Area of polygons (triangles, quadrilaterals)
Area of Polygons
Area of Polygons is the opening topic of Two-Dimensional Measurement in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.11(B), which requires students to apply the formulas for the area of two-dimensional figures to solve problems.
Polygon area formulas include the triangle \(\dfrac12 bh\), the parallelogram \(bh\), and the trapezoid \(\dfrac12(b_1+b_2)h\).
Theory
Each polygon has an area formula based on a base and a perpendicular height:
- Triangle: \(A=\dfrac12 bh\).
- Rectangle / parallelogram: \(A=bh\).
- Trapezoid: \(A=\dfrac12(b_1+b_2)h\), where \(b_1,b_2\) are the parallel sides.
The area formulas:
How to find a polygon's area
- Identify the shape and its base(s) and height.
- Use the perpendicular height, not a slant side.
- Substitute into the formula.
- Rearrange to find a missing dimension if the area is given.
Use \(A=\dfrac12 bh\).
| \(A\) | \(=\) | \(\dfrac12(10)(6)=30\) |
A parallelogram's area is base times height.
| \(A\) | \(=\) | \(8\times 5=40\) |
Use \(A=\dfrac12(b_1+b_2)h\).
| \(A\) | \(=\) | \(\dfrac12(6+10)(4)\) |
| \(=\) | \(\dfrac12(16)(4)=32\) |
Solve \(\dfrac12 bh=24\).
| \(\dfrac12(8)h\) | \(=\) | \(24\) |
| \(4h\) | \(=\) | \(24\) |
| \(h\) | \(=\) | \(6\) |
Common pitfalls
Frequently asked questions
What is the area of a triangle?
\(A=\dfrac12 bh\): one half the base times the perpendicular height.
What is the area of a trapezoid?
\(A=\dfrac12(b_1+b_2)h\): the average of the two parallel sides times the height.
What is the area of a parallelogram?
Base times perpendicular height, \(A=bh\).
What height do you use in area formulas?
Always the perpendicular height to the chosen base, not a slanted side.