Question

HELP I NEED THIS BEFORE 12:00 PM : For the figures below, assume they are made of semicircles, quarter circles and squares. For each shape, find the area and perimeter. Give your answer as a completely simplified exact value in terms of π (no approximations).

Answer 1

Fig1. The area of the shaded part will be given by the area of the square ABCD. This is because the semi circle that has been cut out is of the same size as the one whose diameter is AD. Thus:
Area=length*width
=24*24
=576 cm²
b] The perimeter of the shape will be:
P=length×4
=24×4
=96 cm

Fig2] a) The area of the shaded part will be:
Area=(1/4×π×r²)-(1/2×b×h)
=(1/4×π×12²)-(1/2×12×12)
=(36π-72) cm²

b) Perimeter
Perimeter of the shaded part will be given by:
P=(1/4×π×d)+AC
AC will be found using Pythagorean theorem:
c^2=a^2+b^2
where
c is the hypotenuse
a and b are the sides
thus
c^2=12^2+12^2
c^2=144+144
c=√288
c=AC=√288
Hence:
P=1/4×π×2×12+√288
=6π+√288 cm

Fig3]
Area=(1/2×π×r²)+(1/2×bh)
=(1/2×π×3²)+(1/2×6×6)
=(4.5π+18) cm²

b] Perimeter:
P=(perimeter of the triangle)+(perimeter of semicircle)
Perimeter of triangle is given by:
p=(AB+CA)
AB=BC=6 cm
CA will be calculated by Pythagorean theorem as above:
c²=a²+b2
c²=6²+6²
c²=36+36
c²=72
c=√72

hence:
p=6+√72

The perimeter of the semi circle will be:
p=1/2×π×d
=0.5×π×6
=3π
thus total perimeter will be:
P=3π+6+√72