Question

The circular portion of the following figures are semicircles. for each find perimeter and area

Answer 1

Explanation

To solve this problem, we will use:

(1) Pitagoras Theorem, which states for a right triangle:

$h^2=a^2+b^2.$

Where a and b are the cathetus, and h is the hypotenuse.

(2) The formula for the area of a triangle:

$A_T=\frac{1}{2}\cdot a\cdot b.$

Where a is the height and b is the base.

(3) The formula for the area of a semi-circle:

$A_{SC}=\frac{1}{2}\pi r^2.$

Where r is the radius and π ≅ 3.14.

(4) Length or perimeter of a semi-circle:

$P_{SC}=\pi r.$

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(b) From this figure, we identify:

0. a semicircle of radius r = 5 cm,

,

1. a right triangle with height a = 10 cm and hypotenuse h = √200 cm.

Using Pythagoras Theorem, we have:

$\begin{gathered} (\sqrt{200\text{ }}cm)^2=(10\text{ }cm)^2+b^2, \\ b^2=200\text{ }cm^2-100\text{ }cm^2=100\text{ }cm^2. \\ b=\sqrt{100\text{ }cm^2}=10\text{ }cm. \end{gathered}$

1) The area of the complete figure is the sum of the areas of the semi-circle and triangle:

$A=A_{SC}+A_T.$

Using the formulas and values from above, we get:

$A=\frac{1}{2}\cdot\pi r^2+\frac{1}{2}\cdot a\cdot b\cong\frac{1}{2}\cdot3.14\cdot(5cm)^2+\frac{1}{2}\cdot10cm\cdot10cm\cong89.25cm^2.$

2) The perimeter of the figure is the sum of the length of the sides:

$P=(h+b)+P_S=(\sqrt{200}cm+10cm)+\pi\cdot5cm\cong39.84cm.$

(c) From this figure, we identify:

0. a semi-circle with radius r₁ = 18 cm / 2 = 9 cm,

,

1. two semi-circles with radius r₂ = 9 cm / 2 = 4.5 cm.

1) The area of the figure is given by the sum of the areas of the semi-circles:

$\begin{gathered} A=A_{SC1}+2\times A_{SC2}=\frac{1}{2}\pi r_1^2+2\times(\frac{1}{2}\pi r_2^2) \\ \cong\frac{1}{2}\cdot3.14\cdot(9cm)^2+2\cdot(\frac{1}{2}\cdot3.14\cdot(4.5cm)^2)\cong190.76cm^2. \end{gathered}$

2) The perimeter of the figure is the sum of the perimeters of the semi-circles:

$\begin{gathered} P=P_{SC1}+2\times P_{SC2}=\pi r_1+2\times(\pi r_2) \\ \cong3.14\cdot9cm+2\cdot(3.14\cdot4.5cm)\cong56.52cm. \end{gathered}$Answer

(b) Figure b

• Area ≅ 89.25 cm²

,

• Perimeter ≅ 39.84 cm

(c) Figure c

• Area ≅ 190.76 cm²

,

• Perimeter ≅ 56.52 cm