Example Question - area calculation
Here are examples of questions we've helped users solve.
Solving Linear Equations Involving Area
Da zwei Gleichungen gegeben sind und das Ziel ist, \( y \) zu lösen, kann das folgendermaßen angegangen werden: <p>\( A = 12 \cdot y \) (1)</p> <p>\( A = 52 \cdot y \) (2)</p> Diese beiden Gleichungen scheinen einen gemeinsamen Flächenwert \( A \) zu haben, aber unterschiedliche Multiplikatoren für \( y \). Wenn das der Fall ist, wären die Multiplikatoren gleich, d.h. \( 12 = 52 \), was offensichtlich nicht wahr ist. Daher muss es einen Fehler in der Fragestellung geben, oder es fehlt Kontext bzw. zusätzliche Information, um die Gleichungen angemessen zu lösen. Ohne zusätzliche Information kann nicht bestimmt werden, was \( y \) ist, oder ob \( A \) in beiden Gleichungen gleich ist. Wenn \( A \) nicht gleich sein soll, kann \( y \) für jede Gleichung unterschiedlich sein, aber ohne spezifische Werte für \( A \) kann \( y \) nicht gelöst werden.
Area Calculation of a Geometric Shape
<p>Da kein genauer mathematischer Kontext oder eine geometrische Form angegeben ist, werde ich annehmen, dass die Berechnung die eines Rechtecks ist, da es zwei verschiedene Maßangaben gibt (was bei einem Rechteck zwei Seiten darstellen kann). Der Buchstabe "A" wird oft verwendet, um den Flächeninhalt (Area) zu bezeichnen, und "L" könnte für die Länge (Length) stehen, während "B" für die Breite (Breadth) stehen könnte. Doch ohne Kontext ist dies eine Annahme.</p> <p>Um den Flächeninhalt A eines Rechtecks zu finden, multiplizieren wir die Länge (L) mit der Breite (B):</p> \[ A = L \cdot B \] <p>Setzen wir die gegebenen Maße ein:</p> \[ A = 13\ cm \cdot 18\ cm \] <p>Berechnung des Produkts:</p> \[ A = 234\ cm^2 \] <p>Der Flächeninhalt des Rechtecks beträgt 234 Quadratzentimeter.</p>
Geometry Problem Involving Circular Sector and Area Calculation
La pregunta número 2 pregunta por el área y perímetro del sector circular, cuya fórmula para el área (A) es \( A = \frac{\theta}{360} \cdot \pi r^2 \) y para el perímetro (P) es \( P = 2r + \text{longitud del arco} \), donde la longitud del arco es \( \frac{\theta}{360} \cdot 2\pi r \). Dado que: \( r = 3 \) cm \( \theta = 45 \)° Se sigue que: \( A = \frac{45}{360} \cdot \pi \cdot 3^2 \) <p>\( A = \frac{1}{8} \cdot \pi \cdot 9 \)</p> <p>\( A = \frac{9}{8} \cdot \pi \) cm²</p> Luego, para calcular el perímetro (P): <p>\( P = 2 \cdot 3 + \frac{45}{360} \cdot 2\pi \cdot 3 \)</p> <p>\( P = 6 + \frac{1}{8} \cdot 6\pi \)</p> <p>\( P = 6 + \frac{3}{4} \pi \) cm</p> Por lo tanto, la respuesta al área y perímetro del sector circular es: \( \frac{9}{8} \pi \) cm² y \( 6 + \frac{3}{4} \pi \) cm, respectivamente. La opción correcta es la B.
