📐 geometry

1. **State the problem:** We need to find the amount of paper required to make a paper wrap shaped like a cone with radius $r=3$ cm, height $h=7$ cm, and slant height $l=7.62$ cm.

1. **State the problem:** We need to find the volume of a cylinder with radius $r=9$ meters and height $h=16$ meters. 2. **Formula:** The volume $V$ of a cylinder is given by the f

1. **State the problem:** We need to find the volume of a sphere with a diameter of 30 miles. 2. **Formula:** The volume $V$ of a sphere is given by the formula:

1. The problem is to find the volume of a sphere. 2. The formula for the volume of a sphere is $$V = \frac{4}{3} \pi r^3$$ where $r$ is the radius of the sphere.

1. **State the problem:** Find the surface area of a sphere with a diameter of 30 miles. 2. **Formula:** The surface area $A$ of a sphere is given by the formula:

1. **Problem statement:** Find the length of side $x$ in a right triangle where the angles are $30^\circ$, $60^\circ$, and $90^\circ$, with the side opposite the $30^\circ$ angle g

1. **Problem statement:** A ladder 16 m long is placed with its foot 3 m from a building. We want to find how much further the foot must be moved away from the building to lower th

1. **State the problem:** We have a right-angled triangle with hypotenuse 18.7 cm, one leg 15.2 cm, and we need to find angles $x$ and $y$. 2. **Recall the Pythagorean theorem and

1. **State the problem:** We need to find how many cubic boxes of dimension 50cm × 50cm × 50cm can fit into a storage room of dimensions 4m × 2m × 4m. 2. **Convert units:** Since t

1. **State the problem:** We are given a circle with radius $r = 6.8$ cm. 2. **Find the diameter:** The diameter $d$ of a circle is twice the radius. The formula is:

1. **Problem Statement:** Find the values of $x$ and $y$ that make each quadrilateral a parallelogram. 2. **Key Property:** In a parallelogram, the diagonals bisect each other. Thi

1. The problem asks for the circumference of a circle with a diameter of 17 cm. 2. The formula for the circumference $C$ of a circle is:

1. **Problem Statement:** You need to fill four boxes with digits 1 to 9 (each digit used at most once) in three different ways:

1. **Problem statement:** We have a frustum formed by removing a smaller cone from a larger cone. The smaller cone has height $10$ m and top radius $6$ m. The original large cone h

1. **State the problem:** We have a large cone of height 39 mm and base radius 6 mm. It is split horizontally into a smaller cone on top (height 13 mm, radius 2 mm) and a frustum b

1. **Problem statement:** Given lines ABC and EDC are straight, and EA is parallel to DB. We know EC = 11.5 cm, DC = 4.6 cm, DB = 3.8 cm, and AC = 10.5 cm. We need to find (a) leng

1. **State the problem:** Find the total area of the irregular quadrilateral composed of a rectangle and a right triangle. 2. **Identify shapes and dimensions:**

1. The problem is to find the missing central angle of a puzzle piece that completes a circle and a semicircle from the given pieces with angles 70°, 32°, 136°, 58°, and a quarter

1. **State the problem:** We have a quarter of a solid sphere (Shape S) with volume $576\pi$ cm³. We need to find the surface area of this quarter sphere, rounded to 3 significant

1. **Problem Statement:** Find the orthocenter of the triangle with vertices \(A(2,1,0)\), \(B(5,4,3)\), and \(C(1,6,2)\). 2. **Definition:** The orthocenter is the point where the

1. **Problem Statement:** Find the equation of a plane given certain conditions (e.g., a point and a normal vector). 2. **Formula:** The general equation of a plane is given by $$A