Questions tagged [area]

Area is a quantity that expresses the extent of a two-dimensional or three-dimensional surface or shape.

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In $\triangle ABC$, we have $\angle BAC = 60^\circ$ and $\angle ABC = 45^\circ$… [on hold]

In $\triangle ABC$, we have $\angle BAC = 60^\circ$ and $\angle ABC = 45^\circ$. The bisector of $\angle A$ intersects $\overline{BC}$ at point $T$, and $AT = 24$. What is the area of $\triangle ABC$? ...
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Ratio of area of a triangle to that of its medians

Let a triangle ABC have medians namely l,m,n. I can't figure out how to find the ratio of area of triangle LMN to area of triangle ABC. LMN is the triangle with sides of lengths of the three different ...
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1answer
50 views

How to find an area of a figure that's restricted by $\frac{a^3}{a^2+x^2}, a\neq0, a\in R$ and $2ay=x^2$ using definite integral

I have to find an area using definite integral, where the figure is restricted by two functions, stated in title. Can someone explain how to do that? Here is my try: So I equal the expressions to ...
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Find the area of triangle $SVW$

To find length of side $SV$ I used Pythagoras theorem which gave $SV=17$. To find angle $SVW$ I added $45$ to $90 = 135$ to find side length $SW = 17^2+24^2-2 \cdot 17 \cdot 24 \cdot \cos135=37$ and ...
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Ratio between the width of the intersection of two identical intersecting circles and radius, when the intersection is $\frac{\pi r^2}{2}$

Or more visually, if all sections of the below diagram were equal in area and the circles are identical, what is the ratio of s and r, or what is s in terms of r. I came up with an equation using ...
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1answer
63 views

How is land area calculated when the ellipsoidal shape of the Earth cannot be neglected?

I was curious as to how the land area of a state such as Colorado could be calculated. I understand the area of a 2D rectangle can be calculated using the formula width times length. However, I was ...
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How to find the area of Triangle $\triangle AEF$ in rectangle $\square ABCD$? Is there geometric solution? [on hold]

$E$ is on $\overline {BC} $ , $F$ is on $\overline {CD}$ $\triangle ABE= 40 , \triangle ECF= 60, \triangle ADF= 125 $ How to find the area of Triangle $AEF$ in rectangle $ABCD$? Is there geometric ...
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Area of Portion of Sphere from a inside Cube

The late painter Maqbool Fida Husain once coloured the surface of a huge hollow steel sphere, of radius $1$ metre, using just two colours, Red and Blue. As was his style however, both the red and blue ...
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What does $\int_\mathbb{R^d}$ means in terms of limits

This might be a silly question, but please help me understand that does the limits of the following expression. The left-hand-side is the actual expression, while the right-hand-side is my ...
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1answer
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Find area of cross section of cylinder by the plane $x$

I am working on my scholarship exam practice (assume high school/pre-university math background) and I think I got half way through but I am not sure how I could continue. Let $r$ be a positive ...
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Does angle cause the change of area? [closed]

Suppose there is a square, length =5cm. If one of its angle increases or decreases then it will be rhombus. But will it cause the area to change?
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Triangles area related problem

The question is :- In $\Delta ABC$ , $X$ and $Y$ are points on the sides $AC$ and $BC$ respectively .If $Z$ is on the segment $XY$ such that $ \frac {AX}{XC}=\frac {CY}{YB}=\frac {XZ}{ZY}$ ....
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Area of $y^2=x^2(a^2-x^2)$

Im trying to find the area of $y^2=x^2(a^2-x^2)$ where $a>0$. From my calculations it seems to be $$ 2\int_{0}^{a} x\sqrt{a^2 - x^2}dx=\frac{2}{3}a^3, $$ but I am not sure if it is right.
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Area of a Polygon in a Polygon

I'm dealing with a regular polygon with 7 corners. In this polygon is another polygon defined by connecting one point with the two opposite points of the same polygon. I made a small sketch of the ...
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1answer
82 views

Maximum total area of n non-intersect circles?

Given n points on the x-axis, we give arbitrary radius for each point such that each constructed circle doesn't overlap another constructed circle from another point. Which means these circles do not ...
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Is it possible to cover all circle area with infinite lines starting from the center? [closed]

Is it possible to cover all area of a circle of radius r>0 with infinite lines starting from the center?
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215 views

Area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$

Find the area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$ (using definite integration). I cannot understand how to find this area. I have graphed the lines and found ...
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27 views

Can I say that Integration can be equal to the formula of finding the area of a right triangle?

