In geometry, it is often necessary to calculate the area of a figure. Simple and almost intuitive formulas exist for basic figures like the rectangle or the square. However, they can become more complicated for other configurations. The unit of measurement for area depends on the system used. Indeed, in the metric or international system, it is the square meter, symbol m2. If you are working with imperial measurements, the unit of measurement for area is the square inch or square inch in English and has the symbol in2.
Part 1 of 3: Calculate the area of a square or rectangle
Step 1. Measure the length of the rectangle
The square and the rectangle are particular quadrilaterals. Indeed, the sides of a rectangle are equal two by two and it has at least three right angles. Measure its longest side and write down THE the value found.
Step 2. Measure the width of the rectangle
Note the the value. To get the area of the rectangle, you just need to measure the length of two adjacent sides, which are two segments with a common end.
For the square, only one measurement is enough since all four sides are equal. In other words, the length is equal to the width
Step 3. Multiply the length by the width
If we take the previous notations, the area of a rectangle is equal to L × W.
- Or a rectangle 4 cm long and 3 cm wide. Its area is therefore 4 × 3 or 12 cm2.
- The area formula for the rectangle also works for the square. Given its characteristics and if we note vs the length of a side, the area of a square is given by the formula c × c or more simply vs2. The area of a square with a side of 4 cm is therefore 42 i.e. 16 cm2.
Part 2 of 3: Calculate the area of a circle or polygon
Step 1. Calculate the area of a circle
The formula is Π × r2. The variable r is the radius of the circle, which is the distance from the center to one of the points on the circle. This data can be provided to you in a statement or measured against the ruler. Notice that the formula involves the mathematical constant Π (pi) which is about 3.14.
- Consider a circle with a radius of 4 cm. Its area is therefore Π × 42, that is Π × 16 = 50, 24 cm2. If you don't have a scientific calculator, round up Π at 3, 14.
Step 2. Calculate the area of any triangle
The formula is b × h × 1/2. The variables are the length of the base b and height h. The base is one of the sides of the triangle, and the height is the perpendicular through the base and the opposite vertex. If necessary, draw the height in order to measure it.
- Consider a triangle with a base measuring 4 cm and the associated height measuring 3 cm. The area of the figure is therefore 4 × 3 × 1/2 or 6 cm2.
Step 3. Calculate the area of a parallelogram
This figure is a particular quadrilateral, because its sides are equal two by two. If it has a right angle, then it is a rectangle. The formula for calculating the area of a parallelogram is b × h, with b, the length of one side and h, that of the associated height. This is perpendicular to a side passing through the opposite vertex. Note that for the rectangle, the height is confused with one side of the figure due to the right angle.
- Consider a parallelogram whose length is 4 cm and the associated height is 5 cm. Its area is then 4 × 5 or 20 cm2.
Step 4. Calculate the area of a trapezoid
A trapezoid is a quadrilateral with two parallel sides, called bases, and two non-parallel sides. Its area is given by the formula (B + b) × h × 1/2, with B, the length of the large base, b, that of the small base and h, that of the associated height. This is the segment perpendicular to the two bases. Note that a trapezoid can be divided into simpler figures such as a rectangle and a triangle. This decomposition can help you find its area.
- Consider a trapezoid whose large and small base measure 6 and 4 cm respectively and the height is 5 cm. Its area is therefore 5 × (6 + 4) × 1/2 or 25 cm2.
Step 5. Calculate the area of a regular hexagon
It is a polygon whose six sides are equal. You can calculate its area using different formulas, more or less difficult to remember. One of them is P × a × 1/2, with P, the perimeter of the hexagon and To, the length of the apothem. This is a segment perpendicular to one side and passing through the center of the polygon. The perimeter is simply the sum of the lengths of the sides. If we note vs the length of one side, the perimeter is 6 × c.
- The regular hexagon can be broken down into six equilateral triangles, sideways vs. To find its area, you can calculate that of a triangle and multiply it by six. If we note To, the height of a triangle, its area is therefore c × a × 1/2. Indeed, in a regular hexagon, the apothem merges with a height of a triangle that composes it. The area of the hexagon is therefore 3 × a × c. Notice that we find the formula with the perimeter. If we replace P through 6 × c, then the formula given in the previous point gives 6 × c × a × 1/2 that is 3 × a × c. Consider a hexagon with a side of 4 cm and whose apothem measures 3.5 cm. The perimeter is 24 cm and the area is 24 × 3, 5 × 0, 5 or 42 cm2. Using the other formula, the area is 3 × 3, 5 × 4 or also 42 cm2.
Step 6. Find the area of a regular octagon
It is a polygon whose eight sides are equal. If we note vs the length of one side, the area of the octagon is given by the formula 2 × c2 × (1 + √2). The only variable being the length of the side of the octagon, you just need to measure it to find its area.
- Or an octagon with a side of 4 cm. The area is therefore 2 × 42 × (1 + √2) or 77.25 cm2.
Part 3 of 3: working with imperial units
Step 1. Familiarize yourself with the Anglo-Saxon system
The unit of measurement for area is the square inch and that of length is the inch or inch in English and the symbol in. There are different orders of magnitude. To carry out your calculations, you may need to convert the imperial units to each other or to switch from the imperial system to the metric system and vice versa. The main equivalences are:
- 1 in = 2.54 cm
- 1 foot = 12 in = 30, 48 cm
- 1 yard = 36 in = 91.44 cm
- 1 m = 39.3701 in
- 1 cm = 0.3937 in
Step 2. Convert imperial measures
As in the international system, the Anglo-Saxon system includes variations of the units of area. In addition to the square inch, there is also the square foot or square foot in English and the symbol ft2. To switch between units, multiply the square footage value by 144 to get its square inch equivalent.
- For example, an area of 400 ft2 equivalent to 57,600 in2.
Step 3. Convert square inches to square centimeters
One inch is about 2.54 cm, one square inch is 2.542 i.e. 6.45 cm2. To carry out your conversion, you must therefore multiply the value in square inches by 6.45. For example, we therefore have only 250 in2 equivalent to 1612.5 cm2 down to rounding.
- To do the reverse conversion, note that one centimeter is 0.394 in. So a square centimeter is 0.155 in2. To get a measurement in square inches, you must therefore multiply the value in square centimeters by 0.155. If you are working on higher values, know that a square meter is worth 10,000 cm.2 and a square kilometer is worth 1010 cm2. A square kilometer is therefore equivalent to 1,550,003,100 in2.