Question

exercise 5

1. calculate the area and perimeter of the following shapes. all measurements are in cm.
2. you can use your calculator in this question. calculate the area of each figure. round off your answers to two decimal places.

Explanation:

Step1: Analyze shape (a)

The shape can be divided into rectangles. First, find the height of the upper - most rectangle: $h_1 = 2$, and the length of the bottom rectangle is $l = 10$, width $w_1=2$. The other rectangles on the sides have width $w_2 = 2$.
The area of the bottom rectangle is $A_1=10\times2 = 20$. The two side - rectangles have combined area $A_2 = 2\times(2\times2)=8$. The top rectangle has area $A_3=2\times2 = 4$. So the total area $A=a_1 + a_2+a_3=20 + 8+4=32$ $cm^2$.
For the perimeter, count the outer - side lengths: $P=2 + 2+2+2+2+2 + 10+2=24$ $cm$.

Step2: Analyze shape (b)

The shape can be divided into two rectangles. One rectangle has dimensions $l_1 = 8$ and $w_1 = 2$, and the other has dimensions $l_2=10 - 2=8$ and $w_2 = 5$.
The area of the first rectangle is $A_1=8\times2 = 16$. The area of the second rectangle is $A_2=8\times5 = 40$. So the total area $A = 16+40=56$ $cm^2$.
For the perimeter, $P=10+8+5+2+(10 - 2)+(8 - 5)=36$ $cm$.

Step3: Analyze shape (c)

The shape is a kite. The area of a kite is given by $A=\frac{1}{2}\times d_1\times d_2$, where $d_1=(6 + 6)=12$ and $d_2=(15 + 5)=20$. So $A=\frac{1}{2}\times12\times20 = 120$ $cm^2$.
For the perimeter, use the Pythagorean theorem. In right - triangle $ABD$, the hypotenuse $AB=\sqrt{6^{2}+5^{2}}=\sqrt{36 + 25}=\sqrt{61}$. In right - triangle $BCD$, the hypotenuse $BC=\sqrt{6^{2}+15^{2}}=\sqrt{36+225}=\sqrt{261}$. The perimeter $P = 2(\sqrt{61}+\sqrt{261})\approx2(7.81+16.16)=2\times23.97 = 47.94$ $cm$.

Step4: Analyze shape (a) in part 2

The shape can be divided into two rectangles. One rectangle has dimensions $l_1=4.75$ and $w_1 = 1.75$, and the other has dimensions $l_2=8.25$ and $w_2 = 2$.
The area of the first rectangle is $A_1=4.75\times1.75=\frac{475}{100}\times\frac{175}{100}=\frac{475\times175}{10000}=\frac{83125}{10000}=8.3125$. The area of the second rectangle is $A_2=8.25\times2 = 16.5$. So the total area $A=8.3125+16.5 = 24.8125\approx24.81$ $cm^2$.

Step5: Analyze shape (b) in part 2

The shape can be divided into three rectangles. One rectangle has dimensions $l_1 = 5$ and $w_1 = 1$, another has dimensions $l_2=8 - 1=7$ and $w_2 = 1$, and the third has dimensions $l_3 = 2$ and $w_3 = 1$.
The area of the first rectangle is $A_1=5\times1 = 5$. The area of the second rectangle is $A_2=7\times1 = 7$. The area of the third rectangle is $A_3=2\times1 = 2$. So the total area $A=5 + 7+2=14$ $cm^2$.

Answer:

Shape (a): Area = 32 $cm^2$, Perimeter = 24 $cm$; Shape (b): Area = 56 $cm^2$, Perimeter = 36 $cm$; Shape (c): Area = 120 $cm^2$, Perimeter $\approx47.94$ $cm$; Part 2 - Shape (a): Area $\approx24.81$ $cm^2$; Part 2 - Shape (b): Area = 14 $cm^2$