Let us have a certain function like $f(x)=x$ In the world of integration, can I say that $\int_{0}^bxdx=\frac{1}{2}(b)(f(b))$? Because if you look at graph of function x, it would look like something ...
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0answers
41 views

Surface area of melting ice block

A cubic block of ice is melting and retains its cubic shape as it melts. Its volume (in $\rm m^3$) at time $t$ is given by $$ V = 4000-2000 \cdot e^{0.01t}$$ As the ice melts, the bottom ...
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1answer
20 views

Area of the common region of three circles.

Find the area of the region that is common to the circles $r = 1$, $r = 2 \cos θ$, and $r = 2 \sin θ$. I tried many ways to get the common region, but it seems impossible to eliminate to that point, ...
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Area under curve given a set $\{ [x, y] \in \mathbb{R^2} \ | \ 0 \le x \le 1 \ \& \ 0 \le y \le x \arctan^2 x\}$

Evaluate area under the curve for: $\{ [x, y] \in \mathbb{R^2} \ | \ 0 \le x \le 1 \ \& \ 0 \le y \le x \arctan^2 x\}$ I know that to find the area under a curve of a function from a to b, ...
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find area of kite given side length

Circle with two tangent lines Above is the picture in question. A circle is given, center (-2,4) and a point outside the circle (0,10) is shown. Asked to calculate the area of the quadrilateral ABCD, ...
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Calculating the area between two functions expressed in polar coördinates

I have the following to polar coördinates: $r=1+\cos(\theta)$ and $r=3\cos(\theta)$. The question is to calculate the area in side $r=1+\cos(\theta)$ and outside $r=3\cos(\theta)$. I know I need to ...
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2answers
239 views

Grazing area for a goat around a circle.

I am doing this math question and i am really confused on how to approach it. This is the question: A retired mathematics professor has decided to raise a goat. He owns a silo and a barn. The barns ...
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1answer
47 views

calculating the area in polar coördinates

I have difficulties calculating the area and setting the right boundaries of the following polar coördinates: $$r=2(1+cos(\theta) ) $$ Thanks in advance
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1answer
46 views

Find the area of a sphere inside a function

I want to find the area of the portion of the sphere $x^2+y^2+z^2=4z$ inside the function $x^2+y^2=z$ using double integrals. The graph would be something like this: Because of the nature of this ...
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0answers
70 views

Is this result already a known theorem in geometry?

I have been playing around with geometry and I found that: Let two perpendicular lines intersect at a point that is inside a circle. Then the area of the quadrilateral formed by the vertices made by ...
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1answer
83 views

Find the area limited by 4 curves, using change of variables

I'm trying to show that the area bound by the curves $r^2= 3\cos(2\theta)$, $r^2= 4\cos(2\theta)$, $r^2= 3\sin(2\theta)$, $r^2= 4\sin(2\theta)$ in the first quadrant is equal to $$A= \frac{10 - 7\...
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2answers
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Finding area of triangle given ray of altitude

three altitude intersecting each other at H. Find the area of triangle ABC My attempt feel like cheating a little bit, I let BA = $\dot{\vec{C}}$, AC = $\dot{\vec{B}}$ and CB = $\dot{\vec{A}}$ then ...
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1answer
25 views

Find area under $y= x^2 - x^4$ from x=-1 to x=0 using the Riemann sum

I'm trying to find the area under $y= x^2 - x^4$ from $x=-1$ to $x=0$ using the Riemann sum. This is what I've done so far: $\Delta x = 1/n$ $x_i =-1 + i/n$ $A = R_n = \lim_{n\to \infty} {\sum_{i=...
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1answer
47 views

Maximization problems

I am in a introductory course of calculus in several variables and i have these problems. A farmer wants to build a corral with pentagonal form(not regular) that is formed by the union of a rectangle ...
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using partial derivations to get the approximate error for this problem

The area of a triangle is $A=\frac{1}{2}a*b*sin(c)$ $𝑎*𝑏* sin( 𝑐)$, where $a$, $b$ are two sides of the triangle and $c$ is the included angle. In surveying a triangular plot of land, $a$ and $b$ ...
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Estimating Area Exam Question

Pieces of turf are 1m long by 0.5m wide. Each piece costs £3.79 . $1 * 0.5 = 0.5\text m^2$ a)Estimate the cost of turf required to cover these spaces. i) 9.6m by 2.4m $10 * 2 = 20\text m^2$ ...
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1answer
34 views

Calculate area enclosed by 4 curves

I am trying to find the area enclosed by 4 piecewise smooth curves. As can be seen from the figure, BLACK curve is a segment of a circle, C ...
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4answers
131 views

