Triangle Area Calculator
Free Triangle area Calculator for triangle. Enter values to get step-by-step solutions with formulas and graphs. See charts, tables, and visual results.
Triangle Area Calculator
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Formula: Area = ½ × base × height
Worked example — Area = 12 square meters
Formula
Area = ½ × base × height
A triangle's area equals half the product of its base and the perpendicular height measured from that base to the opposite vertex — a direct consequence of every triangle being exactly half of a parallelogram sharing the same base and height.
Worked Examples
Example 1: Basic base and height
Problem:A triangular garden bed has a base of 6 meters and a height of 4 meters. Find its area.
Solution:Area = ½ × base × height = ½ × 6 × 4 = 12.
Result:Area = 12 square meters
Example 2: Triangular roof section
Problem:A gable roof triangle has a base of 9 meters and a height of 3.5 meters. How much roofing material (area) is needed for that section?
Solution:Area = ½ × 9 × 3.5 = ½ × 31.5 = 15.75.
Result:Area = 15.75 square meters
Frequently Asked Questions
Why is the triangle area formula 'half of base times height'?
Any triangle is exactly half of a parallelogram that shares the same base and height — you can prove this by duplicating the triangle, rotating the copy 180 degrees, and joining it to the original along one side to form a parallelogram with area base × height. Since a parallelogram's area is base × height, each of the two identical triangles that make it up must be exactly half of that.
Which measurement counts as the 'height' of a triangle?
The height (or altitude) is the perpendicular distance from the chosen base to the opposite vertex — not the length of a slanted side. For an obtuse triangle, the height may fall outside the triangle itself, meaning the foot of the perpendicular lands on the extension of the base line rather than between its endpoints, but the formula still applies exactly the same way.
How do I find a triangle's area if I only know the three side lengths, not the height?
Use Heron's Formula: compute the semi-perimeter s = (a+b+c)/2, then Area = √[s(s−a)(s−b)(s−c)]. This avoids needing to measure or calculate a perpendicular height directly and works for any triangle given just its three side lengths.
How is triangle area calculated using two sides and the included angle?
When you know two sides (a, b) and the angle C between them, the area is Area = ½ × a × b × sin(C). This trigonometric form is especially useful in surveying and navigation, where angles are often easier to measure directly than perpendicular heights.
Does the choice of which side to call the 'base' change the area?
No — any of the three sides can be used as the base as long as you pair it with the height measured perpendicular to that specific side. The resulting area is identical no matter which side you choose, because base × height is constant for a given triangle (it always equals twice the true area).
What real-world tasks use the triangle area formula?
Roofers calculate the area of triangular roof sections to order the right amount of shingles, surveyors compute land parcel areas by decomposing irregular plots into triangles, and structural engineers use triangular cross-sections (like truss members) where the base-height area formula determines material load capacity per unit area.
How does triangle area relate to coordinate geometry?
When a triangle's vertices are given as coordinates (x₁,y₁), (x₂,y₂), (x₃,y₃), the Shoelace Formula computes area directly: Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. This coordinate-based approach is what GPS mapping software and CAD programs use internally, since it doesn't require first identifying a base and a perpendicular height by inspection.
Can the area formula help verify if three side lengths can even form a valid triangle?
Yes — before computing area with Heron's Formula, check the triangle inequality: the sum of any two sides must exceed the third side. If this fails, no triangle exists with those three lengths, and Heron's Formula would attempt to take the square root of a negative number, signaling the input is invalid.
What is the shortcut formula for the area of an equilateral triangle?
For an equilateral triangle of side s, the altitude is s√3/2, so substituting into ½ × base × height gives Area = (√3/4)s². A triangle with 10 cm sides therefore has an area of about 43.30 cm². The same result follows from ½ × a × b × sin(C) with a = b = s and C = 60°, since sin 60° = √3/2. It is a handy check when tiling hexagonal or truss patterns built from equal triangles.
How do a triangle's medians divide its area?
A median joins a vertex to the midpoint of the opposite side, so it splits the triangle into two smaller triangles that share the same height and have equal bases, giving each exactly half the total area. Drawing all three medians produces six small triangles of equal area, each one sixth of the whole, and they meet at the centroid, which sits two thirds of the way along each median from its vertex.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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