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Trapezoid area: how to calculate, formula


Using this online calculator, you can find the area of ​​the trapezoid.

Using the online calculator to calculate the area of ​​the trapezoid, you will receive a detailed step-by-step solution of your example, which will allow you to understand the algorithm for solving such problems and consolidate the material covered.

Theory. Trapezoid area

Formula for calculating the area of ​​a trapezoid:

S =(a + b)

where S is the area of ​​the trapezoid,
a, b are the lengths of the bases of the trapezoid,
h is the length of the height of the trapezoid.

You can enter numbers or fractions (-2.4, 5/7,.). Read more in the rules for entering numbers.

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Area of ​​isosceles trapezoid

An isosceles trapezoid is a special case of a trapezoid. Its difference is that such a trapezoid is a convex quadrangle with an axis of symmetry passing through the midpoints of two opposite sides. Its sides are equal.

Isosceles trapezoid

There are several ways to find the area of ​​an isosceles trapezoid.

  • Through the lengths of three sides. In this case, the lengths of the sides will coincide, therefore, are denoted by the same value - c, a and b are the lengths of the bases:
  • If the length of the upper base, the side and the angle at the lower base are known, then the area is calculated as follows:

S = c * sin α * (a + c * cos α)

where a is the upper base, c is the side.

  • If instead of the upper base the length of the lower is known - b, the area is calculated by the formula:

S = c * sin α * (b - c * cos α)

  • If when two bases are known and the angle at the lower base, the area is calculated through the tangent of the angle:

S = ½ * (b2 - a2) * tan α

  • Also, the area is calculated through the diagonals and the angle between them. In this case, the diagonals are equal in length, so each is denoted by the letter d without indices:

  • We calculate the area of ​​the trapezoid, knowing the length of the side, the midline and the angle at the lower base.

Let the side be c, the middle line m, the angle a, then:

Sometimes in an equilateral trapezoid you can enter a circle whose radius is - r.

Trapezoid circle

It is known that a circle can be inscribed in any trapezoid if the sum of the lengths of the bases is equal to the sum of the lengths of its lateral sides. Then the area is found through the radius of the inscribed circle and the angle at the lower base:

The same calculation is made through the diameter D of the inscribed circle (by the way, it coincides with the height of the trapezoid):

Knowing the base and angle, the area of ​​the isosceles trapezoid is calculated as follows:

(this and the following formulas are valid only for trapezoids with an inscribed circle).

Trapezoid in a circle

Through the base and radius of the circle, the area is searched like this:

If only the bases are known, then the area is calculated by the formula:

Through the base and the side line, the area of ​​the trapezoid with an inscribed circle and through the base and the middle line - m is calculated as follows:

Area of ​​a rectangular trapezoid

Rectangular is called a trapezoid, in which one of the sides is perpendicular to the bases. In this case, the lateral side coincides in length with the height of the trapezoid.

A rectangular trapezoid is a square and a triangle. Having found the area of ​​each of the figures, add the results and get the total area of ​​the figure.

Rectangular trapezoid

Also, to calculate the area of ​​a rectangular trapezoid, general formulas for calculating the area of ​​a trapezoid are suitable.

  • If the lengths of the bases and the height (or perpendicular side) are known, then the area is calculated by the formula:

As h (height) can be the side of c. Then the formula looks like this:

  • Another way to calculate the area is to multiply the length of the midline by height:

or the length of the lateral perpendicular side:

  • The next calculation method is through half the product of the diagonals and the sine of the angle between them:

S = ½ * d1 * d2 * sin α

Rectangular trapezoid with perpendicular diagonals

If the diagonals are perpendicular, then the formula simplifies to:

  • Another way of calculating is through a half-perimeter (the sum of the lengths of two opposite sides) and the radius of the inscribed circle.

This formula is valid for reasons. If we take the lengths of the sides, then one of them will be equal to twice the radius. The formula will look like this:

  • If a circle is inscribed in the trapezoid, then the area is calculated in the same way:

where m is the length of the midline.

Area of ​​a curved trapezoid

A curvilinear trapezoid is a flat figure bounded by a graph of a non-negative continuous function y = f (x) defined on the interval [a, b], the abscissa axis and the straight lines x = a, x = b. In fact, its two sides are parallel to each other (the base), the third side is perpendicular to the bases, and the fourth is a curve corresponding to the graph of the function.

Curved trapezoid

The area of ​​a curved trapezoid is sought through the integral according to the Newton-Leibniz formula:

This calculates the area of ​​various types of trapezoid. But, in addition to the properties of the sides, the trapezoid possess the same properties of angles. As with all existing quadrangles, the sum of the internal angles of the trapezoid is 360 degrees. And the sum of the angles adjacent to the side is 180 degrees.

Trapezoid area through the bases and two corners

  • Parallel sides are called the grounds trapezoid.
  • The other two sides are called sides.
  • The line connecting the middle of the sides is called middle line trapezoid.
  • The distance between the bases is called tall trapezoid.
  • A trapezoid with equal sides is called equilateral (or isosceles)
  • A trapezoid, one of the corners of which is straight, is called rectangular.
  • Trapezium midline parallel to the bases and equal to their half-sum.
  • Parallel straight lines intersecting the sides of the corner cut off proportional segments from the sides of the angle.
  • At isosceles trapezoid the angles at the base are equal.
  • In an isosceles trapezoid, the diagonals are equal.
  • If the trapezoid is isoside, then around it you can describe the circle.
  • If the sum of the trapezoid bases is equal to the sum of the sides, then a circle can be entered in it.
  • In the trapezoid of the middle of the bases, the intersection point of the diagonals and the continuation of the sides are on the same line.

A source of information

The bases of the isosceles (isosceles) trapezoid are 8 and 20 centimeters. The side is 10 cm. Find the area of ​​a trapezoid similar to this one, which has a height of 12 cm.

From the top B of the trapezoid ABCD, lower the height BM to the base AD. From peak C to base AD, lower the height CN. Since MBCN is a rectangle, then

The triangles resulting from the fact that we lowered from a smaller base of an equal-sided trapezoid to a larger two heights are equal. In this way,

AD = BC + AM * 2
AM = (AD - BC) / 2
AM = (20 - 8) / 2 = 6 cm

Thus, in the ABM triangle formed by the height, lowered from the smaller base of the trapezoid to the larger, we know the leg and hypotenuse. We will find the remaining leg, which is simultaneously the height of the trapezoid, by the Pythagorean theorem:

BM2 = AB2 - AM2
BM2 = 102 - 62
BM = 8 cm

Since the height of the trapezoid ABCD is 8 cm, and the height of such a trapezoid is 12 cm, the similarity coefficient will be equal to

Since in such figures all geometric dimensions are proportional to each other with a similarity coefficient, we find the area of ​​such a trapezoid. The product of the half sum of the bases of such a trapezoid to a height is expressed in terms of the known geometric dimensions of the original trapezoid and the similarity coefficient:

S sub = (AD * k + BC * k) / 2 * (BM * k)
Spod = (20 * 1.5 + 8 * 1.5) / 2 * (8 * 1.5) = (30 + 12) / 2 * 12 = 252 cm 2