📐 geometry

1. **State the problem:** We have a right circular cone where the slant length $l$ is equal to the diameter of its base. We need to find the volume of this cone. 2. **Known formula

1. **State the problem:** We have an isosceles right triangle with angles 45°, 45°, and 90°. We want to explore the relationship between the lengths of the legs and the hypotenuse.

1. **State the problem:** We have a 45°-45°-90° triangle with angles $m\angle ABC = 45^\circ$ and $m\angle ACB = 45^\circ$. We want to find the lengths of the legs $AB$, $AC$, and

1. Задача: Ребро куба зменшили в 3 рази. У скільки разів зменшився його об’єм? Формула об’єму куба: $$V = a^3$$, де $a$ — довжина ребра.

1. **Problem Statement:** Given parallelogram PURE with diagonals PR and EU intersecting at O. We know diagonals of a parallelogram bisect each other, so $O$ is the midpoint of bot

1. Задача: Знайти об’єм призми, якщо площа основи $S=12$ см², а висота $h=5$ см. Формула об’єму призми: $$V = S \times h$$

1. **Problem statement:** We have an isosceles triangle ABC with AB = AC = 13 cm and BC = 10 cm lying in a horizontal plane.

1. Задача: Знайти об’єм призми, якщо площа основи $S=12$ см², а висота $h=5$ см. Формула об’єму призми: $$V = S \times h$$

1. **Problem Statement:** We are given three triangles: \(\triangle ABC\), \(\triangle DEF\), and \(\triangle IGH\) with side lengths:

1. The problem involves comparing three triangles: \(\triangle ABC\), \(\triangle DEF\), and \(\triangle GHI\) with given side lengths. 2. We want to determine which triangle is si

1. The problem asks: "Where is the shaded area?" This usually refers to identifying the region on a graph or figure that is highlighted or filled. 2. To determine the shaded area,

1. The problem involves solving a triangle using the SSS (Side-Side-Side) theorem and the TABT (The Angle Bisector Theorem). 2. The SSS theorem states that if three sides of one tr

1. **Problem:** Determine if two triangular signs with sides 20 cm, 30 cm, 40 cm and 40 cm, 60 cm, 80 cm are similar. 2. **Formula and rule:** Triangles are similar if their corres

1. **State the problem:** A projector creates a 30cm wide image when it is 50cm from the screen. We want to find the width of the image when the projector is moved back 20cm, makin

1. **State the problem:** We have a sector-shaped water fountain with radius 10.5 m and central angle $\frac{\pi}{2}$ radians. We need to find:

1. **State the problem:** We need to find the total surface area of a composite solid made by joining a cylinder and a cone. 2. **Given data:**

1. **Problem Statement:** Find the surface area of a cylinder and a cone. 2. **Formulas:**

1. **State the problem:** We have a corridor with an actual length of 18 meters, and it is drawn to a scale of 1 : 150. We need to find the length of the corridor in the drawing. 2

1. **Stating the problem:** We have a right triangle with legs of lengths 12 cm and 44 cm, and the hypotenuse labeled as $5y$ cm. We need to find the value of $a$ (assuming $a$ is

1. **Stating the problem:** Given trapezium KLMN with angles \(\angle K = 120^\circ\) and \(\angle M = 110^\circ\), and \(NL = NM\). Point R lies on the extension of \(NL\) beyond

1. **Problem statement:** Given three vertices of a parallelogram $R(-2,-1)$, $S(2,1)$, and $T(0,-3)$, find the coordinates of the fourth vertex. 2. **Formula and rule:** In a para