Perimeter

1. We use millimetres (mm), centimetres (cm), metres (m) and kilometres (km) as units for measuring lengths or distances.

The units are related as follows:

1km = 1000m; 1m = 100cm; 1cm = 10mm

2. Perimeter of a circle is referred to as the circumference. The circumference, C, of a circle of radius, r, can be obtained by using this formula, circumference = 2 × π × r where π = 3.14 or ²²⁄₇.

Area

3. Area of a closed figure is the amount of space it covers. It is measured in square units.

4. Note:

a) 1 cm² = 1 cm × 1 cm

          = 10 mm × 10 mm

          = 100 mm²

b) 1 m² = 1 m × 1 m

          = 100 cm × 100 cm

          = 10 000 cm²

c) 1 km² = 1 km × 1 km

                = 1 000 m × 1 000 m

                = 1 000 000 m²

Hectare (ha) is a unit for measuring large land areas such as farms.

1 ha = 10 000 m²

5. Area of a triangle = ¹⁄₂ × base × height

6.  Area of a parallelogram =  base × height

7. Area of a trapezium = ¹⁄₂ × (sum of parallel sides) × height

8. Area of a circle = π × r², where π = 3.14 or ²²⁄₇.

Volume

Volume of an object is the amount of space it occupies.  We use cubic metre (m³), cubic centimetre (cm³) and millimetre (mm³) as units for measuring volumes.

9. Note:

a) 1 cm³ = 1 cm × 1 cm × 1 cm

= 10 mm × 10 mm × 10 mm

= 1000 mm³

b) 1 m³ = 1 m × 1 m × 1 m

= 100 cm × 100 cm × 100 cm

= 1 000 000 cm³

c) 1 km³ = 1 km × 1 km × 1 km

= 1 000 m × 1 000 m × 1 000 m

= 1 000 000 000 m³

10. Volume of a cuboid = length × breadth × height

11. Volume of a cube = length × length × length

12. A cylinder is a special prism with a circular cross-section. Hence, its volume can be found by multiplying the area of the circular base by its height.

Volume of cylinder = Πr²h

Surface Area

13. A cube has 6 equal faces. Therefore, its surface area is obtained by multiplying the area of one face by 6.

14.The lateral surface of a cylinder is also called the curved surface. If we rolled up a piece of rectangular sheet, the rectangular sheet will become the curved surface of the cylinder.

Area of rectangle = curved surface area of the cylinder

                             = 2πrh

Hence, total surface area of a solid cylinder = area of curved surface + 2 × area of base

                                                                      = 2πrh + 2πr²

Volume and Surface Area of a Pyramid

A pyramid is a solid that has a base with a perpendicular vertex and slant lateral faces.

The base can be any polygon with 3 sides or more (triangle, a square, or a rectangle, etc.)

Volume of pyramid = ¹⁄₃ × base area × height                          Base area = s²        

Surface area of 4 sides = ¹⁄₂ × s × l × 4 sides

                                         = 2sl 

Surface area of pyramid = ¹⁄₂ × base area + total area of slant ∆ faces

                                       = s² + 2sl        

Volume and Surface Area of a Cone

A cone is a solid with a circular base and a vertex.

Arc length = circumference of circular base
Curved surface area of cone = πrl

Volume of cone = ¹⁄₃πr²   

Surface area of a cone
= Area of circle + Area of curved surface                     

= πr² + πrl

r = radius

l = length of slant height =√(r²+h²)

Volume and Surface Area of a Sphere and Hemisphere (Half-Sphere)

Every point on the surface of a sphere is equidistance from the centre.

A hemisphere is half a sphere.

Volume of sphere = ⁴⁄₃πr³                                  

Surface area of a sphere = 4πr²       

Volume of hemisphere = ¹⁄₂  × volume of sphere = ²⁄₃πr³   

Surface area = ¹⁄₂  × spherical surface area + area of circle

= 2πr²+ πr² = 3πr²

Volume and Surface Area of a Prism

Volume of prism = Base area × Height                             

Or  Area of cross-section × Length

Surface area of a prism = Area of all the faces

Or 2 × Cross-Sectional Area + (Perimeter of Cross Section) × length of prism

Volume of triangular prism = ¹⁄₂ × b × h × l

Radian Measure

Radian is another unit of measure for angles, similar to degrees.

One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

Conversion from radians and degrees

2π radians = 360°    

π radians = ^360°⁄₂ = 180°    

1 radian = ^180°⁄_π

Conversion from degree to radians

360°= 2π radians      

180° = π radians

1°= ^π⁄₁₈₀ rad

Conversion between radians and degrees

Part of a Circle

Arc Length

If θ is measured in degree then arc length = ^θ°⁄_{360° × 2πr}

If θ is measured in radian then arc length = ^θ°⁄_{2π × 2πr = rθ}

Area of Sector 

Area of circle = πr²

If θ is measured in degree then area of sector = ^θ°⁄_{360 × πr² = rθ}

If θ is measured in radian then area of sector = ^θ°⁄_{2π × πr² =} ¹⁄_{2 r²θ}

Area of Segment

Area of segment = Area of Sector ‒ Area of Triangle

                               = ^θ°⁄_360° × πr² ‒ ¹⁄₂ r² sinθ  (if θ is a degree)                          

                               = ¹⁄₂ r²θ ‒¹⁄₂ r²sinθ  (if θ is a radian)  

Important: If angles are in radian, change the calculator to radian mode.

For more mensuration formulas and tips, check out our free math eBooks and test papers for secondary school students!

Last Minute Revision for O Level Math?

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