Geometry · Grades 6

Area of a Triangle: Half the Base Times the Height

Quick answer

The area of a triangle is half its base times its height. A triangle with base 10 and height 6 has area 30. The half is there because any triangle is exactly half of a rectangle with the same base and height. The height must be perpendicular to the base — it is not a side unless the triangle has a right angle there.

What you'll learn

  • Find the area of a triangle from its base and height
  • Explain why the formula has a half in it
  • Identify the correct height for any chosen base

The formula

A=12bhA = \tfrac{1}{2} b h
A triangle with base 10 and height 6 A triangle with a base of 10 units and a dashed perpendicular line of 6 units from the top corner down to the base, meeting it at a right angle. base = 10 height = 6
A triangle with base 10 and height 6
A=12(10)(6)=30 square unitsA = \tfrac{1}{2}(10)(6) = 30 \text{ square units}

Multiply the base by the height, then halve it. The order does not matter — halving first is often easier when one of the numbers is even.

Why there is a half

Every triangle is exactly half of a rectangle with the same base and height.

Take any triangle and make a second copy of it. Rotate the copy half a turn and slot it against the first. The two fit together into a rectangle whose width is the base and whose height is the height.

rectangle=bh⟹triangle=12bh\text{rectangle} = b h \quad\Longrightarrow\quad \text{triangle} = \tfrac{1}{2} b h

That is the whole reason for the half, and it holds for every triangle — not only the right-angled ones. The rectangle’s area was already length times width, so nothing new is being introduced.

What counts as the height

This is where most errors happen, and it is worth being precise.

The height is the perpendicular distance from the base to the opposite corner. Perpendicular means at a right angle to the base.

It is not a side of the triangle — unless the triangle happens to have a right angle between that side and the base.

TriangleBase and height
right-angledthe two short sides are a base and height pair
any otherthe height is a line drawn inside or outside the shape

In the figure above the height is drawn as a dashed line, because it is a measurement rather than an edge of the triangle.

Any side can be the base

A triangle has three sides, so it has three base-and-height pairs. All three give the same area.

That is useful: if a question gives you a side and a perpendicular to it, use that pair. There is no need to hunt for a particular “bottom” side, and a triangle drawn on its side is not a harder problem.

When the height falls outside

For a very slanted triangle, the perpendicular from the top corner lands outside the base, on its extension. The formula is unchanged.

This surprises people, but the doubling argument still works: two copies of a slanted triangle still form a rectangle of the same base and height. The picture looks different; the algebra does not.

Worked examples

Common mistakes

Practice problems

  1. Find the area of a triangle with base 88 cm and height 55 cm.

    Answer

    2020 cm²

    Full solution

    12(8)(5)=20\tfrac{1}{2}(8)(5) = 20 cm².

  2. Find the area of a triangle with base 1414 m and height 99 m.

    Answer

    6363 m²

    Full solution

    12(14)(9)=7×9=63\tfrac{1}{2}(14)(9) = 7 \times 9 = 63 m².

  3. A right triangle has legs of 1010 cm and 77 cm. Find its area.

    Hint

    The legs meet at a right angle.

    Answer

    3535 cm²

    Full solution

    The legs are a base and height pair: 12(10)(7)=35\tfrac{1}{2}(10)(7) = 35 cm².

  4. Find the area of a triangle with base 99 cm and height 55 cm.

    Answer

    22.522.5 cm²

    Full solution

    12(9)(5)=452=22.5\tfrac{1}{2}(9)(5) = \tfrac{45}{2} = 22.5 cm².

  5. A triangle has an area of 3636 cm² and a base of 1212 cm. Find its height.

    Answer

    66 cm

    Full solution

    36=12(12)h=6h36 = \tfrac{1}{2}(12)h = 6h, so h=6h = 6 cm.

  6. A triangle has an area of 5050 m² and a height of 1010 m. Find its base.

    Answer

    1010 m

    Full solution

    50=12b(10)=5b50 = \tfrac{1}{2}b(10) = 5b, so b=10b = 10 m.

  7. A triangle has base 66 cm and a sloping side of 55 cm, with a perpendicular height of 44 cm. Find its area.

    Hint

    Only one of those two numbers is the height.

    Answer

    1212 cm²

    Full solution

    The height is 44 cm, not the 55 cm sloping side: 12(6)(4)=12\tfrac{1}{2}(6)(4) = 12 cm².

    Using the 55 would give 1515 cm², which is too big — the slope is longer than the perpendicular.

  8. A shape is a 10×610 \times 6 rectangle with a triangle of base 1010 and height 44 on top. Find the total area.

    Answer

    8080 square units

    Full solution

    (10×6)+12(10)(4)=60+20=80(10 \times 6) + \tfrac{1}{2}(10)(4) = 60 + 20 = 80.

  9. A triangular garden has base 1212 m and height 77 m. Turf costs $8 per square metre. What does it cost to turf?

    Answer

    $336

    Full solution

    Area: 12(12)(7)=42\tfrac{1}{2}(12)(7) = 42 m².

    Cost: 42 \times 8 = \336$.

  10. Ana finds the area of a triangle with base 1010 and height 66 and writes 6060. What went wrong, and what does 6060 actually measure?

    Hint

    What shape has an area of 6060 here?

    Answer

    She left out the half. The area is 3030; her 6060 is the rectangle.

    Full solution

    12(10)(6)=30\tfrac{1}{2}(10)(6) = 30 square units.

    Her 6060 is bhb h — the area of the rectangle with the same base and height. Since two copies of the triangle make exactly that rectangle, her answer is precisely twice the truth.

    That is a useful way to remember it: forgetting the half does not give a random wrong answer, it gives the rectangle.

Frequently asked questions

What is the formula for the area of a triangle?

Half the base times the height. With base 10 and height 6, the area is half of 60, which is 30.

Why is there a half in the formula?

Because a triangle is exactly half of a rectangle with the same base and height. Two copies of any triangle fit together to make that rectangle, so one copy is half of it.

Which side is the base?

Any side you like. Each choice comes with its own height, and all three choices give the same area — so pick whichever pair you have been given.

Is the height always a side of the triangle?

No. The height is the perpendicular distance from the base to the opposite corner. It is only a side when the triangle has a right angle between them.

What if the height falls outside the triangle?

The formula still works. For a very slanted triangle the perpendicular lands outside the base, and half base times height gives the correct area anyway.

What to learn next

Formulas on this page

Key terms in this lesson

Area
Area is the amount of space inside a flat shape, counted in square units such as cm². It comes from multiplying two lengths, which is why the units are squared.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.6.G.A.1GeometryFind the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
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