Composite Area Problems

Composite Area Problems exercise

• Choose the geometric shapes that the pendant is composed of.

  Hints

  Look at the cat ears and count the corners.

  A shape with three corners is a triangle.

  The pendant is composed of six shapes. Two of these shapes are the same.

  Solution

  To determine the area of a composite shape, we first have to decide which shapes it is composed of.

  In the case of Notorious C.A.T.'s pendant, we have:

  • Two ears in the form of right triangles (the blue ones)
  • One rectangular (the yellow one)
  • One half circle (the red one)
  • Two circles (the green ones), which we have to subtract from the sum of the areas of the other shapes, for the eyes of the pendant.

• Calculate the area of the pendant.

  Hints

  Use the lengths given in the description.

  Keep in mind that the radius of a circle is half of the diameter:

  • $r=\frac12 d$
  • $d=2r$

  Plug the given values into the corresponding formula for each shape.

  Solution

  We already know the shapes that make up the pendant:

  • two triangles (the ears)
  • one rectangular
  • one half circle
  • two circles (the eyes)

  Now, we still have to calculate their corresponding areas to get the total area of the pendant.

  Let's start with the ears, the triangles:

  • $A_{\text{triangle}}=\frac12(\text{base})(\text{height})$
  • Plugging in $2$ inches for both the base and the height gives us $A_{\text{triangle}}=\frac12(2)(2)=2~in^2$

  Now the rectangle:

  We only know the width. The height has to be calculated by subtracting $2$, the height of the ears, and $3$, the radius of the half circle, from $9$. So we get $9-2-3=4$ inches for the height. Then we have:

  • $A_{\text{rectangle}}=(\text{width})(\text{height})$
  • Thus we have $A_{\text{rectangle}}=(6)(4)=24~in^2$

  Now we determine the area of the circle with the radius $1$ inch:

  • $A_{\text{circle}}=\pi(\text{radius})^2$
  • Plugging in the given radius, we get $A_{\text{circle}}=\pi(1)^2=\pi~in^2$

  Last we calculate the area of the half circle using the same formula as for the circle. We just have to multiply it with $\frac12$. The radius is the half of the diameter $6$ inches, so $3$ inches:

  • $A_{\text{half circle}}=\frac12 \pi(\text{radius})^2$
  • With the radius $3$ inches, we get $A_{\text{half circle}}=\frac12\pi(3^2)=4.5\pi~in^2$

  All the shape areas are calculated, so we are able to determine the area of the pendant:

  $\begin{array}{rcl} A_{\text{composite area}}&=&2(2~in^2)+24~in^2+4.5\pi~in^2-2(\pi)~in^2\\ &=&28~in^2+2.5\pi~in^2\\ &\approx& 35.85~in^2 \end{array}$

• Determine the area.

  Hints

  You can separate this shape into one rectangle and two triangles.

  Both triangles have the same area.

  The width of the rectangle is $4$ units and the height is $8$ units.

  Use the following formulas:

  • Rectangle: $A_{\text{rectangule}}=(\text{width})(\text{height})$
  • Triangle: $A_{\text{triangle}}=\frac12(\text{base})(\text{height})$

  Solution

  The shape above can be split up into a rectangle and two triangles with the same area. So we use the formulas,

  • Rectangular: $A_{\text{rectangular}}=(\text{width})(\text{height})$
  • Triangle: $A_{\text{triangle}}=\frac12(\text{base})(\text{height})$

  We need to determine the height as well as the width of the rectangle. For this, we count the squares:

  • height: $8$ units, and
  • width: $4$ units.

  This leads to an area of $A_{\text{rectangular}}=(8)(4)=32$ units$^2$.

  Next, we determine the base and the height of the triangle by counting the squares as well:

  • base: $2$ units, and
  • height: $2$ units.

  We plug those values into the formula to get $A_{\text{triangle}}=\frac12(2)(2)=2$ units$^2$.

  Finally, we add both the area of the rectangle and twice the area of the triangle to get the total area of the shape: $A=32+(2)(2)=32+4=36$ units$^2$.

• Calculate the area of the wooden heart.

  Hints

  Use the formula for circle to get the area of the half circle by multiplying it by $\frac12$:

  $A_{\text{half circle}}=\frac12\pi(\text{radius})^2$.

  The radius is half the diameter.

  Don't forget to add the results.

  $\pi$ is approximately equal to $3.14$.

  Solution

  This wooden heart is given by two half circles and one square.

  The half circles have a radius of $\frac22=1$ inch. This is half of the diameter. We plug this in the formula to get,

  $A_{\text{half circle}}=\frac12\pi(\text{radius})^2=\frac12\pi(1)^2=\frac12\pi\approx1.57$ $in^2$.

  Next, we calculate the area of the square, which is given as the square of the side length:

  $A_{\text{square}}=(\text{side})^2=(29)^2=4$ $in^2$.

  Eventually, we are able to calculate the whole area:

  $A=A_{\text{square}}+2~A_{\text{half circle}}=4+(2)(1.57)=4+3.14=7.14~in^2$.

• Find the formula for the area of each given shape.

  Hints

  The area of a half circle is half of the area of a circle.

  $\pi$ is used to determine the area or the circumference of a circle.

  A right triangle is half of a rectangle.

  Solution

  To calculate the areas we want, we need a few formulas:

  • Rectangle: $A_{\text{rectangle}}=(\text{width})(\text{height})$
  • Triangle: $A_{\text{triangle}}=\frac12(\text{base})(\text{height})$
  • Circle: $A_{\text{circle}}=\pi(\text{radius})^2$
  • Half circle: $A_{\text{half circle}}=\frac12\pi(\text{radius})^2$

• Find the correct statements about the colored shape.

  Hints

  Use the following formulas:

  • Rectangle: $A_{\text{rectangular}}=(\text{width})(\text{height})$
  • Triangle: $A_{\text{triangle}}=\frac12(\text{base})(\text{height})$

  The width of the bigger rectangle is the height of the smaller one.

  You need $b$ for each area.

  Solution

  The colored shape is given by a triangle and a rectangle with another rectangle cut out of it. So we have to calculate the area of the bigger rectangle and the smaller rectangle, and then subtract the area of the smaller one from the area of the bigger one.

  So, let's determine each area:

  First, the area of the bigger rectangle:

  • $A_{\text{rectangular}}=(\text{width})(\text{height})$,
  • $A_1=(a)(b)$.

  Similarly, we get the area of smaller rectangle:

  $A_2=(c)(b)$.

  Lastly, we determine the area of the triangle:

  • $A_{\text{triangle}}=\frac12(\text{base})(\text{height})$
  • $A_3=\frac12(b)(d)$

  Almost done. We still have to add and to subtract a little bit to get the area of the colored shape:

  $A=A_1-A_2+A_3$.

  Let's take the results above to get

  $A=(a)(b)-(a)(c)+\frac12(b)(d)$.

  Each term has the same factor $b$. Thus, we can factor it out:

  $A=b\left(a-c+\frac12d\right)$.