Calculating the Combined Area of Geometric Shapes
<p>Let's denote the area of the triangle as A1 and the area of the square as A2.</p> <p>The area of the triangle (A1) can be calculated using the formula for the area of a triangle:</p> <p>A1 = \frac{1}{2} \times base \times height = \frac{1}{2} \times 5 \times 6</p> <p>A1 = \frac{1}{2} \times 30</p> <p>A1 = 15 cm^2</p> <p>The area of the square (A2) can be calculated using the formula for the area of a square:</p> <p>A2 = side^2 = 5^2</p> <p>A2 = 25 cm^2</p> <p>The total area of the given figure is the sum of the areas of the triangle and the square:</p> <p>Total Area = A1 + A2</p> <p>Total Area = 15 cm^2 + 25 cm^2</p> <p>Total Area = 40 cm^2</p>
Area Calculation of Overlapping Squares
Let the side of the larger square be \( x \) cm. The area of the larger square is \( x^2 \) cm\(^2\). Given the side of the smaller square is 10 cm, the area of the smaller square is \( 10^2 = 100 \) cm\(^2\). The difference in area between the shaded part B and the unshaded part A is given as 24 cm\(^2\). We know that the unshaded part A is a smaller square with an area of 100 cm\(^2\) minus the part overlapping with the larger square. Let the side of the overlapped square be \( y \) cm. The area of the shaded part B includes two rectangles and the overlapped square: \( 2 \times x \times y + y^2 \). The difference given is: \( 2xy + y^2 - 100 = 24 \). As the side of the larger square is \( x \) and the side of the overlapped square is \( y \), then \( x = 10 + y \). Replace \( x \) with \( 10 + y \) in the difference equation: \( 2(10 + y)y + y^2 - 100 = 24 \) \( 20y + 2y^2 + y^2 - 100 = 24 \) \( 3y^2 + 20y - 124 = 0 \) Solving this quadratic equation by the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = 20 \), and \( c = -124 \): \( y = \frac{-20 \pm \sqrt{400 + 1488}}{6} \) \( y = \frac{-20 \pm \sqrt{1888}}{6} \) \( y = \frac{-20 \pm 2\sqrt{472}}{6} \) \( y = \frac{-10 \pm \sqrt{472}}{3} \) Since \( y \) must be positive and it is the smaller dimension, we take the positive root: \( y = \frac{-10 + \sqrt{472}}{3} \) Now we find \( x \): \( x = 10 + y \) \( x = 10 + \frac{-10 + \sqrt{472}}{3} \) The area of the unshaded part A is the side of the smaller square squared: Area of A is \( 10^2 = 100 \) cm\(^2\). The area of the shaded part B is the side of the larger square squared minus the area of A. Using the values of \( x \) and \( y \) found above, we can calculate the exact area of B.
Overlapping Square Areas Problem
<p>Let the side of the larger square be \( x \) cm, therefore its area is \( x^2 \) cm².</p> <p>Let the side of the smaller square be \( (x - 10) \) cm, area is \( (x - 10)^2 \) cm².</p> <p>The area of the shaded part A is the difference between the areas of the two squares: \( x^2 - (x - 10)^2 = 24 \) cm².</p> <p>Solving the equation:</p> <p>\( x^2 - (x^2 - 20x + 100) = 24 \)</p> <p>\( x^2 - x^2 + 20x - 100 = 24 \)</p> <p>\( 20x - 100 = 24 \)</p> <p>\( 20x = 124 \)</p> <p>\( x = 6.2 \) cm</p> <p>The side of the smaller square is \( (6.2 - 10) = -3.8 \) cm, which is not possible.</p> <p>There seems to be an error because the side of the smaller square cannot be a negative number. Please verify the information provided in the question.</p>
Finding the Area of a Quadrilateral Within a Rectangle
