📐 geometry

1. **State the problem:** Find the volume of a sphere with a diameter of 8 cm. 2. **Formula:** The volume $V$ of a sphere is given by the formula:

1. **State the problem:** We need to find the width of a rectangular pyramid given its volume, height, and length. 2. **Formula for the volume of a rectangular pyramid:**

1. **State the problem:** We need to find the width of a pyramid given its volume, height, and length. 2. **Given:**

1. **Problem statement:** We have a 400-meter Olympic track composed of two straight sections and two semicircular ends. The straight sections are each 84.39 m long, and the radius

1. **Stating the problem:** We are given a rectangle divided into areas A and B, and three triangular areas labeled 1, 2, and 3. We know that area 2 + area 3 equals half of the com

1. The problem is to understand the definition and properties of a prism. 2. A prism is a 3-dimensional solid characterized by having two identical and parallel faces called bases.

1. **Problem statement:** Given a rectangular prism with volume $V = 200$ cm³, width $w = 4$ cm, height $h = 10$ cm, find the length $l$. 2. **Formula used:** The volume of a recta

1. **Problem statement:** Calculate the volume (V), lateral surface area (LSA), and total surface area (TSA) of a cylinder with diameter $d=10$ cm and height $h=12$ cm.

1. The problem is to mark a point O and draw a line from O towards the north, then draw another line from O making a bearing of 040 degrees. 2. A bearing is measured clockwise from

1. **Problem Statement:** We need to state and prove the relationship between the internal angles of a spherical triangle and the area of that triangle.

1. **Problem statement:** Define a great circle on the sphere $S^2$ and determine the number of intersection points of two distinct great circles $C_1$ and $C_2$ on $S^2$. 2. **Def

1. **State the problem:** We are given a triangle ABC inscribed in a circle with center O. Given angles:

1. **State the problem:** We are given a rectangle with width 16 cm and height 4 cm, and a square whose side length is unknown. The area of the rectangle is equal to the area of th

1. **Problem statement:** Point M is the midpoint of line segment AB. We want to understand what this means and how to find coordinates of M if coordinates of A and B are known. 2.

1. Problem: Given parallelogram MASK, find KS if MA = 9. Step 1: Recall that in a parallelogram, opposite sides are equal.

1. **Problem statement:** Find the equation and length of the altitude from vertex A in triangle ABC with vertices A(2,3), B(4,-1), and C(1,2). 2. **Recall:** The altitude from a v

1. **Problem statement:** We have two circles with centers A and B, radii 5 cm and 8 cm respectively, touching at point C on line ACB. Point D lies on the smaller circle such that

1. Problem statement: Calculate the area of a right triangle given the lengths of its legs. 2. Formula: The area $A$ of a right triangle with legs $a$ and $b$ is given by

1. **Problem:** Find $x$ in triangle ACB where segment GH is parallel to CB. Given: $AG=10$, $GH=2x+6$, $CB=25$, $CH=55$.

1. **Problem Statement:** Examine the diagonals of parallelograms by drawing a rectangle ABCD and its diagonals, measuring the diagonals, and exploring their properties. 2. **Step

1. **Stating the problem:** We have a parallelogram with two given angle measures expressed in terms of $x$: one angle is $4x - 50^\circ$ and the adjacent angle is $x + 20^\circ$.