triangle)

h = Perpendicular height from the base to the opposite vertex

This formula comes from the fact that a triangle is exactly half of a parallelogram with the same base and height. A parallelogram has area = b × h, so the triangle is ½ × b × h. The height must be perpendicular to the base, even if the foot of the altitude falls outside the triangle (for obtuse triangles).

Method 2 — Heron's Formula (three sides):

s = (a + b + c) / 2 → A = √(s × (s − a) × (s − b) × (s − c))

a, b, c = The three side lengths

s = Semi-perimeter (half the total perimeter)

Heron's formula is elegant because it requires no angle measurement. It works for any valid triangle, including obtuse and right triangles. The triangle inequality must be satisfied: the sum of any two sides must exceed the third.

Method 3 — Two Sides and Included Angle (SAS):

A = ½ × p × q × sin(C)

p, q = Two known side lengths

C = The angle between sides p and q (in degrees)

The third side is found via the Law of Cosines: c² = p² + q² − 2pq × cos(C). This allows the perimeter to be computed as well. The SAS formula is derived by expressing the height of the triangle as h = q × sin(C), then substituting into the base × height formula.

Perimeter:

P = a + b + c

How to Use This Calculator

Select your input mode — click the Base & Height, Three Sides (Heron's), or Two Sides + Angle tab depending on what measurements you have available.
Enter your values — type the side lengths or base and height into the input fields. For SAS mode, enter the angle between the two sides in degrees.
Press Calculate — the calculator instantly shows the area, perimeter (if computable), and triangle type.
Review the note — the grey note box below the results shows the step-by-step calculation so you can verify the formula application.
Share or save — use the Copy Link button to generate a URL with your inputs so you can share the exact calculation with others.

Example Calculations

Example 1 — Base and Height

Triangle with base 8 units and height 5 units

Area = ½ × base × height

Area = ½ × 8 × 5 = 20 square units

Area = 20.0000 sq units

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Example 2 — Three Sides: 3-4-5 Right Triangle

Right triangle with sides 3, 4, and 5 (a classic Pythagorean triple)

s = (3 + 4 + 5) / 2 = 6

Area = √(6 × (6 − 3) × (6 − 4) × (6 − 5)) = √(6 × 3 × 2 × 1) = √36 = 6

Perimeter = 3 + 4 + 5 = 12 units. Triangle type: Scalene

Area = 6.0000 sq units · Perimeter = 12 units · Type: Scalene

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Example 3 — Two Sides + Angle (SAS)

Triangle with sides 6 and 8, included angle 30°

Area = ½ × 6 × 8 × sin(30°) = ½ × 6 × 8 × 0.5 = 12

Third side c = √(36 + 64 − 2 × 6 × 8 × cos(30°)) = √(100 − 83.138) = √16.862 ≈ 4.106

Perimeter = 6 + 8 + 4.106 = 18.106 units

Area = 12.0000 sq units · Perimeter ≈ 18.1062 units

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Example 4 — Equilateral Triangle

Equilateral triangle with all sides = 5 units

s = (5 + 5 + 5) / 2 = 7.5

Area = √(7.5 × 2.5 × 2.5 × 2.5) = √117.1875 ≈ 10.8253

Verify: A = (√3 / 4) × 5² = 0.4330 × 25 = 10.8253 ✓

Area = 10.8253 sq units · Perimeter = 15 units · Type: Equilateral

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Frequently Asked Questions

What is Heron's formula and how is it derived?+

Heron's formula calculates the area of a triangle from its three side lengths a, b, c without needing the height. First compute the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s x (s-a) x (s-b) x (s-c)). It was proven by Hero of Alexandria around 60 AD using elementary geometry. The derivation involves dropping an altitude from one vertex and applying the Pythagorean theorem twice to express height in terms of the sides, then simplifying with algebraic manipulation.

How do I find the area of a triangle without the height?+

Use either Heron's formula (if you know all three sides) or the SAS formula (if you know two sides and the angle between them). For Heron's: s = (a+b+c)/2, Area = sqrt(s(s-a)(s-b)(s-c)). For SAS: Area = (1/2) x p x q x sin(C) where C is the included angle. Both approaches are mathematically equivalent when you have the right inputs.

What is the SAS triangle area formula?+

When you know two sides p and q and the included angle C between them, the area is: Area = (1/2) x p x q x sin(C). This comes directly from the definition of the sine function. The height of the triangle from the vertex at angle C can be expressed as h = q x sin(C), so Area = (1/2) x base x height = (1/2) x p x q x sin(C).

How do I find the third side from two sides and an angle?+

Use the Law of Cosines: c^2 = a^2 + b^2 - 2ab x cos(C), where C is the angle between sides a and b. This calculator automatically computes the third side in SAS mode so you get the full perimeter as well.

What is the triangle inequality theorem?+

The triangle inequality theorem states that for any three side lengths to form a valid triangle, the sum of any two sides must be strictly greater than the third side. Check all three: a+b greater than c, a+c greater than b, b+c greater than a. If any condition fails, the sides cannot close into a triangle. This calculator validates the triangle inequality before computing area in SSS mode.

What are equilateral, isosceles, and scalene triangles?+

Equilateral: all three sides equal, all angles 60 degrees. Isosceles: exactly two sides equal, two angles equal. Scalene: no sides equal, no angles equal. This calculator automatically classifies your triangle when you provide three side lengths (SSS or SAS mode).

What is the area of an equilateral triangle?+

For an equilateral triangle with side length a, the area simplifies to A = (sqrt(3) / 4) x a^2. For example, an equilateral triangle with sides 5 units has area = (sqrt(3)/4) x 25 = 10.825 square units. You can verify this using Heron's formula with a = b = c = 5: s = 7.5, Area = sqrt(7.5 x 2.5 x 2.5 x 2.5) = 10.825.

How do you find the area of a right triangle?+

A right triangle has one 90-degree angle. The two sides forming the right angle are called legs. Area = (1/2) x leg1 x leg2. Since the legs are perpendicular, one leg serves as the base and the other is the height. For example, a right triangle with legs 6 and 8 has area = (1/2) x 6 x 8 = 24 square units. The hypotenuse = sqrt(6^2 + 8^2) = 10.

Can I use this calculator for obtuse triangles?+

Yes. The Heron's formula mode works for any valid triangle regardless of angle type - acute, right, or obtuse. The SAS mode also works with obtuse angles since sin(C) is positive for angles between 0 and 180 degrees. The base-height mode works too as long as you measure the true perpendicular height, which for an obtuse triangle may fall outside the triangle's base.

What units does the area result use?+

The area result is in square units corresponding to whatever unit you enter. If you input side lengths in centimetres, the area is in square centimetres. If you input in metres, the area is in square metres. The calculator works with any consistent unit of length - just ensure all inputs are in the same unit.

Triangle Area Calculator

△ What is the Area of a Triangle?

Formulas and Derivation

How to Use This Calculator

Example Calculations

Example 1 — Base and Height

Triangle with base 8 units and height 5 units

Example 2 — Three Sides: 3-4-5 Right Triangle

Right triangle with sides 3, 4, and 5 (a classic Pythagorean triple)

Example 3 — Two Sides + Angle (SAS)

Triangle with sides 6 and 8, included angle 30°

Example 4 — Equilateral Triangle

Equilateral triangle with all sides = 5 units

Frequently Asked Questions

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