Let's denote the length of BE as \( x \), so AE is \( 3x \) because BE is one-third of AE. Given that the area of the rectangle ABCD is \( 112 \, \text{cm}^2 \) and that AE is three times BE, the width of the rectangle is \( 4x \). Assume the height of the rectangle (BF) is \( h \). From \( 2\text{BF} = 5\text{FC} \), we have \( 2h = \frac{5}{7}(4x) \) because FC is the remainder of the length of the rectangle after removing BF, which is \( \frac{7}{2}h \). Solving for \( h \), we get \( h = \frac{5}{7} \cdot \frac{4}{2}x = \frac{10}{7}x \). Now we can find \( x \) from the area of the rectangle: \[ \text{Area} = 4x \cdot h = 112 \] \[ 4x \cdot \frac{10}{7}x = 112 \] \[ \frac{40}{7}x^2 = 112 \] \[ x^2 = \frac{112 \cdot 7}{40} \] \[ x^2 = 19.6 \] \[ x = \sqrt{19.6} \] \[ x = 4.43 \] (approx) Now we calculate \( h \): \[ h = \frac{10}{7} \cdot 4.43 = 6.33 \] (approx) To find the area of EFGF, we need to find the areas of triangles BEG and CFF and subtract them from the area of the rectangle. The triangle BEG has a base of x and height h, and the triangle CFF has a base of 3x and height \( \frac{2}{7}h \). Area of BEG: \[ \text{Area}_{\text{BEG}} = \frac{1}{2} \cdot x \cdot h = \frac{1}{2} \cdot 4.43 \cdot 6.33 \] \[ \text{Area}_{\text{BEG}} = 14.01 \] (approx) Area of CFF: \[ \text{Area}_{\text{CFF}} = \frac{1}{2} \cdot 3x \cdot \frac{2}{7}h = \frac{1}{2} \cdot 3 \cdot 4.43 \cdot \frac{2}{7} \cdot 6.33 \] \[ \text{Area}_{\text{CFF}} = 12.15 \] (approx) Total area of the two triangles: \[ \text{Area}_{\text{TotalTriangles}} = \text{Area}_{\text{BEG}} + \text{Area}_{\text{CFF}} \] \[ \text{Area}_{\text{TotalTriangles}} = 14.01 + 12.15 \] \[ \text{Area}_{\text{TotalTriangles}} = 26.16 \] (approx) Finally, the area of quadrilateral EFGF is: \[ \text{Area}_{\text{EFGF}} = 112 - \text{Area}_{\text{TotalTriangles}} \] \[ \text{Area}_{\text{EFGF}} = 112 - 26.16 \] \[ \text{Area}_{\text{EFGF}} = 85.84 \, \text{cm}^2 \] (approx) Please note that the solution is an approximation due to rounding the value of \( x \) to two decimal places.
Understanding the Phrase 'Rumus Luas Berish Atas (Miring)' in Geometry
It seems like you've shown a part of an image of a handwritten page, which contains a phrase that appears to be in the Indonesian language. The phrase "Rumus Luas berish atas (miring)" translates to "Formula for area of the upper clean side (sloped or slanting)" in English. This phrase alone doesn't provide enough information to solve a specific problem because it seems to be referring to a geometrical formula related to an area calculation of a shape that has an upper side that is sloped or slanted. The phrase might refer to the area calculation of a trapezoid, where the top side is not parallel to the bottom side, or possibly another shape with a slanted side. If you have a specific geometry problem or question, please provide the full context or problem statement so I can help you solve it.
Calculating Area of Square Pyramid and Lateral Faces
The image shows a square pyramid with a slant height of 4.4 inches and a base edge length of 4.5 inches. We are asked to solve for the area of the base and the area of each lateral face. Part A: What is the area of the base? Since the base of the pyramid is a square, the area A of a square is given by the formula \( A = s^2 \) where s is the length of a side. Given that s = 4.5 inches, the area of the base is: \[ A = 4.5^2 = 20.25 \] \[ A = 20.25 \text{ square inches} \] The area of the base is therefore 20.25 square inches. Part A also asks for the area of each lateral face. Each lateral face is a triangle with a base of 4.5 inches and a slant height of 4.4 inches which acts as the altitude in this situation. The area A of a triangle is given by the formula \( A = \frac{1}{2}bh \) where b is the base and h is the height (altitude) of the triangle. Using the provided measurements, the area of one triangular lateral face is: \[ A = \frac{1}{2} \times 4.5 \times 4.4 \] \[ A = \frac{1}{2} \times 19.8 \] \[ A = 9.9 \] \[ A = 9.9 \text{ square inches} \] Thus, the area of each lateral face is 9.9 square inches.
