Triangle Area Calculator

What this triangle area calculator does

This triangle area calculator finds the area of any triangle two ways: from its base and height, or from its three side lengths. Enter the measurements you have, and the tool returns the area in square units, showing the arithmetic so you can check the result.

The base-and-height method uses the classic formula area = ½ × base × height. The three-sides method uses Heron’s formula, which needs no height at all — perfect when you have measured all three edges but cannot easily find the perpendicular height. Both methods give the same answer for the same triangle; you simply use whichever inputs you have on hand.

How the two methods work

Method 1 — base and height

The most direct formula is:

area = ½ × base × height

Here the base is any one side of the triangle, and the height (also called the altitude) is the perpendicular distance from that base straight up to the opposite corner. The height must meet the base at a right angle — it is not the length of a slanted side. The factor of ½ is there because a triangle is exactly half of the rectangle that would enclose it.

Method 2 — three sides (Heron’s formula)

When you know the three side lengths a, b and c but not the height, use Heron’s formula. It works in two steps:

1. Compute the semi-perimeter (half the perimeter): s = (a + b + c) ÷ 2
2. Compute the area: area = √(s(s−a)(s−b)(s−c))

For Heron’s formula to give a real, positive area, the three sides must satisfy the triangle inequality: each side must be less than the sum of the other two. If one side is as long as or longer than the other two combined, the three lengths cannot close into a triangle, and the formula breaks down.

In both methods the units matter: keep every length in the same unit, and the area comes out in that unit squared. Lengths in metres give an area in square metres; lengths in inches give square inches.

Examples

Example 1 — base and height

A triangle has a base of 10 and a height of 6.

area = ½ × 10 × 6 = 30

The area is 30 square units.

Example 2 — three sides (the 3-4-5 triangle)

A triangle has sides of 3, 4 and 5 (perimeter 12).

• Semi-perimeter: s = (3 + 4 + 5) ÷ 2 = 6
• Area: area = √(6 × (6−3) × (6−4) × (6−5)) = √(6 × 3 × 2 × 1) = √36 = 6

The area is 6 square units. Because 3, 4 and 5 also form a right triangle, you can confirm it with the base-and-height method: ½ × 3 × 4 = 6.

Example 3 — checking the triangle inequality

Suppose you try sides of 1, 2 and 5. Check the inequality first: 1 + 2 = 3, which is less than 5, so these lengths cannot form a triangle. Heron’s formula would put a negative number under the square root, so no real area exists. The calculator flags impossible side combinations like this rather than returning a meaningless number.

Quick reference table

What you know	Method	Formula
Base and perpendicular height	Base × height	area = ½ × base × height
All three side lengths (a, b, c)	Heron’s formula	s = (a+b+c) ÷ 2, then area = √(s(s−a)(s−b)(s−c))
Right triangle (two perpendicular legs)	Base × height	area = ½ × leg₁ × leg₂
Three sides that fail a + b less than c	None	No triangle exists — check the triangle inequality

Common uses

• Homework and exams. Geometry students use both formulas constantly; this tool lets you check answers and see the worked steps.
• DIY, flooring and gardening. Triangular patches of lawn, tile, fabric or decking are easy to measure as three sides and feed into Heron’s formula.
• Construction and roofing. Gable ends, trusses and triangular panels are sized by area; base-and-height works well when the span and rise are known.
• Land and survey estimates. Irregular plots are often split into triangles, each measured by its three sides, then added together.
• Crafts and design. Bunting, sails, quilt pieces and signage frequently come down to a triangle’s area.

Tips and common mistakes

• Use the perpendicular height, not a slanted side. The single most common error is plugging a sloping edge into the base-and-height formula. The height must hit the base at 90°.
• Keep units consistent. Mixing metres and centimetres produces a wrong area. Convert everything to one unit before calculating.
• Don’t forget the ½. Leaving out the half doubles your answer; that missing factor is what separates a triangle from its surrounding rectangle.
• Check the triangle inequality before trusting Heron’s result. If a + b is less than or equal to c (for the longest side c), the sides are invalid.
• Confirm with a second method when you can. For right triangles, the two legs are a ready-made base and height, so ½ × leg × leg should match Heron’s answer.

Limitations and notes

This calculator returns the area of a single flat (planar) triangle. It does not handle three-dimensional shapes, curved sides, or triangles defined only by angles without any side length. Heron’s formula needs all three sides; if you know two sides and the angle between them instead, that requires a different (trigonometric) formula not covered here.

Results are rounded for display, so for high-precision engineering or surveying work, treat the output as a close approximation and apply the rounding rules your project requires. The tool runs entirely in your browser — nothing you type is uploaded or stored.

Related tools

• Circle Calculator — area, circumference and radius for round shapes.
• Quadratic Formula Calculator — solve ax² + bx + c = 0, another core algebra-and-geometry routine.
• Percentage Calculator — quick percentage and proportion math to pair with your area figures.

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