The isosceles trapeze is a trapeze at which the opposite nonparallel parties are equal. A number of formulas allow to find the area of a trapeze through its parties, corners, height and. etc. For a case of isosceles trapezes these formulas can become simpler a little.
Instruction
1. The quadrangle at which couple of opposite parties are parallel is called a trapeze. In a trapeze define the bases, the parties, diagonals, height, the average line. Knowing various elements of a trapeze, it is possible to find its area.
2. Sometimes special cases of isosceles trapezes are considered rectangles and squares, but in many sources they do not treat trapezes. One more special case of an isosceles trapeze such geometrical figure with the 3rd equal the party is considered. It call a tripartite trapeze, or a triravnobedrenny trapeze, or, more rare, symtra. Such trapeze can be considered as cutting off of 4 consecutive tops from the regular polygon having 5 or more parties.
3. The trapeze consists of the bases (parallel opposite sides), sides (two other parties), the average line (the piece connecting the middle of sides). The point of intersection of diagonals of a trapeze, a point of intersection of continuations of its sides and the middle of the bases lie on one straight line.
4. That the trapeze was considered as isosceles, at least one of the following conditions has to be carried out. The first: corners at a basis of a trapeze have to be equal: ∠ABC = ∠BCD and ∠BAD = ∠ADC. The second: diagonals of a trapeze have to be equal: AC = BD. The third: if corners between diagonals and the bases are identical, the trapeze is considered isosceles: ∠ABD = ∠ACD, ∠DBC = ∠ACB, ∠CAD = ∠ADB, ∠BAC = ∠BDC. The fourth: the sum of opposite corners is equal 180 °: ∠ABC + ∠ADC = 180 ° and ∠BAD + ∠BCD = 180 °. The fifth: if around a trapeze it is possible to describe a circle, it is considered isosceles.
5. The isosceles trapeze, as well as any other geometrical figure, has a number of invariable properties. The first of them: the sum of the corners adjacent to side of an isosceles trapeze is equal 180 °: ∠ABC + ∠BAD = 180 ° and ∠ADC + ∠BCD = 180 °. The second: if it is possible to enter a circle in an isosceles trapeze, then its side is equal to the average line of a trapeze: AB = CD = m. The third: around an isosceles trapeze it is always possible to describe a circle. The fourth: if diagonals are mutually perpendicular, then height of a trapeze is equal to the half-sum of the bases (average line): h=m. The fifth: if diagonals are mutually perpendicular, then the area of a trapeze is equal to height square: SABCD = h2. The sixth: if it is possible to enter a circle in an isosceles trapeze, then the square of height is equal to the work of bases of a trapeze: h2 = BC • AD. The seventh: the sum of squares of diagonals is equal to the sum of squares of sides plus the doubled work of bases of a trapeze: AC2 + BD2 = AB2 + CD2 + 2BC • AD. The eighth: the straight line passing through the middle of the bases is perpendicular to the bases and is an axis of symmetry of a trapeze: HF ┴ BC ┴ AD. The ninth: height ((CP), lowered from top (C) on the bigger basis (AD), divides it into a big piece (AP) which is equal to the half-sum of the bases and smaller (PD) - it is equal to the semi-difference of the bases: AP=BC+AD/2, PD=AD-BC/2.
6. The most widespread formula for calculation of the area of a trapeze - S = (a+b) of h/2. For a case of an isosceles trapeze it explicitly will not exchange. It is only possible to note that at an isosceles trapeze the corners at any of the bases will be equal (DAB = CDA = x). As its sides are equal too (AB = CD = c), and height of h it is possible to consider on formula h = c*sin (x). Then S = (a+b) of *c*sin (x)/2. Similarly, the area of a trapeze can be written down through the average party of a trapeze: S = mh.
7. Let's consider a special case of an isosceles trapeze when its diagonals are perpendicular. In this case, on property of a trapeze, its height is equal to the half-sum of the bases. Then the area of a trapeze can be calculated on a formula: S = (a+b) ^2/4.
8. Let's consider also one more formula for determination of the area of a trapeze: S = ((a+b)/2) *sqrt (c^2 - ((b-a) ^2+c^2-d^2)/2 (b-a)) ^2), where c and d - trapeze sides. Then in case of an isosceles trapeze when c = d, the formula takes a form: S = ((a+b)/2) *sqrt (c^2-((b-a) ^2/2(b-a)) ^2).
9. Find the area of a trapeze on a formula S=0.5×(a+b) ×h if an and b — lengths of the bases of a trapeze, that is the parallel parties of a quadrangle, and h — trapeze height are known (the smallest distance between the bases). For example, let the trapeze with the bases of a=3 of cm, b=4 of cm and height of h=7 is given see. Then its area will be equal to S=0.5×(3+4) ×7=24.5 to cm².
10. Use the following formula for calculation of the area of a trapeze: S=0.5×AC×BD×sin(β) where AC and BD are trapeze diagonals, and β — a corner between these diagonals. For example, the trapeze with the diagonals of AC=4 cm and BD=6 cm and a corner β=52 °, then sin (52 °) ≈0.79 is set. Substitute values in cm² formula S=0,5×4×6×0,79≈9,5.
11. Count the area of a trapeze when its m — the average line (the piece connecting the middle of the parties of a trapeze) and h — height are known. In this case the area will be equal to S=m×h. For example, let at a trapeze average m=10 line of cm, and h=4 height see. In this case it turns out that the area of the set trapeze is equal to cm² S=10×4=40.
12. Calculate the area of a trapeze, in a case when lengths of its sides and the bases on a formula are given: S=0.5×(a+b) × √ (with²− (((b−a)²+c²−d²) ÷ (2×(b−a)))²), where an and b — the bases of a trapeze, and c and d — its sides. For example, let the trapeze with the bases of 40 cm and 14 cm and sides of 17 cm and 25 cm is given. On the above-stated formula S=0.5×(40+14) × √ (17²− (((14−40)²+17²−25²) ÷(2×(14−40)))²) ≈ 423.7 cm².
13. Calculate the area of an isosceles (ravnoboky) trapeze, that is trapezes at which sides are equal if the circle on a formula is entered in it: S= (4×r²) ÷sin(α) where r is the radius of an inscribed circle, α — a corner at the trapeze basis. Corners at the basis are equal in an isosceles trapeze. For example, let the circle by cm r=3 radius, and a corner is entered in a trapeze at the basis α=30 °, then sin (30 °) =0.5. Substitute values in a formula: S=(4×3²) ÷0.5=72 cm².