# 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 =*l*²+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

- 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.

## 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