Calculating Areas of Triangular Pyramid
The image shows a triangular pyramid, and the task is to find the area of the base and the area of each lateral face. The base of the pyramid is a square, with each side measuring 4.5 inches. The area of a square is calculated by squaring the length of one of its sides. Area of the base (A_base) = side_length^2 A_base = (4.5 in)^2 A_base = 20.25 in^2 For the lateral face (each triangular side of the pyramid), the formula for the area of a triangle is used, which is: Area of a triangle (A_triangle) = (base * height) / 2 From the image, it appears that the calculation for one of the triangular faces has already been made: A_triangle = (4.5 in * 4.4 in) / 2 A_triangle = (19.8 in^2) / 2 A_triangle = 9.9 in^2 Therefore, the area of the base of the pyramid is 20.25 square inches, and the area of each lateral face is 9.9 square inches.
Analysis of a Shaded Area in a Cartesian Coordinate System
Parece que hay una gráfica en la imagen que has proporcionado, pero no puedo ver el texto de las preguntas o los detalles que puedas necesitar resolver. Sin embargo, puedo describir lo que veo y dar una vista general de cómo uno podría abordar un problema relacionado con esta gráfica. La imagen muestra un sistema de coordenadas cartesianas con dos líneas rectas. Una línea es la función identidad, \( f(x) = x \), que es una línea que pasa por el origen con una pendiente de 1. La otra línea parece ser una función lineal con pendiente negativa, pero no tengo información sobre su ecuación exacta. Hay un área sombreada entre las dos líneas, que parece ser un triángulo rectángulo. Para calcular el área de este triángulo, necesitarías determinar la base y la altura del triángulo. La base puede ser la distancia en el eje x entre los puntos donde las dos líneas se cruzan con el eje, y la altura es la distancia en el eje y entre estos mismos puntos de cruce. Si tuvieras las coordenadas exactas de estos puntos de cruce, podrías utilizar la fórmula del área de un triángulo: \[ \text{Área} = \frac{1}{2} \times \text{base} \times \text{altura} \] Si necesitas ayuda para calcular algo más específico, por favor proporción más context o detalles del problema.
Calculation of Colored Areas in Circles
Para resolver la pregunta, necesitamos calcular el área de la parte coloreada de cada una de las figuras que se muestran. Primera figura (círculo con sectores): Podemos ver que el círculo está dividido en 8 sectores iguales, y 4 de ellos están coloreados. Esto significa que la parte coloreada representa la mitad del área total del círculo. La fórmula para calcular el área de un círculo es: \[ A = \pi r^2 \] donde \( r \) es el radio del círculo. El radio del círculo en la figura es la mitad de la longitud del diámetro, así que \( r = \frac{6 \text{ cm}}{2} = 3 \text{ cm} \). Entonces, el área total del círculo es: \[ A = \pi (3 \text{ cm})^2 = \pi \cdot 9 \text{ cm}^2 \] La mitad de esta área corresponde a la parte coloreada: \[ \text{Área coloreada del círculo} = \frac{1}{2} \cdot \pi \cdot 9 \text{ cm}^2 = 4.5\pi \text{ cm}^2 \] Segunda figura (cuadrante de un círculo): La figura parece ser un cuadrante de círculo, que es el 25% (o 1/4) del área de un círculo completo. Primero tenemos que calcular el área del círculo completo usando la misma fórmula \( A = \pi r^2 \) y luego tomar una cuarta parte de esta área. El diámetro del círculo completo del cual el cuadrante es una parte sería de 10 cm, lo que significa que el radio es de \( r = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \). El área total del círculo completo sería: \[ A = \pi (5 \text{ cm})^2 = \pi \cdot 25 \text{ cm}^2 \] Entonces, el área del cuadrante sería un cuarto de esta área: \[ \text{Área del cuadrante} = \frac{1}{4} \cdot \pi \cdot 25 \text{ cm}^2 = 6.25\pi \text{ cm}^2 \] Sumando las áreas coloreadas de ambas figuras obtenemos el área total coloreada: \[ \text{Área total coloreada} = 4.5\pi \text{ cm}^2 + 6.25\pi \text{ cm}^2 = 10.75\pi \text{ cm}^2 \] Para obtener una aproximación numérica, consideramos que \( \pi \approx 3.1416 \), por lo que: \[ \text{Área total coloreada} \approx 10.75 \times 3.1416 \text{ cm}^2 \approx 33.79 \text{ cm}^2 \] Por lo tanto, el área total aproximada de las partes coloreadas de ambas figuras es aproximadamente \( 33.79 \text{ cm}^2 \).
Finding the Total Area of a Composite Rectangular Figure
The image shows a composite rectangular figure, and we are asked to find its area. The shape is somewhat like a staircase, composed of three rectangles put together. To find the total area, we can calculate the area of each rectangle and then sum them up. Looking closely at the image, though blurry, the dimensions of the rectangles are given: 1. The top right rectangle has dimensions 5 cm (width) and 6 cm (height). 2. The middle rectangle has dimensions 8 cm (width, same as the total width at the bottom) and 3 cm (height). 3. The bottom left rectangle is a square with dimensions 3 cm by 3 cm; its size can be inferred by the measurements given for the other parts of the figure. The area of each rectangle is found by multiplying its length by its width. Let's calculate each: 1. Top right rectangle: Area = width × height = 5 cm × 6 cm = 30 cm² 2. Middle rectangle: Area = width × height = 8 cm × 3 cm = 24 cm² 3. Bottom left square: Area = side × side = 3 cm × 3 cm = 9 cm² Now, we sum up the areas of all three rectangles to find the total area of the figure: Total area = Area of top right rectangle + Area of middle rectangle + Area of bottom left square Total area = 30 cm² + 24 cm² + 9 cm² = 63 cm² So, the area of the figure is 63 square centimeters.
Calculating Area of Composite Figure Using Rectangles
To find the area of the composite figure shown in the image, we can divide it into simpler shapes (such as rectangles) and then calculate the area of each before summing them up. From the image, it appears that we can divide the figure into two rectangles: 1. The larger rectangle on the left, which has a width of 3 cm (since the entire bottom length is 8 cm, and the length to the right is 5 cm, the difference is 8 cm - 5 cm = 3 cm) and a height of 6 cm. 2. The smaller rectangle on the right, which has a width of 5 cm and a height of 2 cm (since the entire length on the left is 6 cm, and the topmost length is 4 cm, the difference is 6 cm - 4 cm = 2 cm). Let's calculate their areas: For the larger rectangle: Area = width × height = 3 cm × 6 cm = 18 cm² For the smaller rectangle: Area = width × height = 5 cm × 2 cm = 10 cm² Now, we add the areas of the two rectangles together to get the total area of the figure: Total Area = Area of larger rectangle + Area of smaller rectangle Total Area = 18 cm² + 10 cm² = 28 cm² So, the area of the figure is 28 square centimeters.
Calculating Area of Composite Figure with Rectangles
To find the area of this composite figure, which is a combination of rectangles, we can approach it by breaking down the shape into simpler parts that we can easily calculate the area for and then combine them. From the image, we can see that there is a large rectangle on the right side of the figure with the dimensions of 5 cm by 8 cm. Next to it on the left, there is an upside-down "L" shaped figure which can be broken down into two smaller rectangles - one with the dimensions of 3 cm by 3 cm and the other 2 cm by 5 cm. Let's calculate the area for each part: 1. Large rectangle: 5 cm x 8 cm = 40 cm² 2. Small square (3 cm by 3 cm): 3 cm x 3 cm = 9 cm² 3. Small rectangle (2 cm by 5 cm): 2 cm x 5 cm = 10 cm² Now let's add up the areas of all parts: Total area = Large rectangle area + Small square area + Small rectangle area Total area = 40 cm² + 9 cm² + 10 cm² Total area = 59 cm² So, the area of the figure is 59 square centimeters.
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