📐 geometry

1. **Problem Statement:** Find the volume and total surface area (TSA) of a sphere with a radius of 4 cm. 2. **Formulas:**

1. **Problem Statement:** Prove that the points (4, 3), (6, 4), (5, 6), and (3, 5) are the vertices of a square. 2. **Formula and Rules:** To prove a quadrilateral is a square, we

1. **Problem statement:** Given triangle $AB\Gamma$ with $AB < A\Gamma$, $M$ is the midpoint of $B\Gamma$. A perpendicular from $M$ to $B\Gamma$ meets the extension of the bisector

1. **State the problem:** We need to find the total surface area of a frustum formed by removing a smaller cone from a similar larger cone. The large cone has slant height $l=42$ m

1. **Problem statement:** We are given a square with a diagonal length of 8 cm and need to find the side length of the square in simplest form. 2. **Formula used:** For a square wi

1. The problem asks for the coordinates of point R plotted on a Cartesian plane. 2. Coordinates are given as an ordered pair $(x, y)$ where $x$ is the horizontal position and $y$ i

1. **Problem Statement:** We have a right triangle with legs of lengths 3.6 meters and 2.7 meters. We want to find the length of the hypotenuse (the tube leaning against the wall).

1. **State the problem:** We have a triangle with three angles: 30°, 2x, and x. We need to write an equation for the sum of these angles and find the value of $x$. 2. **Formula use

1. **State the problem:** Find the volume of a sphere with radius $r=9$ units. 2. **Formula:** The volume $V$ of a sphere is given by the formula:

1. Let's state the problem: We want to understand the properties of a coin modeled as a cylinder. 2. A cylinder is a 3D shape with two parallel circular bases connected by a curved

1. The problem is to find the midpoint of a line segment given two points and then find the point above this midpoint. 2. The midpoint formula for two points $A(x_1,y_1)$ and $B(x_

1. The problem asks to find the height of the photo taken, but no numerical data or measurements are provided in the description. 2. To find the height of a photo or an object in a

1. **Problem statement:** (a) Find the central angle $x$ in a regular octagon.

1. **State the problem:** (a) Find the number of sides of a regular polygon if each exterior angle is 15°.

1. **Problem 1: Find the fifth angle of a pentagon given four angles: 97°, 114°, 127°, and 84°.** 2. The sum of interior angles of a polygon with $n$ sides is given by the formula:

1. **Problem statement:** We have a regular pentagon (5-sided polygon) with center O. We need to find the values of angles $x$ and $y$ shown near one vertex. 2. **Formula for inter

1. **Problem statement:** The radius $r$ of a closed cylinder is equal to its height $h$. We need to find the percentage increase in the total surface area when the height increase

1. **State the problem:** We need to find the approximate increase in the volume of a cylinder when its radius increases from 4 cm to 4.20 cm, given the height is 10 cm. 2. **Formu

1. **Problem statement:** We need to find the dimensions (radius $r$ and height $h$) of a closed right circular cylinder that holds a volume of 1 litre (1000 cubic centimeters) and

1. **Problem Statement:** We have a vertical line segment between points $(6,6)$ and $(6,9)$ on the Cartesian plane. We want to find the image of this segment after a dilation cent

1. 問題陳述： 如圖，梯形 ABCD 外切於圓 O，且 AB = CD，AD = 3，BC = 5，求灰色部分的面積。