📐 geometry

1. **State the problem:** Find the lateral area and total surface area of a sphere with radius $r=11$ ft. 2. **Recall formulas:**

1. **Problem Statement:** Find the volume of a sphere with radius $r = 10.5$ miles. 2. **Formula:** The volume $V$ of a sphere is given by the formula:

1. **State the problem:** We need to find the measures of angles \(\angle EBC\), \(\angle ABE\), and \(\angle ABF\) given the rays and angles around point B. 2. **Given information

1. **Problem statement:** Given an inscribed quadrilateral with angles labeled $5x$, $104^\circ$, $2x$, and $110^\circ$, find the value(s) of $x$. 2. **Formula and rule:** In a cyc

1. **Stating the problem:** We have a circle tangent to two rays forming a 70° angle at point Q. The upper tangent touches the circle at R, and the lower tangent touches the circle

1. **State the problem:** We have a circle tangent to two rays from point Q, with tangent points R and P on the circle. The angle at Q is 70° and an expression 83x + 1 is given, li

1. **State the problem:** Find the area of a trapezoid with top base $5$ km, bottom base $9$ km, and height $3$ km. 2. **Formula for the area of a trapezoid:**

1. **Problem statement:** Given a triangle with side lengths $a=45$ mm, $b=60$ mm, and $c=102$ mm, Claudia calculated the angles $\alpha$, $\beta$, and $\gamma$ using the law of co

1. **State the problem:** Find the area of a trapezoid with top base $5$ mi, bottom base $3$ mi, and height $6$ mi. 2. **Formula for the area of a trapezoid:**

1. **Problem Statement:** Find the area of quadrilateral ABCD where AB = 11, BC = 4\sqrt{3}, AD = 12, and there are right angles at B and D. 2. **Understanding the shape:** ABCD ha

1. **State the problem:** We need to find the value of $x$ in a circle where a chord segment $x$ is related to two other segments: one radius segment of length 4.9 and a chord segm

1. **State the problem:** We have points X(-3,3), Y(3,1), Z(4,4), and W(2,-2). We need to find: - Distance from X to line WZ

1. **State the problem:** We need to find the value of $x$ in a circle with a chord and segments labeled 8, 3, and $x$, where two segments labeled 3 are congruent and a perpendicul

1. **Problem statement:** Calculate the volume and surface area of the following solids with given dimensions: - Cube with edge $a=3.5$ cm

1. **Stating the problem:** We are given a geometric figure with two parallel horizontal lines and two pairs of parallel slanted lines forming a central triangle-like shape. The an

1. **State the problem:** We need to find the values of angles $x$ and $y$ given the diagram with some known angles. 2. **Identify known angles and relationships:** From the proble

1. **State the problem:** Given a triangle with angles 3 and 4 measuring 125° and 23° respectively, find the measure of angle 2. 2. **Recall the triangle angle sum rule:** The sum

1. **State the problem:** We have a triangle with interior angles 64° and 27°, and an exterior angle at the left vertex labeled as $97 + x$. We need to find the value of $x$. 2. **

1. **State the problem:** We have an isosceles triangle \(\triangle GHI\) where sides \(GH \cong IG\) and the measure of angle \(\angle I = 51^\circ\). We need to find the measure

1. **State the problem:** We are given a circle with marked arcs of 53°, 60°, and 80°, and two variables $x$ and $y$ representing angles inside the circle. We need to find the valu

1. **State the problem:** We have an obtuse triangle \(\triangle ABC\) with \(CB = 20\), \(\angle A = 30^\circ\), and \(\angle B = 45^\circ\). We want to find the length of side \(