Areas of plane figures
Figure | Area |
|---|---|
Rectangle | length × breadth |
Square | side² |
Triangle | ½ × base × height |
Parallelogram | base × height |
Circle | πr² |
Trapezium | ½ × (sum of parallel sides) × distance between them |
Rhombus | ½ × d₁ × d₂ (product of diagonals) |
General quadrilateral | ½ × d × (h₁ + h₂) |
Trapezium
A trapezium has one pair of parallel sides, a and b, at a distance h apart.
Area = ½ × (a + b) × h
Think of it as the average of the parallel sides multiplied by the height.
Example. A trapezium-shaped field has parallel sides 30 m and 20 m, 12 m apart. Area = ½ × 50 × 12 = 300 m².
Rhombus
The diagonals of a rhombus cut each other at right angles. Each diagonal splits it into two triangles, which gives:
Area = ½ × d₁ × d₂
Example. Diagonals 16 cm and 12 cm: area = ½ × 16 × 12 = 96 cm².
General quadrilateral
Draw a diagonal of length d. The perpendiculars from the other two vertices to it have lengths h₁ and h₂. The quadrilateral is two triangles, so its area is ½ × d × (h₁ + h₂).
Irregular polygons can be split into triangles, rectangles and trapeziums, and the areas added.
Solids: surface area and volume
A solid occupies space. Its surface area is the total area of its faces (measured in square units) and its volume is the space inside it (in cubic units).
Solid | Lateral (curved) surface area | Total surface area | Volume |
|---|---|---|---|
Cuboid (l, b, h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
Cube (side a) | 4a² | 6a² | a³ |
Cylinder (radius r, height h) | 2πrh | 2πr(r + h) | πr²h |
The lateral surface area of a cuboid is the area of its four walls, leaving out the top and bottom. It is what you need to paint the walls of a room.
Example: painting a room. A room is 6 m long, 4 m wide and 3 m high. Area of the four walls = 2 × 3 × (6 + 4) = 60 m².
Example: a cylinder. A closed tank has radius 7 m and height 10 m (take π = 22/7).
Curved surface area = 2 × 22/7 × 7 × 10 = 440 m²
Total surface area = 2 × 22/7 × 7 × (7 + 10) = 748 m²
Volume = 22/7 × 7 × 7 × 10 = 1540 m³
Units of volume and capacity
1 cm³ = 1 mL
1 L = 1000 cm³
1 m³ = 1,000,000 cm³ = 1000 L
Example. How many litres does the tank above hold? 1540 m³ × 1000 = 15,40,000 L.
How changes in size affect volume
If every side of a cube is doubled, its surface area becomes 2² = 4 times and its volume 2³ = 8 times as large.
If the radius of a cylinder is doubled and the height kept the same, the volume becomes 4 times as large, because volume depends on r².
Common mistakes
Using diameter instead of radius in πr².
Mixing units: convert everything to the same unit before calculating.
Using total surface area when the problem needs only the lateral area, such as an open box, a pipe or the walls of a room.
Summary
Trapezium: ½(a + b)h; rhombus: ½d₁d₂.
Cuboid: LSA 2h(l + b), TSA 2(lb + bh + hl), volume lbh.
Cylinder: CSA 2πrh, TSA 2πr(r + h), volume πr²h.
1 m³ = 1000 L, and 1 cm³ = 1 mL.