Area of a triangle formula based on two sides. Area of a triangle
To determine the area of a triangle, you can use different formulas. Of all the methods, the easiest and most frequently used is to multiply the height by the length of the base and then divide the result by two. However, this method is far from the only one. Below you can read how to find the area of a triangle using different formulas.
Separately, we will look at ways to calculate the area of specific types of triangles - rectangular, isosceles and equilateral. We accompany each formula with a short explanation that will help you understand its essence.
Universal methods for finding the area of a triangle
The formulas below use special notation. We will decipher each of them:
- a, b, c – the lengths of the three sides of the figure we are considering;
- r is the radius of the circle that can be inscribed in our triangle;
- R is the radius of the circle that can be described around it;
- α is the magnitude of the angle formed by sides b and c;
- β is the magnitude of the angle between a and c;
- γ is the magnitude of the angle formed by sides a and b;
- h is the height of our triangle, lowered from angle α to side a;
- p – half the sum of sides a, b and c.
It is logically clear why you can find the area of a triangle in this way. The triangle can easily be completed into a parallelogram, in which one side of the triangle will act as a diagonal. The area of a parallelogram is found by multiplying the length of one of its sides by the value of the height drawn to it. The diagonal divides this conditional parallelogram into 2 identical triangles. Therefore, it is quite obvious that the area of our original triangle must be equal to half the area of this auxiliary parallelogram.
S=½ a b sin γ
According to this formula, the area of a triangle is found by multiplying the lengths of its two sides, that is, a and b, by the sine of the angle formed by them. This formula is logically derived from the previous one. If we lower the height from angle β to side b, then, according to the properties of a right triangle, when we multiply the length of side a by the sine of angle γ, we obtain the height of the triangle, that is, h.
The area of the figure in question is found by multiplying half the radius of the circle that can be inscribed in it by its perimeter. In other words, we find the product of the semi-perimeter and the radius of the mentioned circle.
S= a b c/4R
According to this formula, the value we need can be found by dividing the product of the sides of the figure by 4 radii of the circle described around it.
These formulas are universal, as they make it possible to determine the area of any triangle (scalene, isosceles, equilateral, rectangular). This can be done using more complex calculations, which we will not dwell on in detail.
Areas of triangles with specific properties
How to find the area of a right triangle? The peculiarity of this figure is that its two sides are simultaneously its heights. If a and b are legs, and c becomes the hypotenuse, then we find the area like this:
How to find the area of an isosceles triangle? It has two sides with length a and one side with length b. Consequently, its area can be determined by dividing by 2 the product of the square of side a by the sine of angle γ.
How to find the area of an equilateral triangle? In it, the length of all sides is equal to a, and the magnitude of all angles is α. Its height is equal to half the product of the length of side a and the square root of 3. To find the area of a regular triangle, you need to multiply the square of side a by the square root of 3 and divide by 4.
Sometimes in life there are situations when you have to delve into your memory in search of long-forgotten school knowledge. For example, you need to determine the area of a triangular-shaped plot of land, or the time has come for another renovation in an apartment or private house, and you need to calculate how much material will be needed for a surface with a triangular shape. There was a time when you could solve such a problem in a couple of minutes, but now you are desperately trying to remember how to determine the area of a triangle?
Don't worry about it! After all, it is quite normal when a person’s brain decides to transfer long-unused knowledge somewhere to a remote corner, from which sometimes it is not so easy to extract it. So that you don’t have to struggle with searching for forgotten school knowledge to solve such a problem, this article contains various methods that make it easy to find the required area of a triangle.
It is well known that a triangle is a type of polygon that is limited to the minimum possible number of sides. In principle, any polygon can be divided into several triangles by connecting its vertices with segments that do not intersect its sides. Therefore, knowing the triangle, you can calculate the area of almost any figure.
Among all the possible triangles that occur in life, the following particular types can be distinguished: and rectangular.
The easiest way to calculate the area of a triangle is when one of its angles is right, that is, in the case of a right triangle. It is easy to see that it is half a rectangle. Therefore, its area is equal to half the product of the sides that form a right angle with each other.
If we know the height of a triangle, lowered from one of its vertices to the opposite side, and the length of this side, which is called the base, then the area is calculated as half the product of the height and the base. This is written using the following formula:
S = 1/2*b*h, in which
S is the required area of the triangle;
b, h - respectively, the height and base of the triangle.
It is so easy to calculate the area of an isosceles triangle because the height will bisect the opposite side and can be easily measured. If the area is determined, then it is convenient to take the length of one of the sides forming a right angle as the height.
