Area is the amount of surface enclosed by a closed figure. In this chapter, we revisit and extend area formulas to include trapezium, general quadrilateral, special quadrilaterals, and polygons, and also connect area to real-world contexts.
Key Area Formulas
Rectangle: length × breadth (l × b)
Square: side × side (s2)
Triangle: (1/2) × base × height
Parallelogram: base × height
Rhombus: (1/2) × d1 × d2 (where d1, d2 are diagonals)
Trapezium: (1/2) × (a + b) × h (where a and b are parallel sides and h is height)
General quadrilateral: (1/2) × diagonal × (h1 + h2), where h1 and h2 are perpendicular heights from opposite vertices to the diagonal.
Circles:
Area = π × r2
Circumference = 2πr
Area of ring (annulus) = π(R2 - r2), where R = outer radius, r = inner radius.
Surface Area of Solids (brief introduction):
Cube: 6a2 (a = side)
Cuboid: 2(lb + bh + hl)
Cylinder: 2πr(r + h) (total), 2πrh (curved only)
Find the area of a trapezium with parallel sides 8 cm and 12 cm and height 5 cm.
Area = (1/2) × (8 + 12) × 5 = (1/2) × 20 × 5 = 50 sq cm.
A rhombus has diagonals 10 cm and 14 cm. Find its area.
Area = (1/2) × 10 × 14 = 70 sq cm.
A field is in the shape of a quadrilateral with diagonal 24 m and perpendicular heights 8 m and 6 m from opposite vertices.
Area = (1/2) × 24 × (8 + 6) = (1/2) × 24 × 14 = 168 sq m.
A circle has radius 7 cm. Find its area and circumference.
Area = π × 72 = 22/7 × 49 = 154 sq cm.
Circumference = 2 × 22/7 × 7 = 44 cm.
A ring has outer radius 10 cm and inner radius 6 cm. Find its area.
Area = π(102 - 62) = π(100 - 36) = π × 64 = 22/7 × 64 ≈ 201.1 sq cm.
A parallelogram has base 14 cm and height 9 cm. Find its area.
Area = 14 × 9 = 126 sq cm.
The surface area of a cuboid is 94 sq cm. Its dimensions are l=5, b=4, h=3 cm. Verify.
2(lb + bh + hl) = 2(5×4 + 4×3 + 3×5) = 2(20 + 12 + 15) = 2 × 47 = 94 sq cm. Verified.
- Key formulas summary:
- Triangle: (1/2) × b × h
- Trapezium: (1/2)(a+b)h
- Rhombus/Kite: (1/2) × d1 × d2
- Circle area: πr2
- Ring area: π(R2 - r2)
- Cuboid TSA: 2(lb + bh + hl)
- Cube TSA: 6a2
- Cylinder TSA: 2πr(r + h)
Common mistakes
- Using diameter instead of radius in circle formulas.
- Confusing perimeter (length around) with area (surface covered).
- In trapezium, using slant height instead of perpendicular height.
- Forgetting to multiply by 1/2 in the triangle and trapezium formulas.
Summary
Area formulas for different shapes are derived from simpler rectangles and triangles. Always use perpendicular height, and ensure units are consistent (e.g., cm for lengths → sq cm for area). The connection between 2D area and 3D surface area is a natural extension.