TOPIC- HERON’S FORMULA

CLASS-IXPRESENTED BY:-JEEVAN LATAK.V.V.P

04/28/2023 Section 7.2 Nack 2

Formulas for Perimeter• Triangles– Scalene: P = a + b + c– Isosceles: P = b + 2s (base + 2 equal sides)– Equilateral: P = 3s (3 x sides)

• Quadrilaterals– Quadrilateral: P = a + b + c + d (sum of the sides)– Rectangle: P = 2b + 2h or P = 2(b + h)

2 x (base + height)– Square (or rhombus): P = 4s (4 x sides)– Parallelogram: P = 2b + 2s or 2(b + s)

(2 bases + 2 sides)

Q. Find the area of triangle whose base is 9cm and height is 4cm.

SOL. B=9cm H=4cmArea of triangle =½×B×H

=½×9×4 =18cm²

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Example• The triangle in the figure is a right triangle

with right angle at A, and sides as marked.• Find the area of this triangle.• We will take AB as the base. Then the height

would be AC, which we can find with the Pythagorean Theorem:

• So, the area is:

2515

A

B

C

2 225 15 625 225 400 20.AC

20

115 20 15 10 150.2

A

Q. Find the area of an equilateral triangle whose one side given as 4cm.

SOL. A=4cmArea of an equilateraltriangle=√3/4a²=√3/4×4 × 4=4√3cm²

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Heron’s Formula• Heron’s Formula is used to find the

area of a triangle when altitudes are unknown, but all three sides are known.

• If the lengths of the sides of the triangle are a, b, and c, then the area is given by the formula

where s is the semiperimeter:

a b

c( )( )( )A s s a s b s c

1 ( )2

s a b c

Q. Find the area of triangle whose sides are 13cm,14cm, and 15cm.

• Sol. If a,b,c are the sides of a triangle and s is the semi-peremeter, then its area is given by

• A=√s(s-a)(s-b)(s-c)• Here, a=13,b=14,c=15• Therefore s=½(a+b+c)=½(13+14+15)=21• A=√s(s-a)(s-b)(s-c)=√21×(21-13)×(21-14)×(21-15)• A=√21×8x7x6=√7x3x8x7x2x3=√7²×4²×3²=7x4x3=

84cm²

Q. Find the area of a triangle,two sides of which are 8cm and 11cm and the perimeter is 32cm.

Sol. Let a,b,c be the sides of the given triangle and 2s be its perimeter such that a=8cm,b=11cm,and 2s=32cm

i.e. s=16cmNow, a+b+c=2s 8+11+c=32 C=13.Therefore s-a=16-8=8,s-b=16-11=5,and s-c=16-13=3Hence Area of given triangle =√s(s-a)(s-b)(s-c)=√16x8x5x3=√8x8x30=8√30cm²

Q.An isosceles triangle has perimeter 30cm. & each of the equal sides is 12cm.Find the area of the triangle.

• Q.There is a slide in a park.One of its side walls has been painted in some colour with a message”Keep the park green and clean”. If the sides of the park are 15m.,11m. &6m,find the area painted in colour.

PRACTISE QUESTIONS Q.Find the area of a quadrilateral ABCD in which AB=7CM.,BC=6CM.,CD=12CM.,DA=15CM.&AC=9CM.

Q.A rhombus shaped field has green grass for18 cows to graze.If eachside of the rhombus is 30cm. &its longer diagonal is 48m,how much area of grass field will each cow be grazing?

04/28/2023 Section 7.2 Nack 11

Brahmagupta’s Formula for the area of a cyclic* quadrilateral

• Semiperimeter = ½(a + b + c + d)• Area = A = (s - a) (s - b) (s - c) (s – d)

*cyclic quadrilateral can be inscribed in a circle so that all 4 vertices lie on the circle.

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