{"id":9905,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9905"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"area-rhombus","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/area-rhombus\/","title":{"rendered":"Area Rhombus"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Area Of Shapes<\/strong><\/h2>\n

Chapter: <\/strong>Area of Rhombus<\/strong><\/h3>\n

Reference: – What is a Rhombus, Area of Rhombus Using Diagonals (Formula and Derivation), Area of Rhombus Using Base and Height, Finding a Diagonal Given Area and Other Diagonal, Real-Life Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • How to Find the Area of a Rhombus Using Diagonals<\/em><\/li>\n
  • How to Find the Area of a Rhombus Using Base and Height<\/em><\/li>\n
  • How to Find a Missing Diagonal Given Area<\/em><\/li>\n
  • When to Use Each Formula<\/em><\/li>\n<\/ul>\n

    Introduction to Area of Rhombus<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    A rhombus is a parallelogram with all four sides equal. It looks like a slanted square or a diamond shape. Like other quadrilaterals, a rhombus encloses a certain amount of space called its area. Unlike a square, the angles of a rhombus are not necessarily 90°.<\/p>\n

    When we calculate the area of a rhombus, we essentially ask:<\/p>\n

    "How many square units fit inside this diamond shape?"<\/p>\n

    There are two different methods to find the area of a rhombus, and both are useful in different situations.<\/p>\n

    Importance of Area of Rhombus<\/u><\/strong><\/p>\n

      \n
    • Used in kite making and diamond cutting<\/li>\n
    • Essential for tiling and flooring patterns<\/li>\n
    • Helps in solving geometry problems involving quadrilaterals<\/li>\n
    • Foundation for advanced geometry and trigonometry<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      A rhombus with diagonals 6 cm and 8 cm has an area of 24 cm². The same rhombus has a base of 5 cm and height of 4.8 cm, giving the same area.<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Properties of a Rhombus (Quick Review)<\/strong><\/p>\n

        \n
      • All four sides are equal in length<\/li>\n
      • Opposite sides are parallel (it is a parallelogram)<\/li>\n
      • Opposite angles are equal<\/li>\n
      • Adjacent angles are supplementary (sum to 180°)<\/li>\n
      • Diagonals bisect each other at right angles (90°)<\/li>\n
      • Diagonals bisect the interior angles<\/li>\n<\/ul>\n

        2. Area of Rhombus – Method 1 (Using Diagonals)<\/strong><\/p>\n

        This is the most common formula for the area of a rhombus.<\/p>\n

        Formula:<\/strong> Area = (1\/2) × d\u2081 × d\u2082<\/p>\n

        Where d\u2081 and d\u2082 are the lengths of the two diagonals.<\/p>\n

        Derivation:<\/strong> The diagonals of a rhombus divide it into four congruent right triangles. Each triangle has legs equal to half of each diagonal (d\u2081\/2 and d\u2082\/2). The area of one triangle is (1\/2) × (d\u2081\/2) × (d\u2082\/2) = d\u2081d\u2082\/8. Multiplying by 4 triangles gives total area = 4 × (d\u2081d\u2082\/8) = d\u2081d\u2082\/2.<\/p>\n

        Example 1:<\/strong> A rhombus has diagonals 10 cm and 12 cm. Find its area.<\/p>\n

        Area = (1\/2) × 10 × 12 = 60 cm²<\/p>\n

        Example 2:<\/strong> A rhombus has diagonals 14 m and 20 m. Find its area.<\/p>\n

        Area = (1\/2) × 14 × 20 = 140 m²<\/p>\n

        Example 3:<\/strong> A rhombus has diagonals 9 cm and 16 cm. Find its area.<\/p>\n

        Area = (1\/2) × 9 × 16 = 72 cm²<\/p>\n

        3. Area of Rhombus – Method 2 (Using Base and Height)<\/strong><\/p>\n

        Since a rhombus is a parallelogram, we can also use the parallelogram area formula.<\/p>\n

        Formula:<\/strong> Area = base × height<\/p>\n

        Where base is the length of any side, and height is the perpendicular distance between that side and its opposite side.<\/p>\n

        Example 1:<\/strong> A rhombus has side length 8 cm and height 5 cm. Find its area.<\/p>\n

        Area = 8 × 5 = 40 cm²<\/p>\n

        Example 2:<\/strong> A rhombus has side length 12 m and height 7 m. Find its area.<\/p>\n

        Area = 12 × 7 = 84 m²<\/p>\n

        Note:<\/strong> The height is NOT the length of the other side. It is the perpendicular distance between parallel sides.<\/p>\n

        4. Finding a Diagonal Given Area<\/strong><\/p>\n

        If the area and one diagonal are known, we can find the other diagonal.<\/p>\n

        Formula:<\/strong> d\u2082 = (2 × Area) ÷ d\u2081 (or d\u2081 = (2 × Area) ÷ d\u2082)<\/p>\n