All this is of course good, but how to determine whether one of the angles of a triangle is right or not? If the size of our figure is small, then we can use a construction angle, a drawing triangle, a postcard or another object with a rectangular shape.
But what if we have a triangular plot of land? In this case, proceed as follows: count from the top of the supposed right angle on one side a distance multiple of 3 (30 cm, 90 cm, 3 m), and on the other side measure a distance multiple of 4 in the same proportion (40 cm, 160 cm, 4 m). Now you need to measure the distance between the end points of these two segments. If the result is a multiple of 5 (50 cm, 250 cm, 5 m), then we can say that the angle is right.
If the length of each of the three sides of our figure is known, then the area of the triangle can be determined using Heron's formula. In order for it to have a simpler form, a new value is used, which is called semi-perimeter. This is the sum of all the sides of our triangle, divided in half. After the semi-perimeter has been calculated, you can begin to determine the area using the formula:
S = sqrt(p(p-a)(p-b)(p-c)), where
sqrt - square root;
p - semi-perimeter value (p = (a+b+c)/2);
a, b, c - edges (sides) of the triangle.
But what if the triangle has an irregular shape? There are two possible ways here. The first of them is to try to divide such a figure into two right triangles, the sum of the areas of which is calculated separately, and then added. Or, if the angle between two sides and the size of these sides are known, then apply the formula:
S = 0.5 * ab * sinC, where
a,b - sides of the triangle;
c is the size of the angle between these sides.
The latter case is rare in practice, but nevertheless, everything is possible in life, so the above formula will not be superfluous. Good luck with your calculations!
As follows:
S = ½ * a * h,
Where:
S – area of the triangle,
a is the length of its side,
h is the height lowered to this side.
Side length and height must be presented in the same units of measurement. In this case, the area of the triangle will be obtained in the corresponding “ ” units.
Example.
On one side of a scalene triangle 20 cm long, a perpendicular from the opposite vertex 10 cm long is lowered.
The area of the triangle is required.
Solution.
S = ½ * 20 * 10 = 100 (cm²).
If the lengths of any two sides of a scalene triangle and the angle between them are known, then use the formula:
S = ½ * a * b * sinγ,
where: a, b are the lengths of two arbitrary sides, and γ is the value of the angle between them.
In practice, for example, when measuring the area of land, the use of the above formulas is sometimes difficult, since it requires additional construction and measurement of angles.
If you know the lengths of all three sides of a scalene triangle, then use Heron's formula:
S = √(p(p-a)(p-b)(p-c)),
a, b, c – lengths of the sides of the triangle,
p – semi-perimeter: p = (a+b+c)/2.
If, in addition to the lengths of all sides, the radius of the circle inscribed in the triangle is known, then use the following compact formula:
where: r – radius of the inscribed circle (р – semi-perimeter).
To calculate the area of a scalene triangle using the radius of the circumcircle and the length of its sides, use the formula:
where: R – radius of the circumscribed circle.
If the length of one of the sides of the triangle and the values of three angles are known (in principle, two are enough - the value of the third is calculated from the equality of the sum of the three angles of the triangle - 180º), then use the formula:
S = (a² * sinβ * sinγ)/2sinα,
where α is the value of the angle opposite to side a;
β, γ – values of the remaining two angles of the triangle.
A regular triangle is a triangle with three equal sides. It has the following properties: all sides of a regular triangle are equal to each other, and all angles are equal to 60 degrees. A regular triangle is isosceles.
You will need
- Knowledge of geometry.
Instructions
Let a side of a regular triangle with length a=7 be given. Knowing the side of such a triangle, you can easily calculate its area. For this, the following is used: S = (3^(1/2)*a^2)/4. Let's substitute the value a=7 into this formula and get the following: S = (7*7*3^1/2)/4 = 49 * 1.7 / 4 = 20.82. Thus, we found that the area of an equilateral triangle with side a=7 is equal to S=20.82.
If the radius of the circle is given, it will look like this:
S = 3*3^(1/2)*r^2, where r is the radius of the inscribed circle. Let the radius of the inscribed circle be r=4. Let's substitute it into the formula written earlier and get the following expression: S = 3*1.7*4*4 = 81.6. That is, if the radius of the inscribed circle is equal to 4, the area of the equilateral triangle will be equal to 81.6.
With a known radius of the circumscribed circle, the formula for the area of a triangle looks like this: S = 3*3^(1/2)*R^2/4, where R is the radius of the circumscribed circle. Let's assume that R=5, substitute this value into the formula: S = 3*1.7*25/4 = 31.9. It turns out that with a radius of the circumscribed circle equal to 5, the area of the triangle is 31.9.
note
The area of a triangle is always positive, as is the length of a side of a triangle and the radii of the inscribed and circumscribed circles.
