Triangle Calculator

Give this triangle calculator three side lengths and it returns the area, the perimeter, and all three interior angles in seconds. Enter a classic 3-4-5 triangle, for instance, and you get an area of 6 square units, a perimeter of 12, and angles of about 36.87°, 53.13° and 90° - confirming it is a right triangle. If you only know a base and a height, switch modes and the calculator returns the area directly from ½ × base × height. Everything updates instantly as you type, with no button to press.

How to find the area of a triangle

There are two common situations, and this tool handles both. When you know the three side lengths, the calculator uses Heron's formula:

Here s is the semi-perimeter (half the perimeter), and a, b and c are the three sides. When you instead know a base and the perpendicular height, the area is simpler:

Both formulas give the same answer for the same triangle - they just start from different information. The key requirement for the second formula is that the height must be measured perpendicular to the base, not along a slanted side.

How the angles are calculated

When all three sides are known, each angle is found with the law of cosines. The angle opposite side a is:

The same pattern gives angles B and C. The calculator converts each result from radians to degrees, so you read familiar values like 60° or 90°. As a built-in check, the three angles always sum to exactly 180° for any flat (Euclidean) triangle, and the largest angle always sits opposite the longest side.

How to use this triangle calculator

You only need a few numbers. Follow these steps:

1. Pick a mode: choose 3 sides (SSS) if you know all three lengths, or Base & height if you know one side and its perpendicular height.
2. Enter your values: type the side lengths (a, b, c) or the base and height. Any unit works as long as you keep it consistent.
3. Read the headline: the large blue number is the area in square units.
4. Check the angles and breakdown: in SSS mode you also get the three angles, the perimeter, the triangle type, and the altitude to each side.
5. Watch for warnings: if the three sides break the triangle inequality, the calculator tells you they cannot form a triangle instead of returning a wrong number.

Try the preset buttons (Right, Equilateral, Isosceles, Obtuse) to see how different shapes change the angles and area at a glance.

Who this calculator is for

• Students checking geometry or trigonometry homework on area, perimeter and angles.
• Teachers generating quick worked examples with consistent, correct numbers.
• DIY and trades measuring a triangular patch of floor, roof, garden or fabric to estimate material.
• Designers and makers who need the angles of a triangular piece before cutting.
• Anyone who remembers ½ × base × height but wants area and angles straight from three measured sides.

Key terms explained

• Perimeter: the total distance around the triangle, a + b + c.
• Semi-perimeter (s): half the perimeter, used inside Heron's formula.
• Altitude (height): the perpendicular distance from a side to the opposite vertex. Each side has its own altitude.
• Law of cosines: a generalization of the Pythagorean theorem that works for any triangle, used here to recover angles from sides.
• Triangle inequality: the rule that any two sides must together be longer than the third.
• Scalene / isosceles / equilateral: classifications by how many sides are equal (none, two, or all three).
• Acute / right / obtuse: classifications by the largest angle (under 90°, exactly 90°, or over 90°).

Worked example 1: the 3-4-5 right triangle

With sides a = 3, b = 4, c = 5, the semi-perimeter is s = (3 + 4 + 5) / 2 = 6. Heron's formula gives area = √(6 × 3 × 2 × 1) = √36 = 6 square units. The law of cosines puts the angle opposite the longest side (5) at exactly 90°, with the other two angles at roughly 36.87° and 53.13°. Because 3² + 4² = 9 + 16 = 25 = 5², it is the textbook right triangle.

Worked example 2: an equilateral triangle

For sides a = b = c = 6, the semi-perimeter is 9, and the area is √(9 × 3 × 3 × 3) = √243 ≈ 15.59 square units. Every angle is exactly 60°, and the triangle is classified as equilateral. This matches the shortcut for equilateral area, (√3 / 4) × side², which equals (1.732 / 4) × 36 ≈ 15.59.

Worked example 3: area from base and height

Suppose a triangular garden bed has a base of 12 feet and a perpendicular height of 5 feet. The area is ½ × 12 × 5 = 30 square feet. Notice you do not need the slanted sides for the area - only the base and the perpendicular height. To also find the angles, you would measure the three sides and use SSS mode.

Quick reference table

Common triangles and their results, computed with the formulas above (lengths in the same unit, area in square units):

Tips for accurate results

• Keep one unit: mixing feet and inches gives a wrong area. Convert first, then enter.
• Measure the perpendicular height: in base-and-height mode, do not use a slanted side length as the height.
• Use more decimals for small triangles: rounding sides too early can noticeably shift the area.
• Sanity-check with the triangle inequality: if two sides barely exceed the third, the triangle is very "thin" and its area will be small.

Common pitfalls

Most mistakes come from the wrong height or impossible sides. Using a slanted edge instead of the perpendicular height overstates the area, and entering three lengths that violate the triangle inequality describes no real triangle at all. The calculator guards against the second case directly; the first is on the measurer. Another subtle point: a base and height alone fix only the area, never the angles, because the apex can slide along a parallel line without changing either the base or the height.

Related concepts and calculators

Triangles connect to a lot of everyday math. If you are computing the slanted side of a right triangle, the Pythagorean theorem (a² + b² = c²) is the special case of the law of cosines when one angle is 90°. For powers and roots that show up inside these formulas, the Square Root Calculator and Exponent Calculator are handy. For general arithmetic and trigonometric functions, the Scientific Calculator covers sine, cosine and tangent directly. And when you need to scale a triangle up or down while keeping its shape, the Ratio Calculator and Percentage Calculator help with proportional resizing.