Finding the area of a triangle is one of the most common geometry tasks — it appears in school homework, construction planning, land surveying, and computer graphics. The right formula depends on what measurements you have.
This guide covers four methods, explains when to use each one, and walks through examples including the tricky cases that cause mistakes.
Quick Answer
If you have base and height: Area = ½ × base × height
If you have three sides: Use Heron's formula — Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) / 2
If you have two sides and the included angle: Area = ½ × a × b × sin(C)
If you have coordinates: Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
Use our triangle calculator to compute area, angles, and all properties from three sides instantly.
Method 1: Base × Height (Simplest)
Formula: Area = ½ × base × height
This is the most familiar formula. The "height" (or altitude) is the perpendicular distance from the base to the opposite vertex — not the length of a side.
Example 1: Standard triangle
A triangle has a base of 10 cm and a height of 6 cm.
Area = ½ × 10 × 6 = 30 cm²
Example 2: Right triangle
A right triangle has legs of 8 m and 5 m. In a right triangle, one leg is the base and the other is the height.
Area = ½ × 8 × 5 = 20 m²
When to use this method
Use base × height when you can directly measure or know the perpendicular height. This is common in textbook problems and when working with right triangles. It does not work well when you only know the three side lengths, because finding the height from sides requires extra calculation.
Method 2: Heron's Formula (Three Sides)
Heron's formula calculates the area of any triangle when you know all three side lengths. No angles or heights needed.
Formula:
- Calculate the semi-perimeter: s = (a + b + c) / 2
- Area = √[s × (s − a) × (s − b) × (s − c)]
Example 3: Triangle with sides 5, 6, 7
Step 1: Semi-perimeter s = (5 + 6 + 7) / 2 = 18 / 2 = 9
Step 2: Plug into Heron's formula Area = √[9 × (9 − 5) × (9 − 6) × (9 − 7)] Area = √[9 × 4 × 3 × 2] Area = √216 Area ≈ 14.70 units²
Example 4: Triangle with sides 3, 4, 5 (right triangle)
s = (3 + 4 + 5) / 2 = 6 Area = √[6 × 3 × 2 × 1] = √36 = 6 units²
You can verify this with the base × height method: ½ × 3 × 4 = 6. Both give the same result.
Example 5: Nearly flat triangle (edge case)
Sides: 10, 10, 19.9
s = (10 + 10 + 19.9) / 2 = 19.95 Area = √[19.95 × 9.95 × 9.95 × 0.05] Area = √[98.76] Area ≈ 9.94 units²
This triangle is valid but very flat (nearly degenerate). The area is small relative to the side lengths. If the third side were 20 or greater, no triangle could be formed.
When to use Heron's formula
Use Heron's formula when you know all three sides but not the height. This is common in surveying, navigation, and any situation where you can measure distances but not perpendicular heights. The triangle calculator uses Heron's formula internally.
Method 3: Two Sides and Included Angle (Trigonometric)
Formula: Area = ½ × a × b × sin(C)
Here, C is the angle between sides a and b (the "included angle"). This formula is useful in navigation, physics, and whenever angle measurements are available.
Example 6: Two sides with angle
Sides a = 8 and b = 11, with an included angle of 35°.
Area = ½ × 8 × 11 × sin(35°) Area = ½ × 88 × 0.5736 Area ≈ 25.24 units²
Example 7: Right angle case
Sides a = 5 and b = 12, included angle = 90°.
Area = ½ × 5 × 12 × sin(90°) Area = ½ × 60 × 1 Area = 30 units²
Since sin(90°) = 1, this simplifies back to the base × height formula — which makes sense, because in a right triangle the two legs are the base and height.
When to use this method
Use the trigonometric formula when you have two sides and the angle between them. This comes up in physics (force components), engineering (structural analysis), and land measurement when angles are easier to measure than perpendicular distances.
Method 4: Coordinate Geometry (Shoelace Formula)
Formula: Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
This works when you have the (x, y) coordinates of all three vertices.
