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Daily Math Minute

Geometry

Area of Polygons

Finding the area of triangles and other polygons by composing and decomposing shapes.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Area Beyond Rectangles

The area formula for a rectangle, length times width, is straightforward. But what about a parallelogram — a slanted shape where the sides aren't perpendicular? Before reading on, think about whether a parallelogram's area formula should be different from a rectangle's, or whether there's a clever way to turn one into the other.

Picture a parallelogram, and cut a triangle off one slanted end. Slide that triangle to the opposite side, and the shape becomes a rectangle — with the exact same base and the exact same height as the original parallelogram, and no area lost or gained in the move. That means a parallelogram's area uses the very same formula as a rectangle: base times height.

Worked Example — Finding the Area of a Parallelogram

A parallelogram has a base of 8 units and a height of 5 units (the perpendicular distance between the base and its opposite side, not the length of the slanted side). Its area is 8 × 5 = 40 square units.

A triangle's area follows from the same idea: two identical triangles, flipped and joined along their longest side, form exactly one parallelogram with the same base and height. Since the triangle is half of that parallelogram, its area is half of base times height.

Worked Example — Finding the Area of a Triangle

A triangle has a base of 10 units and a height of 6 units. Its area is half of base times height: (10 × 6) ÷ 2 = 30 square units.
A△=12×base×heightA_{\triangle} = \dfrac{1}{2} \times \text{base} \times \text{height}

Geometry Canvas

Construct
Objects
  1. 1.

    P1: a free point, draggable on the plane

  2. 2.

    P2: a free point, draggable on the plane

  3. 3.

    P3: a free point, draggable on the plane

  4. 4.

    poly1: the polygon through P1, P2, P3

Measurements
  • poly1area = 15perimeter = 17.66

Tip

The height of a parallelogram or triangle is always the perpendicular distance from the base to the opposite side or vertex — never the length of a slanted side, even when a slanted side looks like the more obvious measurement to use.

Common Mistakes

  • Using the length of a slanted side as the height in the area formula, instead of the actual perpendicular distance.

    Height always means a measurement made at a right angle to the base — trace straight up or down from the base to the opposite point, not along a slanted edge.

  • Forgetting to divide by 2 when finding a triangle's area, applying the parallelogram formula directly.

    A triangle is exactly half of a matching parallelogram, so its area formula always includes dividing by 2 (or multiplying by 1/2) after multiplying base by height.

Key Takeaways

  • A parallelogram's area equals base times height, provable by rearranging it into a rectangle of the same base and height.
  • A triangle's area is half of base times height, since two matching triangles form one parallelogram.
  • Height is always the perpendicular distance from the base, not the length of a slanted side.

Summary

Composing and decomposing shapes reveals why the parallelogram and triangle area formulas work, not just what they are. The next lesson extends area into three dimensions, finding how much material covers the outside of a solid figure.

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