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Area

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Page 148: How many different ways can you divide a square into 4 parts of equal area?
Page 148: Try to think of different creative ways to divide a square into 4 parts of equal area.
Page 149: Which rectangle requires more rangoli powder — one with sides 7 cm, 4 cm, or one with sides 8 cm, 3 cm?
Page 150: What is the area of each triangle formed by the diagonal of a 7 cm × 4 cm rectangle?
Page 150: Find two rectangles that are examples of regions where Perimeter of Region 1 > Perimeter of Region 2, but Area of…
Page 150: Identify the missing sidelength in the staircase of 4 rectangles (areas 28, 21, 35, 14 in²).
Page 150: Identify the missing sidelengths for the 29 m², 11 m² and 50 m² regions.
Page 151: (i) What measurements do you need to find the area of the path? Once you identify the lengths to be measured, assign…
Page 151: What other measurements are needed to find the area of a crosspath on a 14 m × 12 m plot?
Page 152: Find the area of the spiral tube of width 1, with arms shrinking from 20 to 15 to 10 to 5.
Page 152: If the sidelength of the square is doubled, what is the increase in the areas of regions 1, 2 and 3?
Page 152: Divide a square into 4 parts with two perpendicular lines, then rearrange them into a larger square with a hole.
Page 153: Which has greater area: ΔXDC or ΔYDC, if both rectangles ABDC are identical?
Page 153: Which has greater area: ΔXDC or ΔYBC, if both rectangles ABDC are identical?
Page 153: Find the area of ΔXDC in Fig. 7.1 (height 4, base DC = 5).
Page 153: How do we get the outer rectangle (needed for the area formula) from a given triangle?
Page 154: Does Area = 1/2 × base × height still hold for a triangle whose height falls outside the base (an obtuse triangle)?
Page 155: Find BY, the altitude from B to AC, given AX = 3, BC = 5, AC = 4.
Page 155: Are the 4 triangles formed by the diagonals of a rectangle (regions 1-4) all of equal area?
Page 155: Why does the line from a vertex to the midpoint of the opposite side split a triangle into 2 equal areas?
Page 156: (i) Which of these triangles has the maximum area, and which has the minimum area? (ii) Which of these triangles has…
Page 157: Analyse whether A lies on the perpendicular bisector of BC.
Page 157: Find the areas of 3 triangles: (i) height 3, base 4 (ii) height 3.2, base ~5 (iii) legs 4, 3.
Page 157: Where does the minimum-perimeter triangle's apex sit, among triangles sharing base BC with apex on line l?
Page 158: Find the altitude BY, given AX = 4 (height to BC = 6, obtuse case) and AC = 8.
Page 158: [Śulba-Sūtras] Give a method to transform a rectangle into a triangle of equal area.
Page 158: [Śulba-Sūtras] Give a method to transform a triangle into a rectangle of equal area.
Page 158: (i) If the area of the red region is 49 sq. units, then what is the area of the blue region? (ii) In another version…
Page 159: M, N are midpoints of XY, XZ — what fraction of ΔXYZ's area is ΔXMN?
Page 159: What is the shortest house → river → water-tank path for Gopal, given both points are on the same side of the river?
Page 159: How can any polygon's area be found, by dividing it into triangles?
Page 160: Find the area of quadrilateral ABCD, given diagonal AC = 22 cm and both B, D are 3 cm from AC.
Page 160: Find the shaded region D-F-E-C's area inside an 18 cm × 10 cm rectangle (with E, F splitting two sides).
Page 160: What measurements would you need to find the area of a regular hexagon?
Page 160: What fraction of the square is blue, when an interior point is joined to all 4 corners and opposite triangles are shaded?
Page 161: Give a method to obtain a quadrilateral whose area is half that of a given quadrilateral.
Page 161: How do we convert a parallelogram into a rectangle of equal area, by dissection?
Page 161: Why does the cut-off triangle ΔAXD exactly complete the trapezium ABCX into a rectangle?
Page 162: Can the area of the parallelogram be determined by taking another side as the base and its corresponding height? Can…
Page 162: Observe the parallelograms in the figure below. (i) What can we say about the areas of all these parallelograms?…
Page 162: Find the areas of 4 parallelograms: (i) 7cm×4cm (ii) 5cm×3cm (iii) 5cm×4.8cm (iv) 2cm×4.4cm.
Page 163: Find QN, the height of parallelogram PQRS to side PS, given SR=12, QM=6, PS=7.6.
Page 163: A rectangle and a parallelogram both have sides 5 cm and 4 cm — which has the greater area?
Page 163: Give a method to obtain a rectangle whose area is twice that of a given triangle. What are the different methods…
Page 164: [Śulba-Sūtras] Give a method to obtain a rectangle of the same area as a given triangle.
Page 164: An isosceles triangle can be converted into a rectangle by dissection in a simpler way — how?
Page 164: [Śulba-Sūtras] Give a method to convert a rectangle into an isosceles triangle by dissection.
Page 164: Compare areas: equilateral triangle vs. square (same side), and 2 equilateral triangles vs. that square.
Page 164: How can a rhombus's diagonals be used to dissect it into a rectangle of equal area?
Page 165: Using XW = AC and WZ = BD/2, derive the rhombus area formula.
Page 165: Area of rhombus ABCD can also be determined by finding the areas of ΔADB and ΔCDB. What formula does this give us?…
Page 166: Find the areas of 3 trapeziums drawn on grid paper, by breaking each into a rectangle and triangle(s).
Page 167: Complete Approach 1 (rectangle + triangles) for the trapezium area formula.
Page 167: Draw BG ∥ AD. Will Approach 2 work for any type of trapezium?
Page 168: Two copies of a trapezium are joined along BC — what shape results, and what type of quadrilateral is it?
Page 169: Find the area of a rhombus with diagonals 20 cm and 15 cm.
Page 169: Give a method to convert a rectangle into a rhombus of equal area using dissection.
Page 169: Find the areas of 4 figures: (i) quadrilateral (10,16,7 ft) (ii) trapezium (24,36,14 m) (iii) 4-sided figure (14,10,6 in) (iv) trapezium (12,8,18 ft).
Page 169: [Śulba-Sūtras] Give a method to convert an isosceles trapezium to a rectangle using dissection.
Page 169: Given the trapezium ABCD, how do we find the vertices of the rectangle EFGH?
Page 170: Using the idea of converting a trapezium into a rectangle of equal area, and vice versa, construct a trapezium of…
Page 170: A regular hexagon is split into a trapezium, an equilateral triangle, and a rhombus — find the ratio of their areas.
Page 170: Show area(trapezium ZYXW) = area(ΔZWB), where A is the midpoint of XY and Z,A,B are collinear.
Page 170: What do you think is the area of an A4 sheet? Its sidelengths are 21 cm and 29.7 cm. Now find its area. What do you…
Page 170: Convert to centimeters: (i) 5 in (ii) 7.4 in.
Page 170: Convert to inches: (i) 5.08 cm (ii) 11.43 cm.
Page 171: How many cm² is 1 in²?
Page 171: Using 1 in² = 6.4516 cm², and 1 ft = 12 in, work out: how many cm² is 25 in²? How many in² is 1 ft²? And roughly how…

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