70.6k views
4 votes
The circular portion of the following figures are semicircles. for each find perimeter and area

The circular portion of the following figures are semicircles. for each find perimeter-example-1
User Gburnett
by
8.4k points

1 Answer

2 votes
Step-by-step 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=(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)=(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.

---------------------------------------------------

(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=(1)/(2)\cdot\pi r^2+(1)/(2)\cdot a\cdot b\cong(1)/(2)\cdot3.14\cdot(5cm)^2+(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=(√(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* A_(SC2)=(1)/(2)\pi r_1^2+2*((1)/(2)\pi r_2^2) \\ \cong(1)/(2)\cdot3.14\cdot(9cm)^2+2\cdot((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* P_(SC2)=\pi r_1+2*(\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

User Stratadox
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories