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## Geometry and Measurement

### Plane Geometry - Area and Perimeter of Plane Figures

#### Q.01

'Find the area enclosed by the curve y=|x^{2}-1| and the line y=3.'

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'Find the area S enclosed by the parabola y = -2x^2 + 1 and the line segment AB.'

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'Let 139 a be a constant satisfying 0 ≤ a ≤ 1. Let the parabola y = 1/2 x^2 + 1/2 be represented as C1 and the parabola y = 1/4 x^2 be represented as C2. For a real number a, let the region enclosed by the lines x = a, x = a+1, and C1, C2 be denoted as D, and let the square with vertices (a,0),(a+1,0),(a+1,1),(a,1) be denoted as R.\n1. Calculate the area S of region D.\n2. Calculate the area T of the intersection between square R and region D.\n3. Find the value of a that maximizes T.\n[Source: Central Test] Basic Example 204, Advanced Example 216'

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'Find the arc length and area of the following sectors.'

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'Find the area enclosed by the following curves or lines.'

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'Find the area S enclosed by the parabola and the x-axis.'

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'Find the sum of the areas of the two figures enclosed by the curves y=x^{3}-4x and y=3x^{2}.'

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'Find the area of the triangle OAB with vertices O(0,0), A(2,6), B(4,3).'

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'Find the area of the triangle formed by the lines x-y=0 (1), 2x+y=9 (2), x-4y=0 (3).'

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'Find the area \ S \ of the figure enclosed by the following curves and lines: \ y = x^{2} - 2x, \\quad y = x^{2} + 2x - 3, \\quad x = -1, \\quad x = 0 \'

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'The total area of a plane figure with one length, area (2) of a rhombus is 3 × 3 × 3.14 × 840 ÷ 360=21 × 3.14=65.94(cm²). Moreover, the total area of 8 equilateral triangles is 3.9 × 8=31.2(cm²), so the area surrounded by the thick lines is 65.94+31.2= 97.14(cm²).'

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'What is the total area of the colored part in the 2019th row in square centimeters?'

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'(2) The area of triangle MBP is, 6 ✕ (6+12) ÷ 2 = 54 (cm²) Therefore, the volume of pyramid F-MBP is, 54 ✕ □ ÷ 3 = 18 ✕ □ (cm³). Also, the area of triangle MAQ is, 4 ✕ 12 ÷ 2 = 24 (cm²) and the height of pyramid R-MAQ is, □ ✕'

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'(1) The bottom surface is as shown in the figure on the right. If the area of the rhombus ABCD is taken as 1, then the areas of the 4 triangles ABD, CDB, DAC, BCA all become 1/2. Therefore, the area of triangle AKN is 1/8, the area of triangle CML is 1/18, the areas of triangles DNM and BLK are 1/6 each, hence the area of quadrilateral KLMN is found to be 35/72. Therefore, when the height of the quadrilateral ABCD-EFGH is taken as 1, the volume of the quadrilateral ABCD-EFGH is 1×1=1, and the volume of the frustum O-KLMN is 35/72×1/3=35/216, so the volume of the frustum O-KLMN is 35/216 times the volume of the quadrilateral ABCD-EFGH. O-KLMN is also cut at half the height. Therefore, the cross-section becomes half the size of the quadrilateral KLMN, so the area of the cross-section is 1/4 times the area of quadrilateral KLMN, thus the area of the cross-section is 35/72×1/4=35/288, hence the area of the cross-section is 35/288 times the area of rhombus ABCD.'

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'(4) To find the area of the bottom side (the filled trapezoid section) of the graph drawn in (2). Using the value obtained in (3) for calculation, the distance moved within 60 seconds is 20 × 60 ÷ 2 = 600 meters, from 60 seconds to 120 seconds is 20 × (120-60) = 1200 meters, and from 120 seconds to stopping at 150 seconds is 20 × (150-120) ÷ 2 = 300 meters. Therefore, the total distance required is 600 + 1200 + 300 = 2100 meters.'

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'The angle B of triangle ABC and angle C of triangle ACD are right angles, and the angles marked with • are equal. Point E is the intersection of the extended lines of side BC and AD. The length of side AB is 2 cm, and the length of side BC is 1 cm. (1) What is the area of triangle ACD in square cm?'

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'Answer the following 4 questions. However, please note that the diagram may not be accurate.\n(1) Draw two squares with AC as one side for triangle ABC and BC as one side for a square as shown in diagram 1, and connect two points D and E with a line. In this case, what is the area of triangle CDE in square centimeters?'

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'In 1950, according to Table 2, it can be observed that as of 2020, the Shibuya Educational Academy Makuhari Middle School has a smaller area compared to Kagawa Prefecture, but its area has increased. Additionally, significant increases in area can be seen in Chiba Prefecture, Tokyo, Kanagawa Prefecture, Aichi Prefecture, and others. Please explain the reasons for the increase in the area of these regions on the answer sheet. However, please note that changes in prefectural boundaries or the discovery of uninhabited islands are not included.'

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'Find the area of the following parallelogram: AB = 2, CD = 2, BC = 5, ∠B = 120°'

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'Around a circular pond with a radius of 4m, a flower bed of the same width is to be constructed. To ensure that the area of the flower bed is between 9π square meters and 33π square meters, what should be the width of the flower bed?'

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'Find the area of a triangle given the conditions, and apply it to solid geometry.'

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'How can the area of a rectangle with a perimeter of 20 cm be made to be at least 9 cm^2 and at most 21 cm^2?'

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'Find the area of the following figures: \n(1) Quadrilateral ABCD where AD || BC, AB = 5, BC = 6, DA = 2, and angle ABC = 60 degrees \n(2) Quadrilateral ABCD where AB = 2, BC = square root of 3 plus 1, CD = square root of 2, B = 60 degrees, and C = 75 degrees \n(3) Regular dodecagon with a side length of 1'

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'Find the area of quadrilateral ABCD. The side lengths are AB=5, BC=6, CD=5, DA=3, and the angle is ∠ADC=120°.'

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'In triangle ABC, find the following. Where, let the area of triangle ABC be S. (1) When A=120 degrees, c=8, S=14√3, find a and b. (2) When b=3, c=2, 0<A<90 degrees, S=√5, find sin A and a. (3) When a=13, b=14, c=15, and the length of the perpendicular from vertex A to side BC is h, find S and h.'

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'Practice 5: Comparing the area and perimeter of triangles'

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'Example 135: Area of a Polygon\nTo find the area of a complex polygon, divide it into simple triangles or quadrilaterals, calculate the area of each, and then sum them up.'

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'Find the area enclosed by the PR curve, the line, and the x-axis.'

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'Let A(1,0). As point P moves along the portion of the parabola y=x^2 where -1 ≤ x ≤ 1, find the area of the figure formed by the line segment AP passing through.'

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'Express the area S of the figure enclosed by C and ℓ in terms of m as m moves within the range determined in (1).'

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'Find the area enclosed by the graphs of the two functions y=-x^{2}+x+2 and y=|x|-1.'

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'By mathematics, the area S obtained from the right figure is S = S1 + S2 + S3 = 1/2 * 2 * 1 + 1/2 * 2 (sqrt(3) - 1) + ∫[-1, sqrt(3)] ([-x^2 + x + 2] - [ (2 - sqrt(3)) x + 2 - sqrt(3)]) dx = 1 + sqrt(3) - 1 - ∫[-1, sqrt(3)] (x + 1)(x - sqrt(3)) dx = sqrt(3) - (-1/6)[sqrt(3) - (-1)]^3 = sqrt(3) + 1/6 (10 + 6 sqrt(3)) = 5/3 + 2 sqrt(3)'

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'Find the value of the constant a when the area enclosed by the parabola y=-x(x-2) and the x-axis is divided into two equal parts by the line y=ax. Here, 0<a<2.'

