Question

Find the area of the trapezium field:

Answer 1

Given:-

• OB = 12 cm
• BC = 13 cm
• OC = 5 cm
• OA = 3 cm
• CD = 3 cm

To Find:-

• Area of the trapezium field.

Solution:-

Here we will find the area of triangle and rectangle individually first.

For ∆OCB

• OB = 12 cm
• OC = 5 cm
• BC = 13 cm

Now,

We know:-

Semi - Perimeter of the triangle is given by:-

• $\sf{s = \dfrac{a + b + c}{2}}$

Where a, b, and c are the sides of the triangle.

Hence,

$= \sf{s = \dfrac{12 + 13 + 5}{2}}$

$= \sf{s = \dfrac{30}{2}}$

$= \sf{s = 15}$

Now,

Applying Heron's formula,

We know:-

Heron's formula is as follows:-

• $\sf{Area = \sqrt{s(s - a)(s - b)(s - c)}}$

Putting all the values in the formula:-

Area = √15(15 - 12)(15 - 13)(15 - 5)

Area = √15 × 3 × 2 × 10

Area = √5 × 3 × 3 × 2 × 5 × 2

Area = 5 × 3 × 2

Area = 30 cm²

∴ Area of the triangle is 30 cm².

Now,

For rectangle AOCD,

• AO = 3 cm
• OC = 5 cm
• CD = 3 cm
• AD = OC = 5 cm

Or we can say:-

• Length = 3 cm
• Breadth = 5 cm

We know,

• Area of rectangle = Length × Breadth

Hence,

Area = 3 × 5

Area = 15 cm2

∴ Area of the rectangle = 15 cm².

Finally,

Area of trapezium = Area of triangle + Area od rectangle

Hence,

Area of trapezium = 30 + 15

Area of trapezium = 45 cm²

∴ Area of the trapezium field is 45 cm².

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Additional Information:-

• We use Heron's formula to find the area of a triangle when all the three sides of a triangle is given.

In case of right angled triangle (when all the sides of the triangle is given e.g. in this question the triangle is a right - angled triangle and all the three sides are given) another formula too (other than Heron's formula) can be used to find it's area. The formula is:-

• $\sf{Area = \dfrac{1}{2} \times Length \times Breadth}$

For equilateral triangle too another formula (other than Heron's formula) can be applied to find it's area. The formula is:-

• $\sf{Area = \dfrac{\sqrt{3}}{4} a^2}$

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Answer 2

Answer:

$\pink{ \boxed {\sf\therefore \: Area \:of \: trapezium \: \: A = 45 \: \: {cm}^{2} }}$

Explanation :

Given,

ABCD is a trapezium.

Here,

Height = CO = h = 5 cm

AB and CD are parallel.

length of AB base = (3 + 12) cm = 15 cm

length of CD base = 3 cm

Area of trapezium A = to find

Formula :

The area of a trapezium can be calculated using the lengths of two of its parallel sides and the distance (height) between them.

The formula to calculate the area (A) of a trapezium using base and height is given as,

A = ½ ×(AB + CD) ×h

where,

AB and CD = bases of trapezium, and,

h = height (the perpendicular distance between AB and CD ) 

Solution :

$\sf \: A = \frac{1}{2} \times( AB \times CD) \times h \\ \sf \: \: \: \: \: = \frac{1}{2} \times (15 + 3) \times 5 \\ \: \: \: \: \sf \: = \frac{1}{2} \times 18 \times 5 \\ \: \: \: \: \: \sf = 9 \times 5 \\ \: \: \: \: \: \sf = 45 \: \: {cm}^{2} \\ \\ \green{ \boxed {\sf\therefore \: Area \:of \: trapezium \: \: A = 45 \: \: {cm}^{2} }}$