Lesson 1: Area Calculations for Basic Shapes

Knowing how to calculate areas can help you to assess deck space available for cargo, waterplane areas, and other vessel dimensions. Area calculations are also vital to understanding the principles of ship stability. You will use various area calculations in your Ship Stability course.

The area of a shape can be defined as the amount of surface inside it or enclosed by it. In this lesson we will review the formulae for calculating the areas of some very basic regular shapes like: rectangles, squares, parallelograms, circles, triangles and trapezoids. Here is a Formulae sheet for your reference.

Note: The unit for area is the square of the unit of measurement or unit². The units of measurement must be the same for each side of a given shape when calculating its area.

Rectangle

The area of a rectangle is given by the product of its length (l) and its width (w).

Using variables to represent the dimensions: $A = l \times w$

If the length and width are measured in metres (m) then the units of A will be m²

Example 1: Find the area of a rectangle with length 5m and width 7m

Solution:

We are given that length (l) = 5 m and width (w) = 7m, therefore the area (A) of this rectangle will be:

A = 5 m \times 7 m = 35 m^{2}

Loading the player...

Square

A square is a rectangle whose length is equal to its width. Each side will have the same length: $l = w = s$

Therefore, the area (A) of a square will be given by: $A = l \times w = s \times s = s^{2}$

Example 2: Find the area of a square with each side equal to 8 km

Solution:

A = s^{2} = {(8 k m)}^{2} = 64 k m^{2}

Loading the player...

Parallelogram

A parallelogram is a four-sided shape with two pairs of parallel sides.
When calculating the area of a parallelogram we call one side the base (b) and the perpendicular distance to the opposite parallel side the height (h).
The area of a parallelogram is given by: $A = b \times h$

Example 3: Find the area of this parallelogram

Solution:

We are given that b = 3 m and h = 250 cm. Note that the units of b and h are different, so before we can calculate the area of this parallelogram we must ensure both base and height have the same units. In this example let us change cm to m. Recall

1 c m = \frac{1}{100} m

. Therefore,

h = 250 cm = 250 \times \frac{1}{100} m = 2.5 m

Now, applying the formula to find the area of a parallelogram we get:

A = b \times h = 3 m \times 2.5 m = 7.5 m^{2}

Loading the player...

Triangle

	A triangle is a three sided shape, with 3 angles or vertices. To calculate the area of a triangle we choose one of its sides as its base (b) and then measure the perpendicular distance from the base to the opposite vertex. This distance is the triangle height (h), also called the altitude. We use the base and height to calculate the area: $A = \frac{1}{2} \times b \times h$
This formula can also be derived from the formula for finding the area of a parallelogram (see the adjacent figure). In this example, a triangle is seen to be just half of a parallelogram and hence the area of a triangle is also half of the corresponding parallelogram.

Example 4: Find the area of a triangle with base = 12 m and altitude = 6 m

Solution:

Since

A = \frac{1}{2} \times b \times h

and

b = 12 m and h = 6 m

. Therefore, the area of this triangle will be:

A = \frac{1}{2} \times 12 m \times 6 m = 36 m^{2}

Loading the player...

Circle

The area of a circle with radius 'r' is given by $A = π \times r^{2}$ where $π$ is a constant. The value of $π$ can be approximated as $\frac{22}{7} = 3.142$ (rounding to 3 decimal places). You can also use the $π$ key on your calculator, unless a particular approximated value has been specified for the problem you are solving.

You can also watch a video demonstation on:

Example 5: Find the area of a circle with radius 5 dm. Express the final answer in cm²

Solution:

We know that the area of a circle is given by

A = π r^{2}

There are two ways to solve this problem:

Convert dm into cm before calculating the area.
Calculate area and then convert the units.

Let us take a look at each of these methods.

Method 1: In the last module you learned that 1 dm= 10 cm, hence: $\begin{array}{l} 5 d m = 5 \times 10 c m = 50 c m \\ A = π {(50 cm)}^{2} = 7853.98 {cm}^{2} \end{array}$ Method 2:

$\begin{array}{l} We know that 1 d m^{2} = 1 d m \times 1 d m and 1 d m = 10 c m, therefore \\ 1 d m^{2} = 1 d m \times 1 d m \\ = 10 c m \times 10 c m \\ = 100 c m^{2} \end{array}$

\begin{array}{l} A = π {(5 d m)}^{2} \\ = 78.5398 d m^{2} \\ = 78.5398 \times 100 c m^{2} \\ = 7853.98 c m^{2} \end{array}

\begin{array}{l}  \end{array}

Loading the player...

Trapezoid

A trapezoid is a four-sided figure with one pair of parallel sides. The parallel sides are called bases and the perpendicular distance between them is the height or altitude.
The area of a trapezoid is given by: $A = \frac{1}{2} \times h \times (b_{1} + b_{2}) or \frac{h}{2} (b_{1} + b_{2})$

Example 6: Find the area of the following trapezoid.

Solution:

\begin{array}{l} A = \frac{h}{2} (b_{1} + b_{2}) = \frac{9 c m}{2} (16 c m + 8 c m) \\ = \frac{9 c m}{2} (24 c m) \\ = 4.5 c m \times 24 c m \\ = 108 c m^{2} \end{array}

Loading the player...

Example 7: A barge has a water plane area of 5334 m² and a beam of 42 m. If the bow is an isosceles triangle with base angles 45^o and its stern is semi-circular, find the overall length of the barge. (Use $π = \frac{22}{7}$ )

Solution:

First of all draw a diagram to make it easier to understand the problem:
barge_shape

Now we know that half of the beam length is equal to the radius of the semi-circular stern, which is given as l₁ = 21 m. So, we can calculate the area of the stern by calculating the area of a half circle with a radius of 21 m.

= \frac{1}{2} \times π \times 2 1^{2} = \frac{1}{2} \times \frac{22}{7} \times 21^{2} = 693 m^{2}

Also, since the formula for the area of a rectangle is length times width, we can calculate Area 2:

= 42 \times l_{2} m^{2}

Lastly, considering the bow section (Area 3):

from the geometry of the triangle, l₃ = 21 m, so that:
$= \frac{1}{2} \times 42 \times 21 = 441 m^{2}$

Hence,

\begin{array}{l} T o t a l a r e a = (693 + 42 l_{2} + 441) m^{2} \\ o r, 5334 = 693 + 42 l_{2} + 441 \\ 42 l_{2} = 4200 \\ l_{2} = 100 \end{array}

We already know that, l₁ = 21 m and l₃ = 21 m
Therefore,

T o t a l l e n g t h = 21 m + 100 m + 21 m = 142 m

Loading the player...

Lesson 1: Area Calculations for Basic Shapes

Rectangle

Example 1: Find the area of a rectangle with length 5m and width 7m

Square

Example 2: Find the area of a square with each side equal to 8 km

Parallelogram

Example 3: Find the area of this parallelogram

Triangle

Example 4: Find the area of a triangle with base = 12 m and altitude = 6 m

Circle

Example 5: Find the area of a circle with radius 5 dm. Express the final answer in cm2

Trapezoid

Example 6: Find the area of the following trapezoid.

Example 7: A barge has a water plane area of 5334 m2 and a beam of 42 m. If the bow is an isosceles triangle with base angles 45o and its stern is semi-circular, find the overall length of the barge. (Use π= 22 7 )

Example 5: Find the area of a circle with radius 5 dm. Express the final answer in cm²

Example 7: A barge has a water plane area of 5334 m² and a beam of 42 m. If the bow is an isosceles triangle with base angles 45^o and its stern is semi-circular, find the overall length of the barge. (Use $π = \frac{22}{7}$ )