Triangle Area = ½ × Base × Height

The formula $A = \tfrac{1}{2}bh$ for the area of a triangle isn't an arbitrary rule — it's a direct consequence of the fact that two congruent triangles fit together to form a parallelogram (or a rectangle, in the case of a right triangle).

The Visual Proof

Take any triangle. Make a copy of it, rotate the copy 180°, and place it next to the original. The two triangles together form a parallelogram with the same base $b$ and height $h$ as the original triangle.

The parallelogram has area $bh$ (base times height), so each triangle has half of that:

$A_{\triangle} = \dfrac{bh}{2}$

This works for any triangle — right, acute, or obtuse — because you can always pair it with its 180°-rotated copy.

Step-by-Step Solution

Given: A triangle with base $b = 8$ and height $h = 6$.

Find: The area.

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Step 1 — Apply the formula.

$A = \dfrac{1}{2}bh = \dfrac{1}{2}(8)(6) = \dfrac{48}{2} = 24$

The area is 24 square units.

Step 2 — Verify with a different formula.

If the same triangle has sides $a$, $b$, $c$, you can also use Heron's formula:

$s = \dfrac{a + b + c}{2}, \qquad A = \sqrt{s(s - a)(s - b)(s - c)}$

For example, a 6-8-10 right triangle (which fits our base 8 and height 6 if it's a right triangle): $s = 12$, and:

$A = \sqrt{12 \cdot 6 \cdot 4 \cdot 2} = \sqrt{576} = 24 \;\;\checkmark$

Both methods give the same answer.

Step 3 — Why the height must be perpendicular.

The "height" in $A = \tfrac{1}{2}bh$ is always the perpendicular distance from the opposite vertex to the base, not the slant length of the side. This is critical: tilted side lengths overstate the area.

Step 4 — The general formula using two sides and the included angle.

If you know two sides $a$ and $b$ and the angle $\theta$ between them, the perpendicular height from the apex is $b\sin\theta$, so:

$A = \dfrac{1}{2}\,a\,b\sin\theta$

For example, if $a = 5$, $b = 6$, and the angle between them is 60°:

$A = \tfrac{1}{2}(5)(6)\sin 60° = 15 \times 0.866 = 12.99$

Step 5 — General formula using coordinates.

If the triangle has vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$:

$A = \tfrac{1}{2}\bigl|\,x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\,\bigr|$

This is the shoelace formula, and it works for any triangle (and generalizes to any polygon).

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Answer:

$\boxed{\;A = \dfrac{1}{2}\,b\,h = \dfrac{1}{2}(8)(6) = 24\;}$

The area of a triangle is exactly half the area of the parallelogram (or rectangle) you'd build by combining two copies of it. Always use the perpendicular height, not the slant side.

Try It

• Adjust the base and height with sliders.
• Toggle the show parallelogram option to see how two copies of the triangle make a parallelogram.
• The shaded triangle is the original, the second triangle (faint) is its 180°-rotated copy.