Suryamg Master

Chapter 7: Area

Simulation Topics

Simulation 1: Rectangle and Diagonal Triangle Area

Learning Objective: Explore how rectangle area equals length times width, and triangle area is exactly half of the enclosing rectangle.

Rectangle Area = 28.00 cm squared | Diagonal Triangle Area = 14.00 cm squared

Simulation 2: Parallelogram & Rhombus Area

Learning Objective: Calculate parallelogram area (base times height) and rhombus area (0.5 times product of diagonals).

Parallelogram Area (Base times Height) = 60.00 sq units | Rhombus Area (0.5 times d1 times d2) = 30.00 sq units

Simulation 3: Trapezium Area Calculator

Learning Objective: Compute trapezium area using height and parallel sides a and b: Area = 0.5 times h times (a + b).

Trapezium Area = 0.5 times 8 times (12 + 18) = 120.00 sq ft

Chapter Sections

7.1 Rectangles, Squares & Triangles

Area is measured by counting non-overlapping unit squares contained within a region. Perimeter measures boundary length and is not indicative of area.

Key Area Formulae - Area of a Rectangle = length times width.
- Area of a Triangle = 0.5 times base times height.
- The diagonal of a rectangle divides it into two congruent triangles, each having half the area of the rectangle.

Parallelograms and Rhombuses

By dissection, a parallelogram can be cut along a perpendicular height and rearranged into a rectangle of equal area.

Parallelogram & Rhombus Area Formulae - Area of a Parallelogram = base times height.
- Area of a Rhombus = 0.5 times product of its diagonals (since diagonals are perpendicular bisectors).

Trapeziums & Sulba-SÅ«tras

A trapezium can be split into rectangles and triangles, or combined with a second copy of itself rotated to form a larger parallelogram, yielding the standard formula.

Trapezium Area Formula - Area of a Trapezium = 0.5 times height times sum of parallel sides = 0.5 times h times (a + b).

Real Life Units and Conversions

Areas are measured in cm squared, m squared, in squared, ft squared, acres, and regional units like bigha, gaj, cent, etc.

Conversion Factors - 1 in = 2.54 cm
- 1 ft = 12 in = 30.48 cm
- 1 in squared = 6.4516 cm squared
- 1 acre = 43,560 ft squared

Complete Figure It Out Solutions (Chapter 7)

Figure It Out: Section 7.1 (Page 150)

1. Identify the missing side lengths:
- (i) Given rectangles with areas 28 in squared (width 4 in, length 7 in), 21 in squared (length 7 in, width 3 in), 35 in squared (width 5 in, length 7 in), 14 in squared (length 7 in, width 2 in). Missing lengths are determined using length = Area / width.
- (ii) Rectangles with partial areas 29 m squared and 11 m squared with height 4 m. Total area = 50 m squared. Remaining section area = 50 - (29 + 11) = 10 m squared.

Figure It Out: Triangles (Page 157)

1. Find the areas of the following triangles:
- (i) Base = 4 cm, height = 3 cm. Area = 0.5 times 4 times 3 = 6 cm squared.
- (ii) Base = 5 cm, height = 3.2 cm. Area = 0.5 times 5 times 3.2 = 8 cm squared.
- (iii) Right triangle with legs 3 cm and 4 cm. Area = 0.5 times 3 times 4 = 6 cm squared.

2. Find the length of the altitude BY (Page 158):
Using base AC = 8 units, height AX = 4 units. Area = 0.5 times 8 times 4 = 16. Equating 0.5 times AC times BY = 16 gives BY = 4 units.

3. Find the area of triangle SUB, given it is isosceles, SE is perpendicular to UB, and area of triangle SEB is 24 sq units:
Since triangle SUB is isosceles with altitude SE, SE bisects UB. Thus, triangle SUB is composed of two congruent right triangles (triangle SUE and triangle SEB), each of area 24 sq units. Total area = 24 + 24 = 48 sq units.

Figure It Out: Parallelograms (Page 162-163)

2. Find the areas of the following parallelograms:
- (i) Base = 7 cm, height = 4 cm. Area = 7 times 4 = 28 cm squared.
- (ii) Base = 5 cm, height = 3 cm. Area = 5 times 3 = 15 cm squared.
- (iii) Side = 5 cm, corresponding height = 4.8 cm. Area = 5 times 4.8 = 24 cm squared.

3. Find QN in parallelogram PQRS:
Using base SR = 12 cm and corresponding height QM = 6 cm, Area = 12 times 6 = 72 cm squared. Using side PS = 7.6 cm and altitude QN, Area = 7.6 times QN = 72. Thus, QN = 72 / 7.6 approx 9.47 cm.

Figure It Out: Rhombus & Trapezium (Page 169)

1. Find the area of a rhombus whose diagonals are 20 cm and 15 cm:
Area = 0.5 times product of diagonals = 0.5 times 20 times 15 = 150 cm squared.

3. Find the areas of the following trapezium figures:
- (i) Parallel sides 10 ft and 7 ft, height 16 ft. Area = 0.5 times 16 times (10 + 7) = 8 times 17 = 136 ft squared.
- (ii) Parallel sides 24 m and 36 m, height 14 m. Area = 0.5 times 14 times (24 + 36) = 7 times 60 = 420 m squared.
- (iii) Parallel sides 14 in and 6 in, height 10 in. Area = 0.5 times 10 times (14 + 6) = 5 times 20 = 100 in squared.
- (iv) Parallel sides 12 ft and 18 ft, height 8 ft. Area = 0.5 times 8 times (12 + 18) = 4 times 30 = 120 ft squared.