What is an area?
Area - a characteristic of a closed geometric figure (circle, square, triangle, etc.), which shows its size. Area is measured in square centimeters, meters, etc. Denoted by letter S(square).
How to find the area of a triangle?
S= a h
where a- base length h is the height of the triangle drawn to the base.
Moreover, the base does not have to be at the bottom. That will do too.
If triangle obtuse, then the height falls to the continuation of the base:
If triangle rectangular, then the base and height are its legs:
2. Another formula, which is no less useful, but which for some reason is always forgotten:
S= a b sinα
where a and b two sides of a triangle sinα is the sine of the angle between these sides.
The main condition is that the angle is taken between two known sides.
3. The formula for the area on three sides (Heron's formula):
S=
where a, b and With are the sides of the triangle, and R - semiperimeter. p = (a+b+c)/2.
4. The formula for the area of a triangle in terms of the radius of the circumscribed circle:
S=
where a, b and With are the sides of the triangle, and R- radius of the circumscribed circle.
5. The formula for the area of a triangle in terms of the radius of the inscribed circle:
S= p r
where R - semiperimeter of a triangle, and r- radius of the inscribed circle.
How to find the area of a rectangle?
1. The area of a rectangle is quite simple:
S=a b
No tricks.
How to find the area of a square?
1. Since a square is a rectangle with all sides equal, the same formula applies to it:
S=a a = a2
2. Also, the area of a square can be found through its diagonal:
S= d 2
How to find the area of a parallelogram?
1. The area of a parallelogram is found by the formula:
S=a h
This is due to the fact that if you cut off a right-angled triangle from it on the right and attach it to the left, you get a rectangle:
2. Also, the area of a parallelogram can be found through the angle between the two sides:
S=a b sinα
How to find the area of a rhombus?
A rhombus is essentially a parallelogram in which all sides are equal. Therefore, the same area formulas apply to it.
1. Rhombus area in terms of height:
S=a h
To solve problems in geometry, you need to know formulas - such as the area of a triangle or the area of \u200b\u200ba parallelogram - as well as simple tricks, which we will talk about.
First, let's learn the formulas for the areas of figures. We have specially collected them in a convenient table. Print, learn and apply!
Of course, not all geometry formulas are in our table. For example, to solve problems in geometry and stereometry in the second part of the profile exam in mathematics, other formulas for the area of a triangle are also used. We will definitely tell you about them.
But what if you need to find not the area of a trapezoid or triangle, but the area of some complex figure? There are universal ways! We will show them using examples from the FIPI task bank.
1. How to find the area of a non-standard figure? For example, an arbitrary quadrilateral? A simple technique - let's break this figure into those that we all know about, and find its area - as the sum of the areas of these figures.
Divide this quadrilateral by a horizontal line into two triangles with a common base equal to . The heights of these triangles are equal to and . Then the area of the quadrilateral is equal to the sum of the areas of the two triangles: .
Answer: .
2. In some cases, the area of \u200b\u200bthe figure can be represented as the difference of any areas.
It is not so easy to calculate what the base and height in this triangle are equal to! But we can say that its area is equal to the difference between the areas of a square with a side and three right-angled triangles. See them in the picture? We get: .
Answer: .
3. Sometimes in a task it is necessary to find the area not of the whole figure, but of its part. Usually we are talking about the area of a sector - part of a circle. Find the area of a sector of a circle of radius , whose arc length is equal to .
In this picture we see part of a circle. The area of the whole circle is equal to , since . It remains to find out what part of the circle is depicted. Since the length of the entire circle is (since), and the length of the arc of this sector is equal, therefore, the length of the arc is several times less than the length of the entire circle. The angle on which this arc rests is also times less than a full circle (that is, degrees). This means that the area of the sector will be several times less than the area of the entire circle.
