How large is your garden? How many liters of water does a cylindrical pool hold? Calculating area and volume is one of the most practical mathematical skills there is: it is used in construction, gardening, painting walls, choosing a tank, and in a thousand other everyday situations. In this complete guide, you will find all the formulas for flat shapes and solids, with practical examples, Heron's formula, and applications in everyday life.
Flat shapes: area formulas
The area (or surface) of a flat shape measures the extent of the planar region enclosed by its perimeter. It is expressed in squared units of measurement (m², cm², km², etc.).
Rectangle
Area = base x height
Perimeter = 2 x (base + height)
Example: a rectangular room measuring 5 m x 4 m has an area of 5 x 4 = 20 m² and a perimeter of 2 x (5 + 4) = 18 m.
The rectangle is the most common shape in practical applications: rooms, plots of land, screens, and sheets of paper are all rectangular.
Square
Area = side²
Perimeter = 4 x side
Diagonal = side x √2
Example: a square tile with a 30 cm side has an area of 30² = 900 cm² = 0.09 m².
Triangle
Area = (base x height) / 2
Example: a triangle with base 8 cm and height 5 cm has an area of (8 x 5) / 2 = 20 cm².
This formula works for any type of triangle (equilateral, isosceles, scalene, right), as long as the height is relative to the chosen base.
Right triangle
Area = (leg1 x leg2) / 2
In a right triangle, the two legs serve as base and height for each other, making the calculation particularly simple.
Circle
Area = π x radius²
Circumference = 2 x π x radius = π x diameter
Example: a circle with a 3 m radius has an area of π x 9 = 28.27 m² and a circumference of 2π x 3 = 18.85 m.
The value of π (pi) is approximately 3.14159. For quick calculations, you can use 3.14.
Trapezoid
Area = ((longer base + shorter base) x height) / 2
Example: a trapezoid with a longer base of 10 cm, shorter base of 6 cm, and height of 4 cm has an area of ((10 + 6) x 4) / 2 = 32 cm².
Parallelogram
Area = base x height
Note: the height is not the oblique side, but the perpendicular distance between the two bases.
Rhombus
Area = (longer diagonal x shorter diagonal) / 2
Example: a rhombus with diagonals of 12 cm and 8 cm has an area of (12 x 8) / 2 = 48 cm².
Summary table of flat shape areas
| Shape | Area formula | Required data |
|---|---|---|
| Rectangle | b x h | Base and height |
| Square | l² | Side |
| Triangle | (b x h) / 2 | Base and height |
| Circle | π x r² | Radius |
| Trapezoid | ((B + b) x h) / 2 | Two bases and height |
| Parallelogram | b x h | Base and height |
| Rhombus | (d1 x d2) / 2 | Two diagonals |
| Ellipse | π x a x b | Semi-axes a and b |
| Regular hexagon | (3√3 / 2) x l² | Side |
Heron's formula: area from the perimeter
When you only know the three sides of a triangle (and not the height), you can calculate the area using Heron's formula, attributed to the Greek mathematician Heron of Alexandria (1st century AD).
Formula
Area = √(s x (s-a) x (s-b) x (s-c))
where s = (a + b + c) / 2 is the semi-perimeter and a, b, c are the three sides.
Example
Triangle with sides a = 7 cm, b = 8 cm, c = 9 cm.
- Semi-perimeter: s = (7 + 8 + 9) / 2 = 12
- Area = √(12 x (12-7) x (12-8) x (12-9)) = √(12 x 5 x 4 x 3) = √720 = 26.83 cm²
Heron's formula is particularly useful when measuring triangular plots of land where only the sides are known (measurable with a tape measure) and not the height.
Solids: volume formulas
Volume measures the three-dimensional space occupied by a solid. It is expressed in cubic units (m³, cm³, l). Remember: 1 liter = 1 dm³ = 1,000 cm³ and 1 m³ = 1,000 liters.
Cube
Volume = side³
Total surface area = 6 x side²
Example: a cube with a 2 m side has a volume of 2³ = 8 m³ = 8,000 liters.
Rectangular prism (cuboid)
Volume = length x width x height
Total surface area = 2 x (lw + lh + wh) where l = length, w = width, h = height
Example: a room measuring 5 m x 4 m x 3 m has a volume of 60 m³ = 60,000 liters of air.
Cylinder
Volume = π x radius² x height
Lateral surface area = 2 x π x radius x height
Total surface area = 2 x π x radius x (radius + height)
Example: a cylindrical tank with radius 0.5 m and height 1.5 m has a volume of π x 0.25 x 1.5 = 1.178 m³ = approximately 1,178 liters.
Sphere
Volume = (4/3) x π x radius³
Surface area = 4 x π x radius²
Example: a soccer ball with radius 11 cm has a volume of (4/3) x π x 11³ = 5,575 cm³ = approximately 5.6 liters.
Cone
Volume = (1/3) x π x radius² x height
Lateral surface area = π x radius x slant height
where the slant height g = √(radius² + height²)
Example: a cone with radius 3 cm and height 8 cm has a volume of (1/3) x π x 9 x 8 = 75.4 cm³.