Example 8: Triangle on a coordinate plane
Vertices: A(1, 2), B(4, 7), C(8, 3)
Area = ½ |1(7 − 3) + 4(3 − 2) + 8(2 − 7)| Area = ½ |1(4) + 4(1) + 8(−5)| Area = ½ |4 + 4 − 40| Area = ½ × |−32| Area = ½ × 32 Area = 16 units²
When to use this method
Use the coordinate (shoelace) formula when working in a coordinate system — computer graphics, mapping applications, or any problem where you have vertex positions rather than side lengths.
Triangle Validity: The Triangle Inequality Theorem
Before calculating, verify that three lengths can actually form a triangle. The triangle inequality theorem states:
The sum of any two sides must be greater than the third side.
For sides a, b, c, check all three conditions:
- a + b > c
- a + c > b
- b + c > a
Example: Can 3, 4, 8 form a triangle? 3 + 4 = 7, which is not greater than 8. No, this is not a valid triangle.
Example: Can 5, 5, 9 form a triangle? 5 + 5 = 10 > 9 ✓ | 5 + 9 = 14 > 5 ✓ | 5 + 9 = 14 > 5 ✓ — Yes, valid triangle.
Finding Angles from Sides (Law of Cosines)
When you know three sides, you can find all angles using the law of cosines:
cos(A) = (b² + c² − a²) / (2bc)
Where angle A is opposite side a.
Example 9: Angles of a 5-6-7 triangle
Angle A (opposite side 5): cos(A) = (36 + 49 − 25) / (2 × 6 × 7) = 60 / 84 = 0.7143 A = arccos(0.7143) ≈ 44.42°
Angle B (opposite side 6): cos(B) = (25 + 49 − 36) / (2 × 5 × 7) = 38 / 70 = 0.5429 B = arccos(0.5429) ≈ 57.12°
Angle C = 180° − 44.42° − 57.12° ≈ 78.46°
All angles sum to 180°. The triangle calculator computes all three angles automatically.
Additional Triangle Properties
Beyond area and angles, triangles have several useful properties:
Perimeter
The total length of all sides: P = a + b + c
Heights (Altitudes)
The height to each side can be found from the area:
- Height to side a: hₐ = 2 × Area / a
- Height to side b: h_b = 2 × Area / b
- Height to side c: h_c = 2 × Area / c
Circumradius (R)
The radius of the circle that passes through all three vertices:
R = (a × b × c) / (4 × Area)
Inradius (r)
The radius of the circle inscribed inside the triangle, tangent to all three sides:
r = Area / s (where s is the semi-perimeter)
Medians
A median connects a vertex to the midpoint of the opposite side. The median to side a:
mₐ = ½ √(2b² + 2c² − a²)
Triangle Types at a Glance
Triangles are classified by their sides and their angles:
By sides:
- Equilateral — all three sides equal (all angles = 60°)
- Isosceles — exactly two sides equal (two angles equal)
- Scalene — all three sides different
By angles:
- Acute — all angles less than 90°
- Right — one angle exactly 90°
- Obtuse — one angle greater than 90°
A triangle can be both a side type and an angle type. For example, a 3-4-5 triangle is both scalene (all sides different) and right (one 90° angle). An equilateral triangle is always acute.
Common Mistakes
1. Using a slant side instead of the height
The "height" in the base × height formula must be perpendicular to the base. A common error is using one of the triangle's sides as the height when it is not perpendicular. This only works in right triangles where the legs are the base and height.
2. Forgetting to take the square root in Heron's formula
Heron's formula gives s(s − a)(s − b)(s − c) under a square root. Skipping the √ gives the square of the area, which is a much larger number.
3. Using degrees vs. radians
When using sin(C) for the trigonometric formula, make sure your calculator is in the correct mode. If C = 35°, using radians gives sin(35) ≈ −0.43 (wrong) instead of sin(35°) ≈ 0.57.
4. Assuming the sum of two sides equals the third makes a triangle
If a + b = c (exactly), the three "sides" form a straight line, not a triangle. The inequality must be strict: a + b > c.
5. Rounding intermediate steps
When computing Heron's formula with long decimals, avoid rounding intermediate values. Round only the final answer. Early rounding can cause significant errors, especially for nearly flat triangles.
