📐 geometry

1. **Calculate the area of the sector of a circle with radius 65 mm and sector angle 42°.** - The formula for the area of a sector is $$Area = \frac{\theta}{360} \times \pi r^2$$ w

1. The problem describes a semicircle with a radius of 7.2 meters and a horizontal diameter of 7.2 meters. This indicates possibly a misunderstanding since the diameter should be t

1. Problem statement: We are given an L-shaped polygon with a top horizontal edge of $12$ mm, an interior vertical located $4.3$ mm from the left that drops $2$ mm from the top to

1. **State the problem:** We are given an irregular polygon with labeled sides and need to find its area in mm². 2. **Analyze the shape:** The figure has two vertical segments labe

1. **Problem 10:** Calculate the perimeter of sector OAB with radius $r=7$ cm and area $40$ cm$^2$. 2. The area $A$ of a sector is given by $$A = \frac{1}{2} r^2 \theta$$ where $\t

1. **Problem 6**: A cone has a horizontal base diameter of 24 cm and an axis length of 36 cm with a volume of 3000 cm³. Find the angle of inclination of the axis with the horizonta

1. **State the problem:** We need to determine if a 50 cm ribbon is long enough to go all the way around the cake. 2. **Understanding the shape:** The top of the cake is a circle w

1. **State the problem:** We have a large cone with height $36$ cm and radius $10$ cm.

1. Let's analyze each house diagram carefully. 2. For the first house (top-left):

1. Given that points P, Q, and S lie on a Cartesian plane with PQ parallel to the x-axis and PQ equals 20 times the length of segment S. 2. Point S is at (-6,0), Q is at (k,8), and

1. We are asked to find the volume of an ice cream cone with radius $r=2$ inches and height $h=6$ inches. 2. The volume $V$ of a cone is given by the formula $$V = \frac{1}{3} \pi

1. **Problem statement:** We have points A(4,1) and B(2,5). For each point C with positive integer coordinates $C(x,y)$, define $d_C$ as the shortest distance to travel from A to C

1. **State the problem:** We have a solid formed by removing a hemisphere (radius $2x$ cm) from a cone (base radius $5x$ cm, height $6x$ cm). The volume of this solid is $6948\pi$

1. **Problem statement:** We have a sector OABC of a circle with the center at O. The angle at O (angle AOC) is 60°.

1. **State the problem:** Find the value of angle $x$ in quadrilateral $ABCD$ where angles at $D$ and $C$ are $40^\circ$ and $62^\circ$ respectively, and sides $AB=15.2$ m, $BC=12.

1. **State the problem:** We have a regular pentagon ABCDE and a point F outside the pentagon such that the angle \(\angle AEF = 96^\circ\). We need to find the size of the obtuse

1. The problem asks for the volume of water in a rectangular prism-shaped container. 2. The base dimensions of the container are given as 90 mm by 90 mm, and the height of the wate

1. The problem states we have a rectangle with length 8 meters and height 5 meters. 2. To find the perimeter of a rectangle, use the formula $$P = 2(\text{length} + \text{width})$$

1. Stating the problem: We have a parallelogram ABCD. 2. Draw diagonal $AC$.

1. The problem asks us to calculate the volume of a cone with radius $r=10$ cm and height $h=36$ cm. 2. The formula for the volume of a cone is given by

1. **State the problem:** We need to find the area of quadrilateral EFGH with sides EH = 8 cm (vertical), HG = 14 cm (horizontal), and GF = 2 cm (vertical). Angles EHG and GFE are