Triangle area formula:
what to do when the height is not given

Why the area is half of base times height

Draw any triangle, then copy it, rotate the copy and join it to the original along one side. You get a parallelogram whose area is base times height, and the triangle is exactly half of that. This is the whole origin of the ½ in the formula.

The catch is the word height. It means the perpendicular distance from the base to the opposite corner, not the length of a slanted side. In a right triangle one of the legs happens to be that height, which is why the formula feels easy there and goes wrong elsewhere.

In an obtuse triangle the perpendicular from the apex lands outside the base, on its extension. The formula still works: extend the base line, drop the perpendicular to it and use that distance as h. The base is the side you chose, not the whole line the height touches, and the area is still ½ × b × h.

When you only have three sides: Heron's formula

Surveyors and craftspeople often measure the three sides because they are easy to reach, and the height is not. This method needs no height.

Check with the height: taking 14 as the base, h = 2 × 84 ÷ 14 = 12 cm. The altitude also splits the triangle into a 5-12-13 and a 9-12-15 right triangle, and ½ × 5 × 12 + ½ × 9 × 12 = 30 + 54 = 84.

Two sides and the angle between them

If you know two sides and the angle between them, the height is hidden inside a sine. Height equals the second side times sin of the angle, so the area becomes ½ × a × b × sin C.

The angle must lie between the two sides. Using an angle at a different corner needs a different pair of sides.

When the corners are given as coordinates

For a triangle on a grid, use the shoelace form: take the absolute value of half of x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂).

This approach avoids drawing the figure and scales to any polygon, one extra term per corner.

Which method to use

If more than one route is possible, do two and compare. A mismatch means an input or a sign was wrong.

Special triangles with shortcuts

Some triangles have shorter forms that save a step. They are worth knowing because they come up in tiling, roof framing and paper craft.

• Right triangle: the two legs are base and height, so area = ½ × leg × leg. A 6 by 8 triangle has area 24.
• Equilateral triangle with side s: area = (√3 ÷ 4) × s². For s = 10 that is 0.4330 × 100 = 43.30.
• Isosceles triangle with equal sides a and base b: the height is √(a² − b²/4), then use ½ × b × h. For a = 13 and b = 10, h = √(169 − 25) = 12, so area = 60.
• Triangles on the same base between the same parallel lines have equal area, whatever the position of the apex.

The last point surprises people. Slide the top vertex sideways along a line parallel to the base and the shape changes, but the height does not, so the area stays the same. It is why a triangular plot of land can be reshaped on paper without changing its size.

Units, scale drawings and real jobs

On a scale drawing, a length is scaled by the drawing ratio, but an area is scaled by the ratio squared. If a plan is drawn at 1:50 and the triangle measures 8 cm by 5 cm on the paper, the real base is 4 m and height 2.5 m, so the area is ½ × 4 × 2.5 = 5 m². Multiplying the paper area of 20 cm² by 50 would give 1,000 cm², which is wrong; you need 20 × 2,500 = 50,000 cm², which is the same 5 m².

Triangular gables, sails, gussets and corner shelves are typical jobs. In each case, find the area once, then convert to the unit your supplier uses before you order. For a gable end that is a triangle on top of a rectangle, add the two areas.

Mistakes that waste material

• Using a slanted side as the height in a non-right triangle.
• Forgetting the ½, which doubles the area.
• Measuring in mixed units, such as base in cm and height in m.
• Rounding the semi-perimeter early, which can shift the area noticeably for thin triangles.

For roofing, sail-making or fabric cutting, add an allowance for seams and waste after the area is found.