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'If S1, S2, S3 are taken as shown in the figure, the area S to be found is: S = S1 - (2 S2 - S3) + S3 = S1 - 2 S2 + 2 S3 = ∫[0, 3] { 3x - (3x^2 - 6x) } dx - 2 ∫[0, 2] { - (3x^2 - 6x) } dx + 2 ∫[0, 1] { (-3x^2 + 6x) - 3x } dx = -3 ∫[0, 3] x(x-3) dx + 6 ∫[0, 2] x(x-2) dx - 6 ∫[0, 1] x(x-1) dx = -3 * (-1/6) (3 - 0)^3 + 6 * (-1/6) (2 - 0)^3 - 6 * (-1/6) (1 - 0)^3 = 27/2 - 8 + 1 = 13/2'

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'Find the area of the following figure: (1) a=10, B=30°, C=105° for △ABC'

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'Find the area of triangle ABC. (1) a=3, c=2√2, B=45 degrees (2) a=6, b=5, c=4'

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'Practice finding the area S of the following quadrilateral ABCD (O is the intersection of AC and BD).'

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'Find the area S of the following parallelogram ABCD (O is the intersection point of AC and BD).'

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'Since r>0 and x>0, we have x=(\\sqrt{2}-1) r. The area of the quadrilateral AMON is 2 \\triangle \\mathrm{AMO}=x r=(\\sqrt{2}-1) r^{2}. Therefore, the area of the octagon we seek is 8(\\sqrt{2}-1) r^{2}.'

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'Find the area of the following figure: (3) Quadrilateral ABCD, inscribed in a circle, AB=6, BC=CD=3, ∠B=120°'

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'How can the area of a rectangle with a perimeter of 20 cm be between 9 cm² and 21 cm²?'

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'If the area of triangle ABC is 20√3, find the length of the longest side of triangle ABC.'

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"When the lengths of the three sides are given, to find the area S of triangle ABC, the following steps should be taken:\n(1) Use the cosine rule to find cos A.\n(2) From sin ^ 2 A + cos ^ 2 A = 1, find sin A.\n(3) Substitute into the area formula S = 1/2bc sin A.\nThe area S, expressed in terms of the lengths of the three sides a, b, c, is known as the Heron's formula.\nThe area S of triangle ABC according to Heron's formula is\nWhen 2s = a + b + c,\nS = √(s(s-a)(s-b)(s-c))"

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'How can the area of the quadrilateral ABCD be calculated?'

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'Lake Problem 139 Medium Variable Representation and Area (1)\nFind the area $S$ of the region enclosed by the curve \\left\\{\egin{\overlineray}{l}x=2 \\cos t \\\\ y=\\sin 2 t\\end{\overlineray}\\left(0 \\leqq t \\leqq \\frac{\\pi}{2}\\right)\\right. and the $x$-axis.'

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'(1) On a plane, when the center of a circle with radius r (r ≤ 1) makes one revolution along the side of a square with side length 4, find the area S(r) of the portion of the circle that passes through.'

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'Find the area S(a) of triangle ABC and the minimum value of S(a) as a moves over all real numbers in the given problem.'

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'Therefore, since t=\\frac{1}{2}, the general shape of the curve is as shown in the right figure. Therefore, the area to be determined is'

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'Find the area of the region enclosed by two line segments OP1, OP2, and curve C with respect to points P1 and P2 mentioned above (1).'

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'Find the area S of the figure enclosed by the following curves, lines and x-axis.'

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'In the region in the xy-plane where x^{2}+y^{2} ≤ 2,|x| ≤ 1, find the area S of the part above the curve C: y=x^{3}+x^{2}-x.'

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'Area of a triangle\nThe area of a triangle with vertices O(0,0), A(x1, y1), and B(x2, y2) is given by \ \\frac{1}{2} |x1 y2 - x2 y1| \'

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'Find the value of the constant a when the area enclosed by the parabola y=-x(x-2) and the x-axis is bisected by the line y=ax. Given that 0<a<2.'

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'In the region on the xy-plane where x^{2}+y^{2} ≤ 2,|x| ≤ 1, find the area S of the part above the curve C: y=x^{3}+x^{2}-x.'

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'Find the area enclosed by the tangent lines at the point $(0,0)$ and the point $(2,-2)$ of the parabola $y=-x^{2}+x$. p. 412 EX 155'

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'Find the area enclosed by the curve $x=y^{2}$, the y-axis, and the line $y=2$.'

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'Find the area \ S \ of the curve given by the parametric equations \\( x=2 t+t^{2}, y=t+2 t^{2}(-2 \\leqq t \\leqq 0) \\) and enclosed by the y-axis.'

