Be Careful Units Count Use The Same Units For All Measurements Examples
square = a 2
|one half times the base length times the height of the triangle|
triangle given SAS = a b sin C
triangle given a,b,c = when s = /2
regular polygon = n sin S2 when n = # of sides and S = length from center to a corner
Area is measured in “square” units. The area of a figure is the number of squares required to cover it completely, like tiles on a floor.
Area of a square = side times side. Since each side of a square is the same, it can simply be the length of one side squared.
If a square has one side of 4 inches, the area would be 4 inches times 4 inches, or 16 square inches.
Be sure to use the same units for all measurements. You cannot multiply feet times inches, it doesn’t make a square measurement.
The area of a rectangle is the length on the side times the width. If the width is 4 inches and the length is 6 feet, what is the area?
NOT CORRECT …. 4 times 6 = 24
CORRECT…. 4 inches is the same as 1/3 feet. Area is 1/3 feet times 6 feet = 2 square feet. .
Area Of Compound Shapes
There are many cases where the calculation of a total area requires more than one area to be calculated followed by either an addition, subtraction, or some other combination of operations to find the required area.
Note: In the examples below the units of measurement are not shown and answers and the value of have been rounded to the nearest hundredth.
Area Of A Square Or Rectangle
Before we begin going through the motions to calculate the area of a square or rectangle lets take a quick visit back to perimeter and more specifically how we defined the dimensions of both a square and a rectangle.
Once again the portion shaded grey is the area of each of the object. The question becomes this
Can we use those dimensions when calculating the area? Or do we need to somehow come up with other dimensions in order to get to our answer?
Well as it turns out, those dimensions will work for us. Not only do they come in handy when we are calculating perimeter, but they also work really well when calculating area.
The next question then becomes HOW do we use these dimensions?
Before you move on to see how its done, take a minute to think about it. Maybe even write down some of your thoughts. Once again, this goes back to something we talked about before. If you are able to understand the concept then the process involves less memorization.
Here are the formulas for calculating the area of both a square and a rectangle.
\Large \begin\textbf & \text = \text \times \text\\ \textbf& \text= \text\times \text\end
Take note here as there are a couple things different than dealing with perimeter. The first is that we are now multiplying rather than adding. This ends up leading to our second point.
With area we end up with units that give us an answer using two dimensional units. The best way to understand this is to go through an example.
\Large \text \times \text = ?
Area And Perimeter Of A Parallelogram
The parallelogram has two sets of opposite sides that run parallel to one another. The shape is a quadrangle, so it has four sides: two sides of one length and two sides of another length .
To find out the perimeter of any parallelogram, use this simple formula:
- Perimeter = 2a + 2b
When you need to find the area of a parallelogram, you will need the height . This is the distance between two parallel sides. The base is also required and this is the length of one of the sides.
- Area = b x h
Keep in mind that the b in the area formula is not the same as the b in the perimeter formula. You can use any of the sideswhich were paired as a and b when calculating perimeterthough most often we use a side that is perpendicular to the height.
What Is The Area And Perimeter Of The Square
Area of a square: The area of the square is defined as the region covered by the two-dimensional shape. The units of the area of the square are measured in square units i.e., sq. cm or sq. m.
Perimeter of a square: The perimeter of the square is a measure of the length of the boundaries of the square. The units of the perimeter are measured in cm or m.
Surface Area And Volume Of A Pyramid
A pyramid with a square base and faces made of equilateral triangles is relatively easy to work with.
You will need to know the measurement for one length of the base . The height is the distance from the base to the center point of the pyramid. The side is the length of one face of the pyramid, from the base to the top point.
- Surface Area = 2bs + b2
- Volume = 1/3 b2h
Another way to calculate this is to use the perimeter and the area of the base shape. This can be used on a pyramid that has a rectangular rather than a square base.
- Surface Area = + A
- Volume = 1/3 Ah
Learning About Area In Primary School
Children are introduced to area in Year 4, where they will be asked to find the area of various shapes, simply by counting the 1cm² squares they occupy on paper:
In Year 5 children will be expected to use the formula to work out the area of a rectangle. They will often be given rectangles not drawn to scale, so will need to remember this formula. They also need to estimate the area of irregular shapes.
In Year 6, children will need to work out how to find the area of an irregular shape, such as the following. Often, not all the measurements of each side will be given to make this harder.
