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The area of the given shape is 46.26cm².
How do we calculate the area of a right-angled triangle?
Area of a triangle = (1/2)*base*height.
For a right-angled triangle, the base and height are the two sides making the right angle, that is, the base and the perpendicular.
∴ Area of a right-angled triangle = (1/2)*base*perpendicular.
How do we calculate the area of a semi-circle?
We know the formula for the area of a circle is πr².
A semi-circle is half of the circle, so the area will also be its half.
∴ Area of a semi-circle = (πr²)/2
What is the Pythagoras Theorem?
According to the Pythagoras theorem, the square of the hypotenuse is the sum of the squares of the other two sides. If we say the hypotenuse is 'c' and the other two sides are 'a' and 'b', then by the theorem:
a² + b² = c².
How do we solve the given question?
The given shape consists of one right-angled triangle and a semi-circle. To calculate its area, we first find the area of the right-angled triangle and the semi-circle individually and then add them to calculate the area of the shape.
Area of a right-angled triangle = (1/2)*base*perpendicular,
or, Area of a right-angled triangle = (1/2)*6*6 = 18cm².
We find AC using the Pythagoras theorem,
AC² = AB² + BC² = 6² + 6² = 72.
or, AC = √72 = √(36*2) = 6√2cm
The radius of the semi-circle (r)= (1/2)*diameter = (1/2)*6√2 = 3√2cm.
Area of a semi-circle = (πr²)/2,
or, Area of the semi-circle = (π*(3√2)²)/2 = 18π/2 = 9π.
Now, we can find the total area, by adding the two areas.
Area of the shape = Area of a right-angled triangle + Area of the semi-circle.
or, Area of the shape = 18 + 9π = 18 + 9*3.14 = 18 + 28.26 = 46.26cm²
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