        Example:<\/strong> A rhombus has area 80 cm² and one diagonal is 10 cm. Find the other diagonal.<\/p>\n

        d\u2082 = (2 × 80) ÷ 10 = 160 ÷ 10 = 16 cm<\/p>\n

        5. Finding Side Length from Diagonals<\/strong><\/p>\n

        Since the diagonals are perpendicular and bisect each other, they form four right triangles. The side of the rhombus is the hypotenuse of a right triangle with legs equal to half of each diagonal.<\/p>\n

        Formula:<\/strong> side = √[(d\u2081\/2)² + (d\u2082\/2)²] = (1\/2) × √(d\u2081² + d\u2082²)<\/p>\n

        Example:<\/strong> A rhombus has diagonals 6 cm and 8 cm. Find its side length.<\/p>\n

        Half diagonals = 3 cm and 4 cm
        \nSide = √(3² + 4²) = √(9 + 16) = √25 = 5 cm<\/p>\n

        Solved Examples<\/u><\/strong><\/p>\n

        Example 1 – Area Using Diagonals:<\/strong> Find the area of a rhombus with diagonals 15 cm and 24 cm.<\/p>\n

        Solution:<\/strong> A = (1\/2) × 15 × 24 = (1\/2) × 360 = 180 cm²<\/p>\n

        Answer:<\/strong> 180 cm²<\/p>\n

         <\/p>\n

        Example 2 – Area Using Base and Height:<\/strong> A rhombus has side length 9 m and height 6 m. Find its area.<\/p>\n

        Solution:<\/strong> A = base × height = 9 × 6 = 54 m²<\/p>\n

        Answer:<\/strong> 54 m²<\/p>\n

         <\/p>\n

        Example 3 – Finding Missing Diagonal:<\/strong> A rhombus has area 120 cm² and one diagonal is 20 cm. Find the other diagonal.<\/p>\n

        Solution:<\/strong> d\u2082 = (2 × 120) ÷ 20 = 240 ÷ 20 = 12 cm<\/p>\n

        Answer:<\/strong> 12 cm<\/p>\n

         <\/p>\n

        Example 4 – Finding Side from Diagonals:<\/strong> The diagonals of a rhombus are 10 cm and 24 cm. Find the side length.<\/p>\n

        Solution:<\/strong> Half diagonals = 5 cm and 12 cm
        \nSide = √(5² + 12²) = √(25 + 144) = √169 = 13 cm<\/p>\n

        Answer:<\/strong> 13 cm<\/p>\n

        Common Mistakes to Avoid<\/u><\/strong><\/p>\n

        Mistake 1 – Using side² for area of rhombus<\/strong>
        \nOnly a square has area = side². A non-square rhombus has less area than side².
        \nCorrect understanding: Use A = (1\/2) × d\u2081 × d\u2082 or A = base × height.<\/p>\n

        Mistake 2 – Forgetting the 1\/2 in diagonal formula<\/strong>
        \nA = d\u2081 × d\u2082 (without 1\/2) gives twice the actual area.
        \nCorrect understanding: Always include (1\/2) or divide by 2.<\/p>\n

        Mistake 3 – Using slanted side as height<\/strong>
        \nHeight is the perpendicular distance between parallel sides, not the length of the slanted side.
        \nCorrect understanding: Draw a perpendicular line inside the shape to find height.<\/p>\n

        Mistake 4 – Confusing the diagonals<\/strong>
        \nBoth diagonals are used in the formula, not just one.
        \nCorrect understanding: Multiply both diagonals, then divide by 2.<\/p>\n

        Mistake 5 – Forgetting to take half of diagonals when finding side<\/strong>
        \nSide = √[(d\u2081\/2)² + (d\u2082\/2)²], not √(d\u2081² + d\u2082²).
        \nCorrect understanding: Divide each diagonal by 2 before squaring.<\/p>\n

        Mistake 6 – Thinking all rhombuses are squares<\/strong>
        \nA rhombus can have acute and obtuse angles (like a diamond).
        \nCorrect understanding: A square is a special type of rhombus, but not all rhombuses are squares.<\/p>\n

         <\/p>\n

        Quick Reference Summary<\/u><\/strong><\/p>\n

        Rhombus:<\/strong> Parallelogram with all sides equal<\/p>\n

        Area Formula (Diagonals):<\/strong> A = (1\/2) × d\u2081 × d\u2082<\/p>\n

        Area Formula (Base & Height):<\/strong> A = b × h<\/p>\n

        To find missing diagonal:<\/strong> d\u2082 = (2 × A) ÷ d\u2081<\/p>\n

        To find side from diagonals:<\/strong> s = (1\/2) × √(d\u2081² + d\u2082²)<\/p>\n

        Perimeter of Rhombus:<\/strong> P = 4s<\/p>\n

        Key Fact:<\/strong> In a rhombus, diagonals are perpendicular bisectors of each other<\/p>\n

        Square is a rhombus:<\/strong> When all angles = 90°, A = s² also = (1\/2) × d²<\/p>\n

         <\/p>\n

        <\/p>\n