📐 geometry

1. **Stating the problem:** We have a circle with points W, X, Y, Z on its circumference forming four triangles: WXY, WYZ, XYZ, and WZY. Angles at these points are given or labeled

1. **Problem Statement:** Given a circle with center O and points A, B, C on the circumference, angle AOB at the center is $t$. Angles on the circumference are $\angle ABC = 38^\ci

1. **Problem statement:** Given a circle with diameter AB and a point C on the circumference, angle ABC is 38°. We need to find the measure of angle ACB. 2. **Key property:** In a

1. The problem asks to find the volume of a cube with side length $\frac{4}{9}$ miles. 2. The formula for the volume $V$ of a cube with side length $s$ is:

1. **State the problem:** We need to find the volume of a rectangular prism with side lengths $\frac{3}{5}$ m, $\frac{5}{6}$ m, and $\frac{1}{4}$ m. 2. **Formula for volume of a re

1. **State the problem:** We have a rectangular solid spacer with dimensions length $L=8$ ft, height $H=14$ ft, and width $W=32$ ft.

1. **Problem Statement:** Calculate the volume of titanium needed for one triangular prism bead with a hole, then find the total titanium volume for 500 beads, the weight in pounds

1. **State the problem:** We need to find the volume of sand required to fill a square sandbox with inside dimensions 6 ft by 6 ft and wall height 9 inches, but the sand height is

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: