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Let's look at how to find the area of several common geometrical shapes, such as triangles, circles, parallelograms and rectangles, as well as less common shapes.
The area of a triangle can be found by different methods: using lengths, or usingtrigonometry.
The area of a triangle is
, in which b corresponds to the length of the base of the triangle, and h corresponds to its height directly perpendicular to the base.
Height and base of triangles
Questions may ask you to find the area of a triangle from its dimensions or find an unknown length from its area.
Find the area of a triangle with a base length of 7 cm and height of 50 cm.
Here,
cm and
cm.
Using the area formula,
The area of the triangle is 17.5 cm²
Find the base length and height of a triangle with an area of 9cm², knowing that the height is twice as long as the base.
Let the length of the base b = x, therefore h = 2x.
Substituting known values into the area equation,
Hence,
Since x is a length, -3 is discarded and x = 3.
The length of the base is 3cm and the length of the height is 2 x 3 = 6cm
The area of a triangle can be calculated if 2 lengths and the angle between them is known (case of SAS - side, angle, side)
triangle with sides andangleslabelled a, b, ct
In this case, the formula for the area of a triangle is
.
The known lengths are multiplied by the sin of the angle between them.
The two formulas used to calculate the area of the triangle seem different, but the first equation, using simply lengths, is derived from the second equation usingtrigonometry.
Let us prove this:
Find the area of a triangle using the formula
If the lengths a and b are perpendicular to each other, one is the base of the triangle and the other is the height.Additionally, the angle between them would be 90 ° or π / 2radians.Since
, this formula can be written as
, which is equivalent to the first formula. Here'san example using the trigonometric form of the formula.
What is the area of a triangular backyard to the nearest square metre, when two adjacent fences of the backyard have lengths of 10 and 12m, and the angle between them is 78°?
We will use the trigonometric formula for this question, as we are given 2 sides and 1 angle.
The area of the backyard is 59 m², to the nearest square metre.
The area of a circle can be calculated by using the formula
, in which r represents the radius of the circle (half of the length of the diameter) and π is the irrationalnumber3.14159265 ...
π is a constantratiolinking the circumference of a circle to its diameter.
Anysector of a circlecan also be calculated by taking a fraction of the area.This fraction is
.For example, to calculate the area of a semicircle, the formula
(or
) is used, as a semicircle is a sector consisting of half of a circle.
Find the area of a circle of 3cm radius, leaving your answer in exact form.
We will use the formula
.substituting the known radius,
Hence, the circle has area 9π cm², in exact form.
What is the area of a sector of angle 34 ° of a circle radius 2km?Give your answer to two decimal places.
Since the area of a sector of the circle must be calculated, only a fraction of its area will be taken.
To calculate this, we will take a fraction
of the full area of the circle, since a circle consists of 360 degrees.
Hence, the area of the sector
Since the question asks for an answer accurate to 2 dp, the area of the sector is 1.19 km²
The area of a parallelogram can be calculated from the length of its base and its perpendicular height.The formula linking these together is
, in which b corresponds to the length of the base of the parallelogram and h corresponds to its perpendicular height.This formula is linked to that for the area of a triangle, as a parallelogram can always be split into 2 identical triangles when the parallelogram is separated by its diagonal.
The area of a parallelogram is double the area of one of the triangles it can be split into:
The area of a parallelogram can also be derived from the trigonometric method to find the area of a triangle.
The area of a parallelogram can therefore be calculated as
:, in which a and b are two adjacent lengths, and C is the angle between them.
Parallelogram split into two triangles
Let's look at an example to practice:
If the base of a parallelogram is three times its height, and the area is 108m², find the base.
Let the height = x.The base is three times the height so base = 3x.
Let us formulate an equation for the area in terms of x:
The area of the parallelogram is 3x², or 108m².
Since x is a length it cannot be negative, so x = 6m.
The base is 3x, hence the base = (3) (6) = 18m.
The area of a rectangle is found in a similar method to the area of a parallelogram.The formula for the area of a rectangle is
, where the width and length are perpendicular to each other.
A square is a specific type of rectangle in which all sides are equal.Both w and l are equal to a specific value.Let that value be x.The area of a square is simply
Here's an example:
If the area of a rectangle is 120m², and one of its sides is 6m long, what is its perimeter?
To find the missing length, we will use the formula for the area and rearrange it.
The length of the missing side is 20m.
A rectangle consists of 2 sets of 2 parallel identical sides.
Hence, the perimeter is
m.
To find the area of less common or unknown shapes, it may be necessary to divide this shape into smaller shapes for which the area is easily calculated.重要的是要认识到简单的形状从密苏里州re complex shapes, such as semicircles attached to each end of a rectangle forming a running track shape:
矩形with semicircles
Additionally, regular polygons with n sides can be divided into n equally sized and shaped isosceles triangles.For example, a square (four sides) can be divided into four triangles, as shown.
Square split into four triangles
The Australian 50 cent coin has the shape of a regular dodecagon (12 sides).Eight of these 50 cent coins can fit on an Australian $5 note.What fraction of the note is not covered?
Since we do not know a formula to find the area of the dodecagon directly, to solve this problem, we will need to divide the shapes into smaller, known shapes.
The dodecagon is a regular polygon, and can therefore be split into twelve equally sized isosceles triangles with a vertex in the centre of the dodecagon, as shown.
Dodecagon. Image: public domain
Two of the sides of each triangle have equal lengths, corresponding to the radius of the dodecagon (the length from the centre to any vertex).The angle between those lengths is
degrees.
The dimensions of the coins and the note are unknown.Let radius = r.Hence we can calculate the area of each triangle in terms of r:
units²
There are twelve triangles in each dodecagon, and eight dodecagons on the note.Hence, the area occupied by those eifht dodecagons is:
units²
The area of the note can also be calculated in relation to the radius of the dodecagon. Twocoins fit on the note vertically, and four horizontally.The height of each coin is twice the length of the radius, or 2r.Therefore the height of the note is 4r and the length is 85. Its area is is
The fraction of the note covered by the coins is
Therefore, the fraction of the note which is not covered by the coins is
or
(b is base, h is the perpendicular height) or
(a and b are adjacent sides, C is the angle between)
or
(the base and height are adjacent sides perpendicular to each other so that the sin of the angle between them is 1)
, where a is any side.(explanation) en-pure maths-numerical methods-finding the area-shape2
Use the formula A=(pi)*radius^2
The radius is the length from the centre of the circle to any point on its circumference, and pi is the constant 3.14159265...
Use one of two main formulas:
A=(1/2)*(base)*(perpendicular height)
or A=(1/2)*(a)(b)(sinC), in which a and b are adjacent sides with angle C between them
Since all its sides are equal, A=a^2, in which a is the length of any side of the square
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