Table of Contents
Understanding Area Formulas
In the parent chapter “Area and Volume,” the general idea of area is introduced: area measures how much surface a 2‑dimensional shape covers. This chapter focuses specifically on formulas for finding area of common plane figures and on recognizing when and how to use them.
We will look at:
- Basic polygons (rectangles, triangles, parallelograms, trapezoids)
- Some special quadrilaterals
- Regular polygons
- Circles and sectors
- How to break complicated shapes into simpler ones
Throughout, “units” could be cm, m, inches, etc., and area is always in square units (like $cm^2$, $m^2$).
Area of Rectangles and Squares
A rectangle has opposite sides equal and all angles right angles.
If a rectangle has length $l$ and width $w$, then its area is
$$
A = l \times w.
$$
A square is a special rectangle with all sides equal. If each side is $s$, then
$$
A = s^2.
$$
This formula is often the starting point for other area formulas, because many shapes can be decomposed into rectangles.
Area of Parallelograms
A parallelogram has two pairs of opposite sides parallel. If you know the base and the corresponding height (altitude), you can find its area.
- Let the base be $b$.
- Let the height (perpendicular distance from the base to the opposite side) be $h$.
Then the area is
$$
A = b \times h.
$$
The key point is that $h$ is perpendicular to the base, not just any side length.
Reasoning idea: a parallelogram can be rearranged (by “cutting off” a triangular piece and moving it) to form a rectangle with the same base and height.
Area of Triangles
A triangle can be thought of as half of a parallelogram.
- Let the base be $b$.
- Let the height (altitude from the base to the opposite vertex) be $h$.
Then
$$
A = \frac{1}{2} b h.
$$
Again, the height must be perpendicular to the base. The base can be any side, as long as $h$ is the corresponding perpendicular height.
Right Triangles
For a right triangle, you can use its legs (the two sides that form the right angle) as base and height.
If the legs have lengths $a$ and $b$, then
$$
A = \frac{1}{2} a b.
$$
Triangles with Known Sides: Heron’s Formula (Optional)
If you know the lengths of all three sides $a$, $b$, $c$ of a triangle, but not the height, you can use Heron’s formula.
First compute the semi‑perimeter
$$
s = \frac{a + b + c}{2}.
$$
Then the area is
$$
A = \sqrt{s(s-a)(s-b)(s-c)}.
$$
This is especially useful when the height is difficult to find.
Area of Trapezoids
A trapezoid (or trapezium in some regions) is a quadrilateral with at least one pair of parallel sides, called the bases.
Let:
- $b_1$ and $b_2$ be the lengths of the parallel sides (bases),
- $h$ be the height (perpendicular distance between the bases).
Then the area is
$$
A = \frac{1}{2}(b_1 + b_2)h.
$$
You can think of this as:
- The average of the two bases, $\dfrac{b_1 + b_2}{2}$,
- Multiplied by the height.
So:
$$
A = \text{(average base length)} \times \text{height}.
$$
Areas of Special Quadrilaterals
Some quadrilaterals have their own convenient formulas.
Kite and Rhombus (Using Diagonals)
A kite has two pairs of adjacent equal sides. A rhombus has all four sides equal; it is also a parallelogram.
If a kite or rhombus has diagonals of lengths $d_1$ and $d_2$, then
$$
A = \frac{1}{2} d_1 d_2.
$$
This works because the diagonals in a kite or rhombus are perpendicular and divide the shape into right triangles.
Rhombus (Using Base and Height)
Like any parallelogram, a rhombus with base $b$ and corresponding height $h$ has area
$$
A = b h.
$$
You can choose whichever formula is easier based on the information you are given.
Area of Circles
A circle is determined by its radius.
- Let $r$ be the radius (distance from the center to any point on the circle).
- Let $\pi$ denote the constant approximately $3.14$ or more precisely $3.14159\ldots$.
The area of a circle is
$$
A = \pi r^2.
$$
If you know the diameter $d$ instead (distance across the circle through its center), then $r = \dfrac{d}{2}$, so
$$
A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}.
$$
Area of Sectors of Circles
A sector is the “slice” of a circle formed by two radii and the arc between them.
Let:
- $r$ be the radius of the circle,
- $\theta$ be the central angle of the sector.
You can express $\theta$ in degrees or radians. The formula depends on which unit you use.
If $\theta$ Is in Degrees
A full circle is $360^\circ$. So a sector with angle $\theta^\circ$ has area:
$$
A = \frac{\theta}{360^\circ} \cdot \pi r^2.
$$
If $\theta$ Is in Radians
A full circle is $2\pi$ radians. So a sector with angle $\theta$ radians has area:
$$
A = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2} r^2 \theta.
$$
The radian form $A = \dfrac{1}{2} r^2 \theta$ is particularly common.
Area of Regular Polygons
A regular polygon has all sides equal and all angles equal (equilateral triangle, square, regular pentagon, etc.).
There are several ways to find the area of a regular polygon. A general and useful formula uses the perimeter and the apothem.
- Let $P$ be the perimeter (sum of all side lengths).
- Let $a$ be the apothem (distance from the center to any side, measured perpendicularly).
Then the area is
$$
A = \frac{1}{2} P a.
$$
This formula works for any regular polygon (with $3$ or more sides).
Regular Polygon with Known Side Length and Number of Sides
Let:
- $n$ be the number of sides,
- $s$ be the length of each side.
Then $P = n s$.
If you can find the apothem $a$, you can substitute to get
$$
A = \frac{1}{2} n s a.
$$
In many problems, you will be given enough information (such as side length and radius, or angles) to compute the apothem using geometry or trigonometry, but that detailed work belongs to other chapters.
Using Decomposition and Composition
Many shapes you meet in problems will not be a single basic shape. To find their area, you often use a strategy of decomposition (breaking into simpler pieces) or composition (fitting them into a larger simple shape).
Decomposition
Break the figure into familiar shapes whose areas you know:
- Split into rectangles and triangles.
- Break a complex polygon into trapezoids.
- Cut off corners to form simpler shapes.
Then add the areas of the parts:
$$
A_{\text{total}} = A_1 + A_2 + A_3 + \dots
$$
Composition (Subtracting Areas)
Sometimes it is easier to start with a large simple shape and subtract unwanted parts.
For example:
- Find the area of a rectangle, then subtract the area of cut‑out rectangles or triangles.
- For a ring‑shaped region (an annulus), find the area of the large circle, then subtract the inner circle.
In general:
$$
A_{\text{desired}} = A_{\text{large}} - A_{\text{removed}}.
$$
Choosing the Right Formula
When you face an area problem, a common sequence of thoughts is:
- Identify the shape (or shapes) involved.
- List what you know (side lengths, heights, diagonals, angles, radius, etc.).
- Match information to formulas:
- Base and height → triangle, parallelogram, trapezoid.
- Diagonals → kite or rhombus.
- Radius or diameter → circle (or sector).
- Perimeter and apothem of a regular shape → regular polygon formula.
- Check units and ensure your answer is in square units.
Being familiar with these area formulas and when to apply them is essential for solving geometric problems and for applications in many other areas of mathematics and real life.