Which Formula Should I Use?
| What you know | Formula | Complexity |
|---|---|---|
| Base and perpendicular height | ½ × b × h | Simplest |
| Three side lengths | Heron's formula | Moderate |
| Two sides + included angle | ½ × a × b × sin(C) | Moderate |
| Three vertex coordinates | Shoelace formula | Moderate |
| Only one side and one angle | Not enough information | — |
Decision rule: Start with whatever measurements you have. If you have three sides and no angles, use Heron's formula or the triangle calculator directly. If you have a right triangle, base × height is fastest.
Worked Example: Full Triangle Solution
Problem: A triangular garden plot has sides of 8 m, 11 m, and 14 m. Find its area, all angles, the required fencing (perimeter), and the radius of the largest circular fountain that fits inside (inradius).
Perimeter: 8 + 11 + 14 = 33 m of fencing needed
Semi-perimeter: s = 33 / 2 = 16.5
Area (Heron's): Area = √[16.5 × 8.5 × 5.5 × 2.5] Area = √[16.5 × 8.5 × 5.5 × 2.5] Area = √[1928.4375] Area ≈ 43.91 m²
Inradius: r = 43.91 / 16.5 ≈ 2.66 m — the largest inscribed circle has a radius of about 2.66 meters
Angles (law of cosines):
- Angle opposite 14 m: cos(A) = (64 + 121 − 196) / (2 × 8 × 11) = −11/176 → A ≈ 93.6°
- Angle opposite 11 m: cos(B) = (64 + 196 − 121) / (2 × 8 × 14) = 139/224 → B ≈ 51.7°
- Angle opposite 8 m: C ≈ 180° − 93.6° − 51.7° ≈ 34.7°
This is an obtuse scalene triangle (one angle > 90°, all sides different). You can verify all of these results instantly with the triangle calculator.
Frequently Asked Questions
What is the simplest formula for triangle area?
Area = ½ × base × height. This is the easiest formula when you know the base and the perpendicular height. For right triangles, the two legs serve as base and height directly.
How do you find the area of a triangle with only three sides?
Use Heron's formula. First calculate the semi-perimeter s = (a + b + c) / 2, then Area = √[s(s − a)(s − b)(s − c)]. This works for any valid triangle, regardless of shape.
What is Heron's formula used for?
Heron's formula calculates the area of a triangle from its three side lengths alone. It is useful when you cannot easily measure the height, such as in land surveying, navigation, or when working with coordinate-free measurements.
How do you find the height of a triangle from the area?
Rearrange the area formula: height = 2 × Area / base. Calculate the area first (using any method), then divide twice the area by the base length to get the corresponding height.
What is the triangle inequality theorem?
It states that the sum of any two sides of a triangle must be greater than the third side. If this condition fails for any pair, the three lengths cannot form a triangle.
How do you calculate triangle angles from sides?
Use the law of cosines: cos(A) = (b² + c² − a²) / (2bc), where A is the angle opposite side a. Apply this for each angle, or find two angles and subtract from 180° for the third.
What is the difference between circumradius and inradius?
The circumradius (R) is the radius of the circle passing through all three vertices. The inradius (r) is the radius of the largest circle that fits inside the triangle, touching all three sides. For an equilateral triangle with side a, R = a/√3 and r = a/(2√3).
Can a triangle have two right angles?
No. The interior angles of a triangle always sum to exactly 180°. Two right angles would total 180° already, leaving 0° for the third angle, which is impossible.
What is a degenerate triangle?
A degenerate triangle has zero area — its three vertices are collinear (on the same line). This happens when one side exactly equals the sum of the other two (e.g., sides 3, 4, 7). Technically, it is not a triangle at all.
How accurate is Heron's formula for very flat triangles?
Heron's formula can lose numerical precision for extremely flat (nearly degenerate) triangles due to floating-point arithmetic. For such cases, the Kahan variation of Heron's formula provides better accuracy: sort sides so a ≥ b ≥ c, then Area = ¼ √[(a + (b + c))(c − (a − b))(c + (a − b))(a + (b − c))].
What units does triangle area use?
Area is always in square units of whatever unit the sides use. If sides are in centimeters, the area is in cm². If sides are in feet, the area is in ft². Never mix units — convert all sides to the same unit first.
Is there a formula for the area of a triangle inscribed in a circle?
Yes. If the triangle has sides a, b, c and is inscribed in a circle of radius R: Area = (a × b × c) / (4R). Conversely, R = (a × b × c) / (4 × Area).