Pyramid
Volume = (1/3) x Base area x height
This formula works for any pyramid, regardless of the base shape (square, triangular, hexagonal, etc.).
Example: a pyramid with a square base of side 6 cm and height 10 cm has a volume of (1/3) x 36 x 10 = 120 cm³.
Summary table of solid volumes
| Solid | Volume formula | Required data |
|---|---|---|
| Cube | l³ | Side |
| Rectangular prism | l x w x h | Length, width, height |
| Cylinder | π x r² x h | Radius and height |
| Sphere | (4/3) x π x r³ | Radius |
| Cone | (1/3) x π x r² x h | Radius and height |
| Pyramid | (1/3) x Ab x h | Base area and height |
| Truncated cone | (π x h / 3) x (R² + r² + Rr) | Base radii and height |
| Prism | Ab x h | Base area and height |
Conversions between area and volume units
Area units
| From | To | Factor |
|---|---|---|
| 1 m² | cm² | 10,000 |
| 1 km² | m² | 1,000,000 |
| 1 hectare (ha) | m² | 10,000 |
| 1 are (a) | m² | 100 |
Volume units
| From | To | Factor |
|---|---|---|
| 1 m³ | liters | 1,000 |
| 1 dm³ | liters | 1 |
| 1 liter | cm³ (ml) | 1,000 |
| 1 m³ | cm³ | 1,000,000 |
Practical applications in everyday life
Calculating paint for walls
To find out how much paint you need for a room, calculate the total wall surface area:
Wall surface = room perimeter x height - area of doors and windows
Example: room 4 m x 5 m, 2.7 m high, with one door (0.9 x 2.1 m) and one window (1.2 x 1.4 m).
- Perimeter: 2 x (4 + 5) = 18 m
- Gross wall surface: 18 x 2.7 = 48.6 m²
- Door area: 0.9 x 2.1 = 1.89 m²
- Window area: 1.2 x 1.4 = 1.68 m²
- Net surface: 48.6 - 1.89 - 1.68 = 45.03 m²
- With a coverage of 10 m²/liter and 2 coats: 45.03 x 2 / 10 = 9 liters of paint
Calculating pool capacity
For a rectangular pool measuring 8 m x 4 m with an average depth of 1.5 m:
Volume = 8 x 4 x 1.5 = 48 m³ = 48,000 liters
For a circular pool with a 4 m diameter and 1.2 m depth:
Volume = π x 2² x 1.2 = 15.08 m³ = approximately 15,080 liters
Calculating soil for a pot
A truncated cone-shaped pot with top radius 20 cm, bottom radius 15 cm, and height 30 cm:
Volume = (π x 30 / 3) x (20² + 15² + 20 x 15) = (π x 10) x (400 + 225 + 300) = 10π x 925 = 29,060 cm³ = approximately 29 liters
Calculating the area of an irregular plot of land
For irregularly shaped plots, the most practical method is to divide them into triangles, calculate the area of each (using Heron's formula if only the sides are needed) and add the results.
Relationships between shapes: useful formulas
Some geometric relationships are particularly useful:
- The volume of a cone is exactly 1/3 of the cylinder with the same base and height
- The volume of a pyramid is exactly 1/3 of the prism with the same base and height
- The volume of a hemisphere is exactly 2/3 of the circumscribed cylinder (discovered by Archimedes)
- The area of a circle is exactly π/4 of the area of the circumscribed square
- The ratio between volume and surface area is maximized in a sphere (the sphere is the most efficient solid)
Frequently asked questions (FAQ)
What is the formula for the area of a circle?
Area = π x r², where r is the radius. If you know the diameter d, the radius is d/2, so Area = π x (d/2)² = π x d² / 4.
How do you calculate the volume of a cylinder?
Volume = π x r² x h, where r is the base radius and h is the height. To get the result in liters, use measurements in decimeters (1 dm³ = 1 liter).
How do you calculate the area of a triangle knowing only the sides?
Use Heron's formula: calculate the semi-perimeter s = (a+b+c)/2, then Area = √(s(s-a)(s-b)(s-c)).
How many liters are in 1 cubic meter?
1 cubic meter = 1,000 liters. Equivalently, 1 liter = 1 cubic decimeter = 1,000 cubic centimeters.
How do you calculate the surface area of a sphere?
Surface area = 4 x π x r². A sphere with radius 10 cm has a surface area of 4π x 100 = 1,256.6 cm².
What is the difference between area and perimeter?
The perimeter measures the length of the outline of a shape (in meters, cm, etc.). The area measures the surface enclosed by the outline (in m², cm², etc.). They are different quantities with different units of measurement.
How do you calculate the area of a trapezoid?
Area = ((longer base + shorter base) x height) / 2. The height is the perpendicular distance between the two parallel bases.
Conclusion
The formulas for calculating areas and volumes are indispensable tools both in the study of geometry and in practical life. From painting a room to choosing a pool, from designing a garden to calculating material for a DIY project, knowing how to calculate surfaces and capacities allows you to plan precisely and save materials and money. Use our area and volume calculator for instant results: select the shape, enter the dimensions, and get the area, perimeter, volume, and surface area with all calculation steps shown.
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