To solve geometry problems, you need to know formulas - such as the area of a triangle or the area of a parallelogram - as well as simple techniques that we will cover.
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 Unified State Exam in mathematics, other formulas for the area of a triangle are 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 divide this figure into those that we know everything about, and find its area - as the sum of the areas of these figures.
Divide this quadrilateral with a horizontal line into two triangles with a common base equal to . The heights of these triangles are equal 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 a figure can be represented as the difference of some areas.
It is not so easy to calculate what the base and height of 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 triangles. Do you see them in the picture? We get: .
Answer: .
3. Sometimes in a task you need to find the area of not the entire figure, but part of it. 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 entire circle is equal to . It remains to find out which part of the circle is depicted. Since the length of the entire circle is equal (since ), and the length of the arc of a given sector is equal , therefore, the length of the arc is several times less than the length of the entire circle. The angle at which this arc rests is also a factor of less than a full circle (that is, degrees). This means that the area of the sector will be several times smaller than the area of the entire circle.
Area formula is necessary to determine the area of a figure, which is a real-valued function defined on a certain class of figures of the Euclidean plane and satisfying 4 conditions:
- Positivity - Area cannot be less than zero;
- Normalization - a square with side unit has area 1;
- Congruence - congruent figures have equal area;
- Additivity - the area of the union of 2 figures without common internal points is equal to the sum of the areas of these figures.
| Geometric figure | Formula | Drawing |
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The result of adding the distances between the midpoints of opposite sides of a convex quadrilateral will be equal to its semi-perimeter. |
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Circle sector. The area of a sector of a circle is equal to the product of its arc and half its radius. |
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Circle segment. To obtain the area of segment ASB, it is enough to subtract the area of triangle AOB from the area of sector AOB. |
S = 1 / 2 R(s - AC) |
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The area of the ellipse is equal to the product of the lengths of the major and minor semi-axes of the ellipse and the number pi. |
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Ellipse. Another option for calculating the area of an ellipse is through two of its radii. |
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Triangle. Through the base and height. Formula for the area of a circle using its radius and diameter. |
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Square . Through his side. The area of a square is equal to the square of the length of its side. |
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Square. Through its diagonals. The area of a square is equal to half the square of the length of its diagonal. |
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Regular polygon. To determine the area of a regular polygon, it is necessary to divide it into equal triangles that would have a common vertex at the center of the inscribed circle. |
S= r p = 1/2 r n a |
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,