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Calculator Answers

Triangle Area Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A triangle area calculator is a mathematical utility that computes the two-dimensional space enclosed by a triangle. The calculator supports multiple methods of computation depending on the available input values, including the standard base and height formula, Heron's formula using three side lengths, and trigonometric methods using two side lengths and the included angle. It displays the step-by-step mathematical work for the selected calculation method, helping students and educators analyze geometric problems.

Pick a method, enter the matching values, and the calculator returns the triangle's area. Supports base and height, three sides via Heron's formula, and two sides with the included angle.

See also: Area Calculator – Find the Area of a Shape, Trapezoid Calculator, and Right Triangle Calculator.

Quick Answer

Calculate the area of a triangle. Enter the base and height, three side lengths, or two sides and an angle to find the area.

Method

Length of one side. · e.g. 10

Perpendicular distance from base to opposite vertex. · e.g. 6

Base × height ÷ 2 is the fastest when you have height. Heron's formula works for any triangle from its three sides.

Triangle area

Area

30

½ × 10 × 6

Base10
Height6
Area30

Area = base × height ÷ 2.

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Examples

Base 10, height 6

area 30

Sides 5, 6, 7 (Heron)

area ≈ 14.697 · perimeter 18 · s = 9

Sides 8 and 10 with 60° between

area ≈ 34.641

How it works

Formula · Three methods: Area = (base x height) / 2; Heron's Area = sqrt(s(s-a)(s-b)(s-c)) where s is half the perimeter; and Area = (1/2) a b sin(C) for two sides and the included angle.

Triangle area can be found three common ways. The right choice depends on what you know about the triangle, but all three give the same answer for any given triangle.

Base & height · A = base × height ÷ 2

Heron · A = √(s(s−a)(s−b)(s−c)), s = (a+b+c) ÷ 2

SAS · A = ½ × a × b × sin(angle)

Heron mode also validates the triangle inequality.

Three ways to find a triangle's area

Which formula you need depends entirely on what you already know, and this calculator carries all three rather than assuming you have a height.

  • A base and a height. Area is half the base times the height. This is the one most people learn first, and it is the only one of the three that needs a height at all.
  • Three sides, using Heron's formula. Half the perimeter is called s, and the area is the square root of s(s − a)(s − b)(s − c). No angle and no height required, which is what makes it useful for a measured plot of land where all you have is a tape measure.
  • Two sides and the angle between them. Area is one half times a times b times the sine of the included angle. The angle has to be the one between the two sides you entered; any other angle gives the wrong answer.

Working an area out by hand

One triangle, all three ways, so you can see they agree. Take a right triangle with sides 6, 8 and 10.

  1. Base and height. The two shorter sides meet at the right angle, so one is the base and the other is the height. Half of 8 × 6 is 24.
  2. Heron. The perimeter is 6 + 8 + 10 = 24, so s is 12. Then 12 × (12 − 6) × (12 − 8) × (12 − 10) = 12 × 6 × 4 × 2 = 576, and the square root of 576 is 24.
  3. Two sides and the angle. The angle between the sides of 6 and 8 is 90 degrees, whose sine is 1. Half of 6 × 8 × 1 is 24.

All three give 24 square units, because they are three routes to the same quantity rather than three different definitions of area.

Edge cases, and what the answer means

  • Sides that cannot close. Give Heron's method sides of 1, 2 and 10 and there is no triangle to have an area: the two short sides together are shorter than the long one, so they never meet. This is the triangle inequality, and the calculator says so rather than returning a number.
  • Sides that close exactly. Sides of 3, 4 and 7 add up precisely, so the shape collapses to a straight line with zero area. Heron would return 0 here, which is arithmetically right but geometrically useless, so this is rejected as well rather than reported as a valid triangle.
  • Angles at the limits. The included angle has to be greater than 0 and less than 180 degrees. At either end the two sides fold flat onto each other and there is no triangle left.
  • Zero or negative lengths. Rejected on every method. A side of zero is not a short side, it is a missing one.

Your measurements stay in the page

Every measurement you enter is processed by this page in your browser. The side lengths, the angle and the result are not sent to a server, not stored after you close the tab, and not visible to anyone else. There is no account and nothing to sign up for.

