Triangle calculator - area and angles

Do these three sides actually form a triangle?

Before calculating anything, check the triangle inequality. One test is enough: the two shorter sides must add up to more than the longest side. If that holds, the other two conditions hold automatically.

Take sides 2, 3 and 6: 2 + 3 = 5, which is less than 6. Two short sticks cannot reach each other when their far ends are 6 units apart, so there is no triangle and the calculator reports an error.

The borderline case is sides 3, 4 and 7. Here 3 + 4 = 7 exactly, so all three corners lie on one straight line and the area would be zero. The calculator does not accept such a "flat" triangle, because the inequality requires a strictly greater sum.

Heron's formula step by step: sides 5, 6 and 7

The beauty of Heron's formula is that you need neither the height nor any angle. Let's solve a triangle with sides a = 5, b = 6 and c = 7 metres.

• 1. Perimeter: 5 + 6 + 7 = 18 m, semi-perimeter s = 18 ÷ 2 = 9.
• 2. Differences: s - a = 9 - 5 = 4, s - b = 9 - 6 = 3, s - c = 9 - 7 = 2.
• 3. Product: 9 × 4 × 3 × 2 = 216.
• 4. Square root: √216 ≈ 14.70 m².
• 5. Height to side a: h = 2 × area ÷ a = 2 × 14.70 ÷ 5 ≈ 5.88 m.

How the angles come from the law of cosines

The largest angle always sits opposite the longest side, so start with angle C opposite c = 7. The law of cosines gives cos C = (a² + b² - c²) ÷ (2ab) = (25 + 36 - 49) ÷ (2 × 5 × 6) = 12 ÷ 60 = 0.2. The inverse function arccos 0.2 gives C ≈ 78.46°.

The same way you get A ≈ 44.42° and B ≈ 57.12°. As a check, add them: 44.42 + 57.12 + 78.46 = 180°. If your hand calculation sums to, say, 179.9°, the cause is usually rounding of intermediate values rather than a mistake.

Classifying triangles

By sides a triangle is equilateral (all sides equal, all angles 60°), isosceles (two sides equal, and the angles opposite them equal too) or scalene. By angles you can classify it without computing any angle: compare the square of the longest side with the sum of the squares of the other two.

• c² < a² + b²: acute. Example 5, 6, 7: 49 < 61, largest angle 78.46°.
• c² = a² + b²: right. Example 6, 8, 10: 100 = 36 + 64, angle 90°.
• c² > a² + b²: obtuse. Example 4, 5, 8: 64 > 41, largest angle ≈ 125.1°.
• Isosceles example 5, 5, 6: area 12, both base angles 53.13°.

In practice: plots, gables and a tape measure

The triangle calculator shines when an angle is hard to measure but a length is easy to measure with a tape. A triangular corner of a plot with sides 12, 15 and 20 m has an area of ≈ 89.67 m². An irregular garden or lawn can be split into triangles along diagonals: measure only straight lines and add the areas up.

A house gable with an 8 m base and two 5 m sloping edges is isosceles: the area is 12 m² and the calculator's height to the base is 3 m. That is the surface you need when estimating cladding boards or paint.

Be careful with very flat triangles. Sides 10, 10 and 19.9 give an area of 9.94, while 10, 10 and 19.8 already give 13.97. A 10 cm measuring error shifts the result by almost 40%, so for such shapes it is better to measure the height directly.