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'Solve the following problem. Find the area \ S \ of the curve represented by \\( x = 2t + t^2, y = t + 2t^2 (-2 ≤ t ≤ 0) \\) and the shape enclosed by the \ y \ axis.'

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"Therefore, let \ \\triangle ABC \'s area be denoted as \ S \, then\nS = S_{1} + S_{2} - S_{3} = 6\\sqrt{3} + (\\sqrt{6} + \\sqrt{2}) - \\frac{3(\\sqrt{6} - \\sqrt{2})}{2} = \\frac{5\\sqrt{2} + 12\\sqrt{3} - \\sqrt{6}}{2}\n\nRepresenting 3 points \ A, B, C \ in Cartesian coordinates,\nA(3, 3\\sqrt{3}), B(-2, 2\\sqrt{3}), C(-\\sqrt{2}, -\\sqrt{2})\n\n\\overrightarrow{AB} = (-5, -\\sqrt{3}), \\overrightarrow{AC} = (-\\sqrt{2} - 3, -\\sqrt{2} - 3\\sqrt{3})\nS = \\frac{1}{2}| -5(-\\sqrt{2} - 3\\sqrt{3}) - (-\\sqrt{3})(-\\sqrt{2} - 3) | = \\frac{5\\sqrt{2} + 12\\sqrt{3} - \\sqrt{6}}{2}"

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'Find the area S enclosed by the following curve or line. where the constant a in (2) satisfies 0 < a < 1.'

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'From the equation given by (3), 2x ^ 2-2xy + y ^ 2 = 4, we can conclude y ^ 2-2xy + 2x ^ 2-4 = 0. Therefore, y = x ±√(4-x ^ 2) (-2≤x≤2). From the diagram, the area is calculated as S =∫_(-2)^ 2 {x +√(4-x ^ 2)} = 2∫_(-2)^ 2√(4-x ^ 2)dx = 2 * π * 2 ^ 2 = 4π'

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'Find the area enclosed by the curve represented by x=cos(2t) and y=t*sin(t) in the coordinate plane using the parameter t (0≤t≤2π).'

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'Using the area formula given in the example, find the area of the region enclosed by curve $C$ represented by the polar equation $r=1+\\sin \\frac{\\theta}{2}(0 \\leqq \\theta \\leqq \\pi)$ and the x-axis.'

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'Find the area of the shape enclosed by the curve y² = (x + 3)x²'

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'Find the area S enclosed by the following curve and lines.'

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'Find the maximum area S of trapezoid ABCD. Where AD // BC, AB=AD=CD=a, and BC>a.'

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'Find the area S enclosed by the following curve and lines: (1) x=-1-y^{2}, y=-1, y=2, y-axis'

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'Calculate the area of the triangle formed by the points P = (1, 1), Q = (4, 5), and R = (7, 2).'

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'Find the area S enclosed by the following curve and straight lines: y=√x, x-axis, x=1, x=2.'

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'Ancient Greeks used geometry to calculate areas and volumes.'

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'Please calculate the area of an isosceles triangle with one side of 10cm and the other side of 15cm.'

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"Probability of a hundred yen coin fitting into a tile When throwing a hundred yen coin (diameter 2.2 cm) onto a large floor covered with square tiles of side 3 cm, let's consider the probability of it fitting completely onto one tile."

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'When the points P, Q, R on a plane are not collinear, the area of the triangle with them as 3 vertices is denoted by △PQR. Also, when P, Q, R are collinear, the area is defined as △PQR=0. Let A, B, C be 3 points on the plane, with △ABC=1. Find the area of the feasible range of the movement of point X on the plane while satisfying 2≤△ABX+△BCX+△CAX≤3. [University of Tokyo]'

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'Find the area of a quadrilateral inscribed in a circle with a radius of 106 (2)\nIn the quadrilateral ABCD inscribed in a circle with a radius of 106, if AB=5, BC=4, CD=4, DA=2, find the area S of the quadrilateral ABCD.'

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Find the area $S$ of the triangle $riangle \mathrm{OAB}$ in each of the following cases. (2) When the vertices are the 3 points \( \mathrm{O}(0,0), \mathrm{A}(1,-3), \mathrm{B}(2,2) \), let \overrightarrow{\mathrm{OA}}=ec{a} and \overrightarrow{\mathrm{OB}}=ec{b} .