All formulas for the area of plane figures
Area of an isosceles trapezoid
1. The formula for the area of an isosceles trapezoid in terms of sides and angle
a - lower base
b - top base
c - equal sides
α - angle at the lower base
The formula for the area of an isosceles trapezoid in terms of the sides, (S):
The formula for the area of an isosceles trapezoid in terms of sides and angle, (S):
2. The formula for the area of an isosceles trapezoid in terms of the radius of the inscribed circle
R- radius of the inscribed circle
D- diameter of the inscribed circle
O - inscribed circle center
H- height of the trapezoid
α, β - trapezoid angles
The formula for the area of an isosceles trapezoid in terms of the radius of the inscribed circle, (S):
FAIR, for an inscribed circle in an isosceles trapezoid:
3. The formula for the area of an isosceles trapezoid in terms of the diagonals and the angle between them
d-diagonal of a trapezoid
α,β- angles between diagonals
The formula for the area of an isosceles trapezoid in terms of the diagonals and the angle between them, (S):
4. The formula for the area of an isosceles trapezoid through the midline, lateral side and angle at the base
c- side
m- middle line of the trapezoid
α, β - angles at the base
The formula for the area of an isosceles trapezoid in terms of the midline, lateral side and angle at the base,
(S): 
5. The formula for the area of an isosceles trapezoid in terms of bases and height
a - bottom base
b - top base
h - the height of the trapezoid
The formula for the area of an isosceles trapezoid in terms of bases and height, (S):
Area of a triangle given a side and two angles, formula.
a, b, c - sides of the triangle
α, β, γ - opposite angles
Area of a triangle through a side and two angles (S):
The formula for the area of a regular polygon
a - polygon side
n - number of sides
Area of a regular polygon, (S):
The (Heronian) formula for the area of a triangle in terms of the semi-perimeter (S):
The area of an equilateral triangle is:
Formulas for calculating the area of an equilateral triangle.
a - side of the triangle
h - height
How to calculate the area of an isosceles triangle?
b - the base of the triangle
a - equal sides
h - height
3. The formula for the area of a trapezoid in terms of four sides
a - bottom base
b - top base
c, d - sides
The radius of the circumscribed circle of the trapezoid on the sides and diagonals
a - the sides of the trapezoid
c - bottom base
b - top base
d - diagonal
h - height
The formula for the radius of the circumscribed circle of a trapezoid, (R)
find the radius of the circumscribed circle of an isosceles triangle along the sides
Knowing the sides of an isosceles triangle, you can use the formula to find the radius of the circumscribed circle around this triangle.
a, b - sides of the triangle
Radius of the circumscribed circle of an isosceles triangle (R):
Radius of an inscribed circle in a hexagon
a - side of the hexagon
Radius of an inscribed circle in a hexagon, (r):
Radius of an inscribed circle in a rhombus
r - radius of the inscribed circle
a - side of the rhombus
D, d - diagonals
h - diamond height
Radius of an inscribed circle in an isosceles trapezoid
c - lower base
b - top base
a - sides
h - height
Radius of an inscribed circle in a right triangle
a, b - legs of the triangle
c - hypotenuse
Radius of an inscribed circle in an isosceles triangle
a, b - sides of the triangle
Prove that the area of the inscribed quadrilateral is
\/(p - a)(p - b) (p - c) (p - d),
where p is the semi-perimeter and a, b, c and d are the sides of the quadrilateral.
Prove that the area of a quadrilateral inscribed in a circle is
1/2 (ab + cb) sin α, where a, b, c and d are the sides of the quadrilateral and α is the angle between sides a and b.
S = √[ a ƀ c d] sin ½ (α + β). - Read more on FB.ru:
The area of an arbitrary quadrilateral (Fig. 1.13) can be expressed in terms of its sides a, b, c and the sum of a pair of opposite angles:
where p is the semiperimeter of the quadrilateral.
The area of a quadrangle inscribed in a circle () (Fig. 1.14, a) is calculated using the Brahmagupta formula
and described (Fig. 1.14, b) () - according to the formula
If the quadrilateral is inscribed and described at the same time (Fig. 1.14, c), then the formula becomes quite simple:
Peak Formula
To estimate the area of a polygon on checkered paper, it is enough to calculate how many cells this polygon covers (we take the area of \u200b\u200bthe cell as a unit). More precisely, if S is the area of the polygon, is the number of cells that lie entirely inside the polygon, and is the number of cells that have at least one common point with the interior of the polygon.
We will consider below only such polygons, all of whose vertices lie at the nodes of the checkered paper - in those where the grid lines intersect. It turns out that for such polygons, you can specify the following formula:
where is the area, r is the number of nodes that lie strictly inside the polygon.
This formula is called the “Peak formula” after the mathematician who discovered it in 1899.