A good way of finding the area of a shape like this is to split the shape up into smaller shapes and then work out the area of each of these. The areas of the smaller shapes can then be added up to find the answer.
Children in Year 6 also learn to calculate the area of parallelograms and triangles .
Children will sometimes be asked to solve worded puzzles or investigations about area, where no pictorial representation is given, for example:
A rectangle has a perimeter of 36cm. What could the area of this shape be?
There is more than one possible answer to this question. One way of working out a possible answer would be to draw a rectangle and then work out what the sides could be if the perimeter is 36cm.
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Example : Surface Area Of A Cube
Work out the surface area of the cube:
Calculate the area of each face.
Since it is a cube, all of the faces are the squares and of equal size.
Area of a square formula: \text \times \text
So, the area of each face =9 \times 9=81cm^2
Add the areas together.
There are 6 faces, each with an area of 81cm^2
6 \times 81=486
Write the answer, including the units.
The measurements of the cube are in cm so the area will be measured in cm^2
Total surface area = 486cm^2
S On How To Calculate Area Of Rectangle
The steps on how to calculate the area of a rectangle are mentioned below. If you follow these steps properly, you will never get errors in your solutions.
Step 1: Write down the dimensions of the given rectangle from the questions.
Step 2: Put the values in the area of rectangle formula, i.e., length x width.
Step 3: Multiply the values and get the product.
Step 4: Write the result in square units
To further understand how to calculate the area of a rectangle, consider the following example. Well calculate the area of a rectangle with a length of 20 units and a width of 5 units.
Step 1: Given the length of the rectangle = 20 units, breadth of the rectangle = 5 units
Step 2: Area of rectangle formula = length x breadth
Step 3: 20 x 5 = 100
Step 4: The area of the rectangle is 100 square units.
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Area Of A Circle Calculator
The area of a circle calculator helps you compute the surface of a circle given a diameter or radius. Our tool works both ways – no matter if you’re looking for an area to radius calculator or a radius to the area one, you’ve found the right place
We’ll give you a tour of the most essential pieces of information regarding the area of a circle, its diameter, and its radius. We’ll learn how to find the area of a circle, talk about the area of a circle formula, and discuss the other branches of mathematics that use the very same equation.
Perimeter And Area Of Square
The Perimeter and Area of the Square are used to measure the length of the boundary and space occupied by the square. These are two important formulas used in Mensuration. Perimeter and Area of the Square formulas are used in the 2-D geometry.
Square is a regular quadrilateral where are the sides and angles are equal. The concepts of the Perimeter and Area Square formula, Derivation, Properties, are explained here. The solved examples with clear cut explanations are provided in this article. Students can understand how and where to use the formulas of Area and Perimeter of Square.
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What Is Area In Math Area Definition
Simply speaking, area is the size of a surface. In other words, it may be defined as the space occupied by a flat shape. To understand the concept, it’s usually helpful to think about the area as the amount of paint necessary to cover the surface. Look at the picture below – all the figures have the same area, 12 square units:
There are many useful formulas to calculate the area of simple shapes. In the sections below you’ll find not only the well-known formulas for triangles, rectangles, and circles, but also other shapes, such as parallelograms, kites or annulus.
We hope that after this explanation you won’t have any problems defining what an area in math is!
Units For Measuring Area
We measure area using squares. We use different sizes of squares depending on how big or small an area is.
|Dont forget the wee 2|
|We write square sizes using a small 2 next to the unit.We write mm2, cm2, m2 , km2, cm2We can say 63 millimeters squared or 63 square millimeters|
We could use small squares to measure large areas. The only problem with this is that we would end up having to use very big numbers. For example, a field might be measured at 5,000,000,000 square millimeters when 5,000 square meters would be a much easier size to say, write, and visualize.
You will probably hear more units for measuring area square inches, square feet, square yards, square miles, acres, hectares are all units used for measuring area.
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Calculating Area Using The Grid Method
When a shape is drawn on a scaled grid you can find the area by counting the number of grid squares inside the shape.
In this example there are 10 grid squares inside the rectangle.
In order to find an area value using the grid method, we need to know the size that a grid square represents.
This example uses centimetres, but the same method applies for any unit of length or distance. You could, for example be using inches, metres, miles, feet etc.
In this example each grid square has a width of 1cm and a height of 1cm. In other words each grid square is one ‘square centimetre’.
Count the grid squares inside the large square to find its area..
There are 16 small squares so the area of the large square is 16 square centimetres.
In mathematics we abbreviate ‘square centimetres’ to cm2. The 2 means squared.
Each grid square is 1cm2.
The area of the large square is 16cm2.
Counting squares on a grid to find the area works for all shapes as long as the grid sizes are known. However, this method becomes more challenging when shapes do not fit the grid exactly or when you need to count fractions of grid squares.
In this example the square does not fit exactly onto the grid.
We can still calculate the area by counting grid squares.
- There are 25 full grid squares .
- 10 half grid squares 10 half squares is the same as 5 full squares.
- There is also 1 quarter square .
- Add the whole squares and fractions together: 25 + 5 + 0.25 = 30.25.
You can also write this as 30¼cm2.
How To Calculate Area
Well, of course, it depends on the shape! Below you’ll find formulas for all sixteen shapes featured in our area calculator. For the sake of clarity, we’ll list the equations only – their images, explanations and derivations may be found in the separate paragraphs below .
Are you ready? Here are the most important and useful area formulas for sixteen geometric shapes:
- Square area formula: A = a²
- Rectangle area formula: A = a * b
- Triangle area formulas:
Want to change the area unit? Simply click on the unit name and a drop-down list will appear.
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Explanation Of The Area Of A Circle Formula
Take a circle and divide it into equally sized sectors and rearrange these as shown below. Notice how, as the sectors become smaller, the shape becomes more like a rectangle. Note: There is no limit to how small these sectors could be and to how closely they could resemble a rectangle when arranged.
Assuming we know that the circumference of a circle is equal to 2r we can add dimensions to the rectangle as shown below. Using the area of a rectangle area formula, area = width x height we can see how our circle, re-configured as a rectangle, can be shown to have an area that approximates to r x r or r2
Circle Sectors Rearranged
Circle Sectors Rearranged Starting to Look Like a Rectangle
Area And Perimeter Of A Triangle
The triangle is one of the simplest shapes and calculating the perimeter of this three-sided form is rather easy. You will need to know the lengths of all three sides to measure the full perimeter.
- Perimeter = a + b + c
To find out the triangle’s area, you will need only the length of the base and the height , which is measured from the base to the peak of the triangle. This formula works for any triangle, no matter if the sides are equal or not.
- Area = 1/2 bh
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Area Of An Octagon Formula
To find the area, all you need to do is know the side length and the formula below:
The octagon area may also be calculated from:
A perimeter in octagon case is simply 8 * a. And what is an apothem? An apothem is a distance from the center of the polygon to the mid-point of a side. At the same time, it’s the height of a triangle made by taking a line from the vertices of the octagon to its center. That triangle – one of eight congruent ones – is an isosceles triangle, so it’s height may be calculated using e.g., Pythagoras’ theorem, from the formula:
h = * a / 4
So finally we obtain the first equation:
Practice Surface Area Questions
1. Find the surface area of the rectangular prism below.
Calculating the area of each face, we have:
Total surface area: 16+16+24+24+6+6=92cm^2
2. Calculate the surface area of the triangular prism.
Calculating the area of each face, we have:
Total surface area = 30+30+96+104+40=300cm^2
3. Work out the surface area of the prism.
Calculating the area of each face, we have:
Total surface area = 20+20+160+40+100+100=440mm^2
4. Find the surface area of the cylinder. Give your answer to 3 significant figures.
\begin\text& =2 \pi rh\\\\& =2 \times \pi \times 6.5 \times 9\\\\& =117\pi\end \begin\text& =\pi r^\\\\& =\pi \times 6.5^\\\\& =\frac\pi\end Total surface area: 117\pi+\frac\pi+\frac\pi=\frac\pi \frac\pi=633.0309197 Surface area = 633m^2 \
5. Work out the surface area of the cone. Give your answer to 3 decimal place.
\begin\text& =\pi rl\\\\& =\pi \times 2 \times 2.7\\\\& =5.4\pi\end \begin\text& =\pi r^\\\\& =\pi \times 2^\\\\& =4\pi\end Total surface area: =5.4\pi+4\pi=9.4\pi Surface area = 29.5cm^2 \
6. Calculate the surface area of the sphere. Give your answer to 2 decimal place.
\begin\text& =4 \pi r^\\\\& =4 \times \pi \times 0.4^\\\\& =0.64\pi \\\\& =2.010619298\end Surface area = 2.01m^2 \
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How To Find The Area Of A Triangle
Assume that we know two sides and the angle between them: