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.
If l = length and b = breadth of a rectangle, then
perimeter = 2 (l +b)
length of diagonal = l²+b²
area = l×b.
If a is the length of side of a square, then
perimeter = 4a
length of diagonal = 2 a
area = a²
Area of a parallelogram = base×height.
Area of a trapezium = (1/2) (sum of parallel sides)×height
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).
- Find the area of an isosceles right triangle, if one of the equal sides is 14 cm long.
- 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]
- 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.
- The area of a triangle is 48 cm2. If a side and the corresponding altitude are in the ratio 3:2, find their lengths.
- 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.]
- 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.
- It costs Rs 936 to fence a square field at Rs 7.80 per meter. Find the area of the square field.
- 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?
- 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.
- 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.]
- 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.]
- 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.
- 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.]
- In the following diagram, all measurements are in centimeters. Calculate
the area of the shaded region.
- If the area of a circle is 78.5 cm², find its circumference. Take = 3.14
- Find the circumference of the circle whose area is 16 times the area of the circle with diameter 7 cm.
- 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.
- 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.]
- 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.
- 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.
- 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.
Answers1. 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