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Area Calculator – Find the Area of a ShapeFind the area of a shape from one tool. Supports circle, triangle, rectangle, and other common 2D geometric shapes.Trapezoid CalculatorCalculate the area and perimeter of a trapezoid from the two parallel bases, the height, and the two non-parallel sides.Right Triangle CalculatorSolve a right triangle from any of two legs, leg + hypotenuse, leg + acute angle, or hypotenuse + acute angle. Returns all sides, both acute angles, area, and perimeter.Perimeter CalculatorCalculate the perimeter (or circumference) of common 2D shapes: rectangle, triangle, circle, trapezoid, parallelogram, and regular polygon.

Frequently asked questions

Square units of whatever you entered. Put the sides in centimetres and the area is in square centimetres; put them in feet and it is in square feet. The calculator does not convert between units, so all three sides need to be measured in the same one before you start.

Only for a shape that is not really a triangle. If the three sides add up exactly, as 3, 4 and 7 do, the figure flattens into a straight line and Heron's formula returns zero. The calculator treats that as an invalid triangle rather than a valid area, because a zero here means there is no shape rather than a very small one.

Use the area calculator when you need a different shape, or several shapes in one sitting: it covers rectangles, circles, trapezoids and more, with one method each. Use this page when the shape is a triangle and you want a choice of method, because this is the only one that offers base and height, Heron, and two sides with the included angle together.

Heron's formula finds the area from the three side lengths alone, with no height and no angle. Take half the perimeter and call it s, then the area is the square root of s(s − a)(s − b)(s − c). For sides 6, 8 and 10 the perimeter is 24, so s is 12, and 12 × 6 × 4 × 2 = 576 whose square root is 24. It is the method to reach for when you have measured a shape rather than drawn it, because measuring three sides is easy and measuring a perpendicular height is not.

Yes, in two ways. If you know all three sides, use Heron's formula. If you know two sides and the angle between them, the area is one half times the two sides times the sine of that angle. The height is only needed for the base-times-height method, and it is often the hardest thing to measure, which is why the other two exist.

The angle between the two sides you entered, called the included angle. Any other angle in the triangle will give a wrong answer that still looks plausible, which makes this the easiest mistake to make silently. If the two sides meet at a right angle, the sine is 1 and the formula reduces to half the product of the sides.

Because they cannot form a triangle. Two sides have to be longer together than the third, or they never meet. Sides of 1, 2 and 10 fail badly. Sides of 3, 4 and 7 fail exactly: they add up precisely, so the shape flattens into a straight line with zero area. Both are refused rather than answered, since a zero here means no shape rather than a small one.

They do, when they are given the same triangle. A 6-8-10 triangle gives 24 square units by base and height, by Heron, and by two sides and the included angle. They are three routes to one quantity, not three different definitions, so a disagreement between them means an input was measured or entered wrongly.

Multiply the base by the height and divide by 2. The height must be the perpendicular distance from the base to the opposite vertex, not the slant side. For base 10 and height 6, the area is 10 × 6 ÷ 2 = 30 square units.

Heron's formula gives the area of any triangle from its three side lengths: A = √(s(s − a)(s − b)(s − c)), where s = (a + b + c) ÷ 2 is the semiperimeter. It works for scalene, isosceles, and equilateral triangles without needing a height.

When you know two sides and the angle between them, area = ½ × a × b × sin(angle). The angle must be the one between the two sides, not opposite either. For a = 8, b = 10, angle = 60°: ½ × 8 × 10 × sin(60°) ≈ 34.641 square units.

For three lengths to actually form a triangle, the sum of any two sides must be greater than the third. If you enter 3, 4, and 100, no triangle is possible because 3 + 4 < 100. The calculator flags this case in Heron mode.

A triangle's interior angle is always strictly between 0° and 180° (the three angles sum to exactly 180°). Values at or beyond the boundary describe a degenerate triangle with zero area or a self-intersecting shape that is not a triangle.

Yes, any of the three methods works for right triangles. For a right triangle, base × height ÷ 2 is the simplest because the two legs are already perpendicular. For more right-triangle math (legs, hypotenuse, angles), use the right triangle calculator.