Helpful advice
The radius of the inscribed and circumscribed circles in an equilateral triangle differs by a factor of two, knowing this, you can remember only one formula, for example, through the radius of the inscribed circle, and derive the second, knowing this statement.
If the length of one of the sides of a triangle and the values of the adjacent angles are known, its area can be calculated in several ways. Each of the calculation formulas involves the use of trigonometric functions, but this should not be intimidating - to calculate them, it is enough to have access to the Internet, not to mention the presence of a built-in calculator in the operating system.
Instructions
The first option for calculating the area (S) from the known length of one of the sides (A) and the values of the adjacent angles (α and β) involves calculating these angles. The area in this case will be the square of the length of the known side, divided by twice the cotangents of the known angles: S = A*A/(2*(ctg(α)+ctg(β))). For example, if the length of a known side is 15 cm, and the adjacent angles are 40° and 60°, then the area calculation will look like this: 15*15/(2*(ctg(40)+ctg(60))) = 225/(2*(-0.895082918+3.12460562)) = 225/4.4590454 = 50.4592305 square centimeters.
The second option for calculating area uses sines of known angles instead of cotangents. In this version, the area is equal to the square of the length of the known side, multiplied by the sines of each of the angles and divided by double the sine of the sum of these angles: S = A*A*sin(α)*sin(β)/(2*sin(α + β) ). For example, for the same triangle with a known side of 15 cm, and adjacent angles of 40° and 60°, the area calculation will look like this: (15*15*sin(40)*sin(60))/(2* sin(40+60)) = 225*0.74511316*(-0.304810621)/(2*(-0.506365641)) = -51.1016411/-1.01273128 = 50.4592305 square centimeters.
The third option for calculating the area of a triangle uses the tangents of the angles. The area will be equal to the square of the length of the known side, multiplied by the tangents of each of the angles and divided by double the sum of the tangents of these angles: S = A*A*tg(α)*tg(β)/2(tg(α)+tg(β) ). For example, for a triangle used in the previous steps with a side of 15 cm and adjacent angles of 40° and 60°, the calculation of the area will look like this: (15*15*tg(40)*tg(60))/(2*(tg (40)+tg(60)) = (225*(-1.11721493)*0.320040389)/(2*(-1.11721493+0.320040389)) = -80.4496277/-1.59434908 = 50.4592305 square centimeters.
Practical calculations can be made, for example, using the Google search engine calculator. To do this, just substitute numerical values into the formulas and enter them in the search query field.
Tip 4: How to find the area of a triangle and a rectangle
Triangle and rectangle are the two simplest plane geometric figures in Euclidean geometry. Inside the perimeters formed by the sides of these polygons, there is a certain section of the plane, the area of which can be determined in many ways. The choice of method in each specific case will depend on the known parameters of the figures.
Area of a geometric figure- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface limited by the closed contour of this figure). The size of the area is expressed by the number of square units contained in it.
Triangle area formulas
- Formula for the area of a triangle by side and height
Area of a triangle equal to half the product of the length of a side of a triangle and the length of the altitude drawn to this side - Formula for the area of a triangle based on three sides and the radius of the circumcircle
- Formula for the area of a triangle based on three sides and the radius of the inscribed circle
Area of a triangle is equal to the product of the semi-perimeter of the triangle and the radius of the inscribed circle. where S is the area of the triangle,
- lengths of the sides of the triangle,
- height of the triangle,
- the angle between the sides and,
- radius of the inscribed circle,
R - radius of the circumscribed circle,
Square area formulas
- Formula for the area of a square by side length
Square area equal to the square of the length of its side. - Formula for the area of a square along the diagonal length
Square area equal to half the square of the length of its diagonal.S= 1 2 2 where S is the area of the square,
- length of the side of the square,
- length of the diagonal of the square.
Rectangle area formula
- Area of a rectangle equal to the product of the lengths of its two adjacent sides
where S is the area of the rectangle,
- lengths of the sides of the rectangle.
Parallelogram area formulas
- Formula for the area of a parallelogram based on side length and height
Area of a parallelogram - Formula for the area of a parallelogram based on two sides and the angle between them
Area of a parallelogram is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.a b sin α
where S is the area of the parallelogram,
- lengths of the sides of the parallelogram,
- length of parallelogram height,
- the angle between the sides of the parallelogram.
