📐 geometry

1. **State the problem:** We are given a rectangular prism container with a height of $11 \frac{1}{2}$ inches and a length of $7 \frac{1}{2}$ inches. We need to find the width of t

1. The problem involves a sequence of geometric transformations on shapes A, P, M, and others, including reflections, translations, and rotations. 2. For problem 7, we start with s

1. **Problem statement:** We need to find the volume of a wedge cut from a circular cylinder of radius 4 by two planes. One plane is perpendicular to the cylinder's axis, and the o

1. **Stating the problem:** You are asked to identify types of transformations (translation, reflection, rotation) applied to geometric figures on grids and describe combined trans

1. **State the problem:** We need to find the total area of a composite shape consisting of a rectangle and a right triangle. 2. **Identify the shapes and their dimensions:**

1. **State the problem:** Calculate the area of the given polygon which consists of a rectangle and a triangular section attached to its right side. 2. **Identify the dimensions:**

1. **Problem Statement:** We have a quadrilateral ABCD with sides AB = 9.8 cm, BC = 9.5 cm, CD = 5.2 cm, AD = 12.3 cm, and angle \(\angle DAB = 40.5^\circ\). We need to find:

1. **Problem Statement:** We have a quadrilateral ABCD with sides AB = 9.8 cm, BC = 9.5 cm, CD = 5.2 cm, diagonal AC = 12.3 cm, and angle \(\angle BAD = 40.5^\circ\). We need to fi

1. **Problem:** Given that □ROCK is a rhombus, and angles m∠KCR = 7x - 9 and m∠OCR = 5x + 5, find the value of $x$. 2. **Understanding the problem:** In a rhombus, the diagonals bi

1. **Problem statement:** We have a right triangle ABC with a right angle at B. Side AB is labeled as $x$, side BC is $4\sqrt{2}$, and point E lies on AC such that AE is $3\sqrt{2}

1. **Problem statement:** We need to find the size of angle $x$ in a triangle where the other two angles are $62^\circ$ and $135^\circ$. 2. **Angle facts used:**

1. **Problem statement:** We need to find the size of angle $x$ in a diagram where three angles are given: $63^\circ$, $x$, and $134^\circ$. The angles are arranged around a triang

1. **State the problem:** We need to find the size of angle $x$ in a triangle formed by two intersecting lines with given angles $63^\circ$ and $134^\circ$. 2. **Identify angle fac

1. **Problem Statement:** Find the sum of the interior angles of a regular polygon with 7 sides (a heptagon). 2. **Formula:** The sum of the interior angles $S$ of a polygon with $

1. The problem asks for the sum of the interior angles of a polygon with 7 sides (a heptagon). 2. The formula to find the sum of interior angles of any polygon with $n$ sides is:

1. **Problem 1:** Find the area of the region enclosed between two circles given by $$x^2 + y^2 = 1$$

1. The problem asks for the sum of the interior angles of a polygon with seven sides, called a heptagon. 2. The formula to find the sum of interior angles of any polygon with $n$ s

1. **Problem statement:** We have an irregular pentagon with internal angles 68°, 101°, 116°, and 21°, and a triangle sharing one side with the pentagon. We need to find the size o

1. **State the problem:** We have a triangle sharing a side with an irregular pentagon. We need to find the size of angle $g$ in the triangle. 2. **Identify known angles:** The pen

1. **Problem statement:** We need to find the size of angle $g$ in a figure where a triangle shares a side with an irregular pentagon. Given angles are 68°, 77°, 116°, 101°, and 21

1. **State the problem:** We need to find the size of angle $k$ in a non-convex quadrilateral with given angles $29^\circ$, $244^\circ$, and $35^\circ$.\n\n2. **Recall the formula: