Perimeter and Area of Plane Figures

• Perimeter of a closed plane figure is the length of its boundary.
• Area of a closed plane figure is the measure of the region (surface) enclosed by its boundary.
• Pythagoras theorem
  In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
• Area of a triangle
  Area of a triangle = (1/2) × base × height
  Also, Area of a triangle = [s (s -a)(s -b)(s-c)] where a, b, c are the lengths of the sides and s = (a +b +c)/2
  Area of an equilateral triangle = 3a²/4, where a is the side.
• Rectangle
  If l = length and b = breadth of a rectangle, then
  perimeter = 2 (l +b)
  length of diagonal = l²+b²
  area = l×b.
• Square
  If a is the length of side of a square, then
  perimeter = 4a
  length of diagonal = 2 a
  area = a²
• Parallelogram
  Area of a parallelogram = base×height.
• Trapezium
  Area of a trapezium = (1/2) (sum of parallel sides)×height
• Circle
  If r is the radius of a circle, then
  length of a diameter = 2r
  circumference = 2r
  area = r²
  Take = 22/7 (unless given otherwise)
• Circular ring (track)
  If R and r are the radii of two concentric circles then
  area of the circular ring (track) = (R²-r²).

Exercise

1. Find the area of an isosceles right triangle, if one of the equal sides is 14 cm long.
2. The sides of a triangle are 975 m, 1050 m and 1125 m. If the field is sold at the rate of Rs 1000 per hectare, find its selling price. [1 hectare = 10000 m²]
3. ABC is an equilateral triangle with side 10 cm. If BD = 7 cm and CD = 5 cm, find the area of the shaded region correct to 2 decimal places.
                              
4. The area of a triangle is 48 cm2. If a side and the corresponding altitude are in the ratio 3:2, find their lengths.
5. Find the area of a regular hexagon of side 6 cm. You can leave the answer in surds.
   [Hint. Join the vertices of the regular hexagon with its center. It is divided into 6 equilateral triangle each of side 6 cm.]
6. In the adjacent 20-sided polygon, all adjacent sides are at right angles. The breadth of the shaded region is 1 cm through out. Find
   (i) the area enclosed in the polygon.
   (ii) the perimeter of the polygon.
7. It costs Rs 936 to fence a square field at Rs 7.80 per meter. Find the area of the square field.
8. A person walks at 3 km/hr. How long will he take to go round a square ground 5 times, the area of which being 2025 m²?
9. In the following figure, ABCDEFG is a heptagon which is symmetrical about the line passing through the vertex A and the mid-point M of the side ED. All measurements are in centimeters. Find the perimeter and the area enclosed by the heptagon.
                    
10. The area of a square plot is 1764 m². Find the length of its one side and one diagonal.
    [Hint. Length of diagonal = 2 x length of a side.]
11. ABCD is a parallelogram with sides AB = 12 cm, BC = 10 cm and diagonal AC = 16 cm. Find the area of the parallelogram. Also find the distance between its shorter sides.
    [Hint. Find area of ABC by Heron formula.
    Area of parallelogram ABCD = 2×area of ABC.]
12. The parallel sides of an isosceles trapezium are in the ratio 2:3. If its height is 4 cm and area is 60 cm², find
    (i) the lengths of parallel sides
    (ii) the perimeter of the trapezium.
13. In the following diagram, all measurements are in centimeters. Find
    (i) the length of AB.
    (ii) the area of the trapezium ABCD.
    [Hint. From C, draw CN perpendicular to AD.]
                      
14. In the following diagram, all measurements are in centimeters. Calculate the area of the shaded region.
                   
15. If the area of a circle is 78.5 cm², find its circumference. Take = 3.14
16. Find the circumference of the circle whose area is 16 times the area of the circle with diameter 7 cm.
17. Find the circumference of the circle whose area is equal to the sum of the areas of three circles with radius 2 cm, 3 cm and 6 cm.
18. From a square cardboard, a circle of biggest area was cut out. If the area of the circle is 154 cm², calculate the original area of the cardboard.
    [Hint. Side of the square board = diameter of the circle.]
19. A bucket is raised from a well by means of a rope which is wound round a wheel of diameter 77 cm. Given that the bucket ascents in 1 minute 28 seconds with a uniform speed of 1.1 m/sec. Calculate the number of revolutions the wheel makes in raising the bucket.
20. A road 3.5 m wide surrounds a circular park whose circumference is 88 m. Find the cost of paving the road at the rate of Rs 60 per square meter.
21. The following sketch shows a running tract 3.5 m wide all around which consists of two straight paths and two semi-circular rings. Find the area of the track.
                       

Answers

1. 98 cm²          2. Rs 47250        3. 27.05 cm²
4. Side = 12 cm, altitude = 8 cm   5. 543 cm²
6. (i) 37 cm²      (ii) 76 cm            7. 900 m²          8.18 minutes
9. 40 cm; 62.15 cm².                     10. 42 m; 59.4 m
11. 119.8 cm²; 11.98 cm              12. (i) 12 cm, 18 cm    (ii) 40 cm
13. (i) 8 cm       (ii) 40 cm²           14. 54 cm2        15. 31.4 cm
16. 88 cm          17. 44 cm          18. 196 cm²      19. 40
20. Rs 20790    21. 1480.5 m²