Formulas for the area of a rhombus
- Formula for the area of a rhombus based on side length and height
Area of a rhombus equal to the product of the length of its side and the length of the height lowered to this side. - Formula for the area of a rhombus based on side length and angle
Area of a rhombus is equal to the product of the square of the length of its side and the sine of the angle between the sides of the rhombus. - Formula for the area of a rhombus based on the lengths of its diagonals
Area of a rhombus equal to half the product of the lengths of its diagonals. where S is the area of the rhombus,
- length of the side of the rhombus,
- length of the height of the rhombus,
- the angle between the sides of the rhombus,
1, 2 - lengths of diagonals.
Trapezoid area formulas
- Heron's formula for trapezoid
Where S is the area of the trapezoid,
- lengths of the bases of the trapezoid,
- lengths of the sides of the trapezoid,
Concept of area
The concept of the area of any geometric figure, in particular a triangle, will be associated with a figure such as a square. For the unit area of any geometric figure we will take the area of a square whose side is equal to one. For completeness, let us recall two basic properties for the concept of areas of geometric figures.
Property 1: If geometric figures are equal, then their areas are also equal.
Property 2: Any figure can be divided into several figures. Moreover, the area of the original figure is equal to the sum of the areas of all its constituent figures.
Let's look at an example.
Example 1
Obviously, one of the sides of the triangle is a diagonal of a rectangle, one side of which has a length of $5$ (since there are $5$ cells), and the other is $6$ (since there are $6$ cells). Therefore, the area of this triangle will be equal to half of such a rectangle. The area of the rectangle is
Then the area of the triangle is equal to
Answer: $15$.
Next, we will consider several methods for finding the areas of triangles, namely using the height and base, using Heron’s formula and the area of an equilateral triangle.
How to find the area of a triangle using its height and base
Theorem 1
The area of a triangle can be found as half the product of the length of a side and the height to that side.
Mathematically it looks like this
$S=\frac(1)(2)αh$
where $a$ is the length of the side, $h$ is the height drawn to it.
Proof.
Consider a triangle $ABC$ in which $AC=α$. The height $BH$ is drawn to this side, which is equal to $h$. Let's build it up to the square $AXYC$ as in Figure 2.
The area of rectangle $AXBH$ is $h\cdot AH$, and the area of rectangle $HBYC$ is $h\cdot HC$. Then
$S_ABH=\frac(1)(2)h\cdot AH$, $S_CBH=\frac(1)(2)h\cdot HC$
Therefore, the required area of the triangle, by property 2, is equal to
$S=S_ABH+S_CBH=\frac(1)(2)h\cdot AH+\frac(1)(2)h\cdot HC=\frac(1)(2)h\cdot (AH+HC)=\ frac(1)(2)αh$
The theorem has been proven.
Example 2
Find the area of the triangle in the figure below if the cell has an area equal to one
The base of this triangle is equal to $9$ (since $9$ is $9$ squares). The height is also $9$. Then, by Theorem 1, we get
$S=\frac(1)(2)\cdot 9\cdot 9=40.5$
Answer: $40.5$.
Heron's formula
Theorem 2
If we are given three sides of a triangle $α$, $β$ and $γ$, then its area can be found as follows
$S=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
here $ρ$ means the semi-perimeter of this triangle.
Proof.
Consider the following figure:
By the Pythagorean theorem, from the triangle $ABH$ we obtain
From the triangle $CBH$, according to the Pythagorean theorem, we have
$h^2=α^2-(β-x)^2$
$h^2=α^2-β^2+2βx-x^2$
From these two relations we obtain the equality
$γ^2-x^2=α^2-β^2+2βx-x^2$
$x=\frac(γ^2-α^2+β^2)(2β)$
$h^2=γ^2-(\frac(γ^2-α^2+β^2)(2β))^2$
$h^2=\frac((α^2-(γ-β)^2)((γ+β)^2-α^2))(4β^2)$
$h^2=\frac((α-γ+β)(α+γ-β)(γ+β-α)(γ+β+α))(4β^2)$
Since $ρ=\frac(α+β+γ)(2)$, then $α+β+γ=2ρ$, which means
$h^2=\frac(2ρ(2ρ-2γ)(2ρ-2β)(2ρ-2α))(4β^2)$
$h^2=\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2 )$
$h=\sqrt(\frac(4ρ(ρ-α)(ρ-β)(ρ-γ))(β^2))$
$h=\frac(2)(β)\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
By Theorem 1, we get
$S=\frac(1)(2) βh=\frac(β)(2)\cdot \frac(2)(β) \sqrt(ρ(ρ-α)(ρ-β)(ρ-γ) )=\sqrt(ρ(ρ-α)(ρ-β)(ρ-γ))$
