Area of polygons (triangles, quadrilaterals)
Area of Polygons
Area of Polygons is the opening topic of Two-Dimensional Measurement in the California Common Core State Standards. It is aligned to Standard G-GPE.7, which requires students to use coordinates and formulas to compute the areas of triangles and rectangles and other polygons.
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.