Rectangle and square
P = a · b, L = 2(a + b)Area90
- Perimeter39
- Diagonal14.151
- Area of a square with side a144
Mathematics
Geometry
Areas and perimeters of shapes, volumes of solids, the Pythagorean theorem and trigonometry.
Most practical calculations — how much paint the walls need, how many liters a tank holds, whether an angle is a right angle — come down to a handful of area and volume formulas. The results come with the quantities derived from them: the diagonal, the slant height, the height and the circumradius.
Calculators
10Triangle from three sides
Heron's formulaArea26.8328
- Perimeter24
- Height to side a7.6665
- Angle opposite side c73.4°
- Circumradius4.6957
Pythagorean theorem
c = √(a² + b²)Hypotenuse5
- Area of a triangle6
- Perimeter12
- Angle at side a53.13°
Circle
P = πr², L = 2πrArea78.5398
- Perimeter31.4159
- Diameter10
- Area of a semicircle39.2699
Trapezoid
P = (a + b) · h / 2Area32
- Midsegment8
- Area of the triangle a · h20
Cuboid
V = a · b · cVolume9,000
- Surface area2,700
- Diagonal39.0512
- Volume in liters (dimensions in cm)9
Cylinder
V = πr²hVolume502.6548
- Lateral area251.3274
- Total surface area351.8584
- Volume in liters (dimensions in cm)0.5027
Sphere and cone
V = 4/3 πr³ ; V = πr²h / 3Volume of a sphere904.7787
- Surface area of a sphere452.3893
- Volume of a cone376.9911
- Slant height of the cone11.6619
Trigonometry of an angle
sin, cos, tan (in degrees)sin0.5
- cos0.866025
- tan0.57735
- In radians0.523599
- Complementary angle60°
Angle from the inverse function
arcsin, arccos, arctanAngle30°
- In radians0.523599
- sin of the angle0.5
- cos of the angle0.866025
Tables and cheat sheets
3Areas and perimeters of plane shapes
h — height, r — radius, d — diagonal| Shape | Area | Perimeter |
|---|---|---|
| square | a² | 4a |
| rectangle | a · b | 2(a + b) |
| triangle | a · h / 2 | a + b + c |
| equilateral triangle | a²√3 / 4 | 3a |
| parallelogram | a · h | 2(a + b) |
| rhombus | d₁ · d₂ / 2 | 4a |
| trapezoid | (a + b) · h / 2 | a + b + c + d |
| circle | πr² | 2πr |
| circular sector | πr² · α / 360° | 2r + 2πr · α / 360° |
Volumes and surface areas of solids
Pp — base area, Pb — lateral area, l — slant height| Solid | Volume | Surface area |
|---|---|---|
| cube | a³ | 6a² |
| cuboid | a · b · c | 2(ab + ac + bc) |
| prism | Pp · h | 2Pp + Pb |
| cylinder | πr² · h | 2πr(r + h) |
| pyramid | Pp · h / 3 | Pp + Pb |
| cone | πr² · h / 3 | πr(r + l) |
| sphere | 4/3 · πr³ | 4πr² |
Special angles
exact and approximate values| Angle | sin | cos | tan | Radians |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | 1/2 = 0,5 | √3/2 ≈ 0,866 | √3/3 ≈ 0,577 | π/6 |
| 45° | √2/2 ≈ 0,707 | √2/2 ≈ 0,707 | 1 | π/4 |
| 60° | √3/2 ≈ 0,866 | 1/2 = 0,5 | √3 ≈ 1,732 | π/3 |
| 90° | 1 | 0 | does not exist | π/2 |
| 180° | 0 | −1 | 0 | π |
Worked examples with full solutions
2How much paint for the walls of a room
A room 5 × 4 m, 2.7 m high. We take off 3 m² of windows and doors. The paint covers 10 m² per liter and we apply two coats.
- Perimeter of the room
2 · (5 + 4) = 18 m - Wall area
18 · 2,7 = 48,6 m² - Subtract the openings
48,6 − 3 = 45,6 m² - Two coats
45.6 · 2 = 91.2 m² to paint - Divide by the coverage
91.2 / 10 = 9.12 liters
Answer: About 9.2 liters of paint are needed, or 10 liters if you buy a little spare.
Volume of a cylindrical tank
A tank has a radius of 40 cm and a height of 1.2 m. How many liters does it hold?
- Put the units on the same footing
r = 40 cm = 0.4 m, h = 1.2 m - Formula for the volume of a cylinder
V = πr²h = π · 0,16 · 1,2 - Calculate
V = 0,6032 m³ - Convert to liters
1 m³ = 1,000 liters → 603.2 liters
Answer: The tank holds about 603 liters.
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See also
NumbersGCD and LCM, prime factors, powers, number bases, Roman numerals and combinatorics.Equations and sequencesLinear and quadratic equations, systems of two equations, arithmetic and geometric sequences.StatisticsMean, median, quartiles, standard deviation, weighted mean and points converted into grades.