Area is the amount of two-dimensional space enclosed within a shape. Whether you're tiling a floor, painting a wall, buying carpet, or solving a geometry problem, knowing how to compute area is one of the most practical math skills you can have. This guide covers all the major shapes with clear formulas, step-by-step examples, and an interactive calculator.
What Is Area?
Area measures the size of a surface — how much space a flat shape covers. It is always expressed in square units: square centimeters (cm²), square meters (m²), square feet (ft²), and so on. The "square" in the unit reflects the two-dimensional nature of area — you are multiplying two length measurements together.
Area is the space inside a shape (measured in square units).
Perimeter is the total distance around the outside edge of a shape (measured in linear units).
A rectangle 4 m × 3 m has an area of 12 m² (space inside) and a perimeter of 14 m (distance around the outside).
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1. Area of a Square
Measure one side → square it
Since all sides of a square are equal, you only need one measurement. Multiply the side length by itself.
2. Area of a Rectangle
3. Area of a Triangle
h = height — the perpendicular distance from the base to the opposite vertex
The height must be perpendicular (90°) to the base — not the slant side
The height (h) in the triangle formula is not a side of the triangle — it is the perpendicular distance from the base to the opposite vertex. For a right triangle, the two legs serve as base and height. For other triangles, you may need to draw the height outside the triangle (for obtuse triangles).
Heron's Formula — When You Only Know the Three Sides
If you know all three side lengths but not the height, use Heron's formula:
a, b, c = lengths of the three sides
4. Area of a Circle
π ≈ 3.14159265...
If you know the diameter d: r = d ÷ 2, so A = π × (d/2)²
5. Area of a Trapezoid
b = length of the bottom parallel side
h = perpendicular height between the two parallel sides
6. Area of a Parallelogram
h = perpendicular height (NOT the slant side length)
A parallelogram with the same base and height as a rectangle has the same area.
7. Area of an Ellipse
b = semi-minor axis (half the shortest diameter)
When a = b, this reduces to the circle formula A = πr²
8. Area of a Rhombus
d₂ = length of the second diagonal
The diagonals of a rhombus always bisect each other at right angles.
All Formulas at a Glance
| Shape | Formula | Variables |
|---|---|---|
| Square | A = s² | s = side length |
| Rectangle | A = l × w | l = length, w = width |
| Triangle | A = ½ × b × h | b = base, h = perpendicular height |
| Circle | A = π × r² | r = radius |
| Trapezoid | A = ½ × (a+b) × h | a,b = parallel sides, h = height |
| Parallelogram | A = b × h | b = base, h = perpendicular height |
| Ellipse | A = π × a × b | a,b = semi-axes |
| Rhombus | A = ½ × d₁ × d₂ | d₁, d₂ = diagonals |
| Regular Hexagon | A = (3√3 ÷ 2) × s² | s = side length |
| Sector of circle | A = ½ × r² × θ | r = radius, θ = angle in radians |
Area Unit Conversions
Area units are squared — so conversions are not linear. For example, 1 m = 100 cm, but 1 m² = 10,000 cm² (100²).
| From | To | Multiply by |
|---|---|---|
| 1 m² | cm² | 10,000 |
| 1 m² | mm² | 1,000,000 |
| 1 km² | m² | 1,000,000 |
| 1 hectare (ha) | m² | 10,000 |
| 1 ft² | m² | 0.0929 |
| 1 in² | cm² | 6.4516 |
| 1 acre | m² | 4,046.86 |
Practice Problems
| Problem | Answer |
|---|---|
| Square with side 9 cm | 81 cm² |
| Rectangle 15 m × 6 m | 90 m² |
| Triangle base 12 cm, height 8 cm | 48 cm² |
| Circle radius 5 m | ≈ 78.54 m² |
| Trapezoid: top 4 cm, bottom 10 cm, height 5 cm | 35 cm² |
| Parallelogram base 14 cm, height 7 cm | 98 cm² |
| Rhombus diagonals 8 cm and 12 cm | 48 cm² |
1. Area is always in square units (cm², m², ft²)
2. Square: s² | Rectangle: l × w
3. Triangle: ½ × b × h — height must be perpendicular
4. Circle: π × r² — if you have diameter, divide by 2 first
5. Trapezoid: ½ × (a + b) × h — average the parallel sides
6. Area unit conversions are squared — 1 m = 100 cm but 1 m² = 10,000 cm²
7. When only three sides are known for a triangle, use Heron's formula
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Our Area Calculator supports all major shapes with unit conversions and step-by-step solutions.
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