Revision and important questions: Area

Area20 min readLesson 5 of 6

5.1 The formula box Learn each formula with the meaning of its letters and the condition under which it applies. Rectangle: length \times width. Square: side \times side. Triangle: ½ \times base \times height, with the height perpendicular to the chosen base. Parallelogram: base \times height. Rhombus: ½ \times product of the diagonals. Trapezium: ½ \times height \times sum of the parallel sides. Polygon: split into triangles and add their areas. Conversions: 1 in = 2.54 cm, so 1 in² = 6.4516 cm²; 1 ft = 12 in; 1 acre = 43,560 ft²; 1 m² = 10,000 cm².

A_{rect} = l w A_{tri} = \frac{1}{2} bh

Rectangle and triangle areas

A_{para} = bh A_{rhombus} = \frac{1}{2} d_{1} d_{2}

Parallelogram and rhombus areas

A_{trap} = \frac{1}{2} h (a + b)

Trapezium area

5.2 Reading the formula box The rectangle formula is the seed of the others. The triangle takes half a rectangle, the parallelogram becomes a rectangle after a cut, the rhombus is four right triangles or a rectangle of dimensions half its diagonals, and the trapezium is a rectangle plus two triangles. Each formula therefore records a dissection, not a separate rule to memorise blindly. When a figure looks unfamiliar, look for a way to cut it into rectangles and triangles and use the formulas you already trust. Before substituting, write every length in the same unit; a single stray conversion is the commonest cause of a wrong area. Then read the answer back into the figure and check that it is sensible.

5.3 Common mistakes Using a slanted side as the height. The height must be perpendicular to the base. Forgetting the ½ in the triangle formula, which doubles or halves the answer. Mixing units: multiplying a length in inches by a length in centimetres without converting first. Adding two triangles that overlap, as in a crosspath, without subtracting the overlap once. Thinking that doubling every side doubles the area. The area is multiplied by 4, because both dimensions double. Using the same side twice in a parallelogram instead of pairing a side with its own perpendicular height. Treating the diagonal of a rectangle as a side when computing perimeter or area of a triangle. Rounding unit conversions too early. Keep exact factors until the final step. Using the diagonal of a rectangle as both length and width when multiplying for area.

5.4 Ten exam-style questions with answers A rectangle is 12 cm by 5 cm. Find its area and perimeter. Answer: area 60 cm², perimeter 34 cm. A triangle has base 8 cm and height 5 cm. Find its area. Answer: ½ \times 8 \times 5 = 20 cm². A parallelogram has base 6.5 cm and height 4 cm. Find its area. Answer: 26 cm². A rhombus has diagonals 18 cm and 12 cm. Find its area. Answer: ½ \times 18 \times 12 = 108 cm². A trapezium has parallel sides 9 cm and 15 cm and height 6 cm. Find its area. Answer: ½ \times 6 \times 24 = 72 cm². A triangle has area 42 cm² and base 12 cm. Find its height. Answer: ½ \times 12 \times h = 42, so h = 7 cm. A rectangle is 9 cm by 4 cm. What is the area of each triangle formed by a diagonal? Answer: 36 \div 2 = 18 cm². One square inch is 6.4516 cm². Convert 5 in² to cm². Answer: 5 \times 6.4516 = 32.258 cm². A square has side 3 cm; its side is doubled. By what factor does the area increase? Answer: the area becomes 4 times larger, so the increase is 3 times the original area. A rectangle is 30 cm by 12 cm. Find the total area of the two triangles formed by a diagonal. Answer: the rectangle's area is 360 cm², so the two triangles together have the same 360 cm² and each triangle has 180 cm².

\frac{1}{2} \times 12 \times h = 42 \Rightarrow h = 7

Exam question 6: finding the height from the area

5 \times 6.4516 = 32.258

Exam question 8: converting square inches to square centimetres

5.5 Exam strategy For a figure made of rectangles, label every unknown side and work from the rectangle whose area and one side are both known. For a composite shape, either add the pieces or subtract the unshaded parts from the whole, whichever needs fewer steps. Write the formula before substituting numbers, and keep the units visible on every line. In questions about doubling, remember that area scales by the square of the length factor. A useful final check is to read the answer back into the figure. Does a path area of 124 m² fit inside a 300 m² park? Does a triangle's height come out shorter than its longest side? Does a trapezium's area lie between the area of the rectangle formed by its shorter parallel side and the area formed by its longer one? These sanity checks catch swapped bases, missing halves and unit slips before the examination hall does, and they take only a few seconds each. When a question supplies a diagram, redraw it roughly and mark every known length before choosing a formula; those ten seconds usually prevent a wrong base, a missed half or a forgotten unit.

in hinglish

Revision ke liye formula box yaad rakho: rectangle l \times w, square side², triangle ½ \times base \times height (height perpendicular ho), parallelogram base \times height, rhombus ½ \times d₁ \times d₂, trapezium ½ \times height \times (a + b) . Polygon ko triangles mein tod kar areas jodo. Conversions mein 1 in² = 6.4516 cm², 1 acre = 43,560 ft². Common mistakes : slanted side ko height samajhna, triangle mein ½ bhoolna, units mix karna, crosspath ka overlap do baar ginna, side double hone par area ko double samajhna (asal mein 4 guna), parallelogram mein side ka apna height use na karna, aur conversion factor ko jaldi round karna. Exam mein pehle formula likho, phir values daalo, aur units check karo. Composite figure mein pehle pieces pehchaano, phir areas jodo ya subtract karo; aur conversion factor ka square tabhi lagao jab area convert kar rahe ho.