Definite integral of $1/(2\sin^4x + 3\cos^2x)$

I have $f = \frac 1 {(2\sin^4x + 3\cos^2x)}$ which area should be calculated from $0$ to $\frac{3\pi}2 $. I noticed that $$\int_0^{\frac{3\pi}2} f \,dx= 3\int_0^{\frac{\pi}2} f \,dx$$ I tried to ...
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Calculating how much light gets through steel mesh (commonly used to make cages)

I have expanded steel mesh that I use to make garden cages: I would like to know how much sunlight the mesh lets through. I think I need to calculate the area of the mesh's negative space. And then ...
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1answer
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find the area enclosed by $ f(x)=x+\sin(x)$ and its inverse from $x=0$ to $x=2$ [closed]

I don't have a single clue to start, and we cant find the inverse so we must use some properties, but which ones? thanks
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3answers
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I Need Help in a Challenge [closed]

My teacher challenged me with the question below: $$\sqrt{\frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2}\ast \left ( \frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2} - \frac{\sqrt{41}}{1} \right )\ast \left ( \frac{\...
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2answers
138 views

How can i Prove that the gray area is the same as white area? [duplicate]

A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area? I just want any hint for solving this question....
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1answer
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Find area of cylinder $x^2 + y^2 = r^2$ that satisfies $0 \le z \le y$

I think I can imagine the shape of surface area. This is what I did: $$\begin{eqnarray*} \text{Surface area} & = & \int_{0}^{\pi} r y \sqrt {2} \, d\theta \\ & = & \int_{0}^{\pi} r\...
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How to calculate area of triangle-like structure of blocks? (Pick's Theorem seems insufficient.)

Given the following structure, is there a formula to calculate the number of blocks? (EDIT: and I am really looking for a solution for any BASE and HEIGHT.) At first, it would seem that this is a ...
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1answer
44 views

How do you find area of the loop in the graph of $x(x^2+y^2)=(x^2-y^2)$

The graph of the given equation is $x(x^2+y^2)=(x^2-y^2)$"> I believe I have to use (r,θ) coordinates but I do not know how to integrate this in (r,θ).
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1answer
23 views

Differential area for the lateral surface of frustum of a cone

I am studying Fluid Mechanics and I needed a differential area element of the side or lateral surface of a frustum. This frustum is cut from a cone. In solution manual of the book I study, ...
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3answers
46 views

Area defined by $x^2+y^2 \leq 1$ and $y\geq x(x^2-16)$

Area defined by $x^2+y^2 \leq 1$ and $y\geq x(x^2-16)$ One very obvious way would be to find the points of intersection which would be messy and subject to many conditions. I was trying to solve ...
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1answer
44 views

Find the diameter of a circle subtended by an angle

The question doesn't state whether its subtended at the center or circumference, but I not sure if it matters The sector a circle subtended by an angle of $22.5$ degrees has an area of $\frac{9\pi}{4}...
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Integrating an absolute value function to find area between curves $[∫^{-1.02}_{-2.84}(|cos(5.7x-10)|+1.7)dx] $

I'm trying to find the area between the curves $5.7 e^{-0.7x-3}+1.3 $ and $-|cos(5.7x-10)|+1.7 $ from $-2.84$ to $-1.02$ After graphing this and finding the upper and lower functions, it lead me to ...
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4answers
48 views

Why is my approach for showing $r^2 \frac{\theta}{2}$ equals the area of a circular sector incorrect? Do we need calculus?

I know that the area of the sector of circle can be computed using the following two formulas $$\pi r^2 \frac{\theta }{360} \space \space \text{ (degrees case)}$$ $$or$$ $$r^2 \frac{\theta }{2} \...
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1answer
60 views

Area bound by $y =\ln x +\tan^{-1} x$?

I'm not really sure how to do this without having a real graph using a graphing calculator. Adding the integrals would no work since some areas are overlapping, and even subtracting doesn't since one ...
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1answer
46 views

Finding $\iiint 6z\,dx\,dy\,dz$ over $\lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$

I want to calculate integral : $\iiint 6z\,dx\,dy\,dz$. The area is $$\Omega= \lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$$ My problem is this, that I don't know what is ...
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1answer
32 views

Determine the area enclosed by the curve

Determine the area enclosed by the curve with two polar equations: $r= \sqrt 2 \sin(\alpha)$ $r^2 = \sin(2\alpha)$ I have no clue how to do this. A formula we are given is the one below but I'm ...