📐 geometry

1. **Problem statement:** We have a frustum formed by removing a smaller cone from a larger cone. The smaller cone has height $10$ m and top radius $6$ m. The original large cone h

1. **State the problem:** We have a large cone of height 39 mm and base radius 6 mm. It is split horizontally into a smaller cone on top (height 13 mm, radius 2 mm) and a frustum b

1. **Problem statement:** Given lines ABC and EDC are straight, and EA is parallel to DB. We know EC = 11.5 cm, DC = 4.6 cm, DB = 3.8 cm, and AC = 10.5 cm. We need to find (a) leng

1. **State the problem:** Find the total area of the irregular quadrilateral composed of a rectangle and a right triangle. 2. **Identify shapes and dimensions:**

1. The problem is to find the missing central angle of a puzzle piece that completes a circle and a semicircle from the given pieces with angles 70°, 32°, 136°, 58°, and a quarter

1. **State the problem:** We have a quarter of a solid sphere (Shape S) with volume $576\pi$ cm³. We need to find the surface area of this quarter sphere, rounded to 3 significant

1. **Problem Statement:** Find the orthocenter of the triangle with vertices \(A(2,1,0)\), \(B(5,4,3)\), and \(C(1,6,2)\). 2. **Definition:** The orthocenter is the point where the

1. **Problem Statement:** Find the equation of a plane given certain conditions (e.g., a point and a normal vector). 2. **Formula:** The general equation of a plane is given by $$A

1. **State the problem:** We need to find the total surface area of a solid hemisphere with radius $8$ cm. 2. **Recall the formulas:**

1. The problem asks to find the circumference of a circle with a diameter of 158 cm, rounded to 1 decimal place. 2. The formula for the circumference $C$ of a circle is:

1. The problem asks for the circumference of a circle with a diameter of 13 cm. 2. The formula for the circumference $C$ of a circle is given by:

1. **Problem statement:** Find the following:

1. **Problem statement:** We need to find the length of the diagonal $DG$ inside the cuboid. 2. **Given dimensions:**

1. The problem asks for the degree of rotation that maps a regular hexagon onto itself when rotated about its center. 2. A regular hexagon has 6 equal sides and 6 equal angles.

1. **State the problem:** Find the perimeter of rectangle PQRS with vertices P(4,1), Q(4,7), R(8,7), and S(8,1). 2. **Recall the formula for perimeter of a rectangle:**

1. The problem asks to identify all triangles similar to triangle T from the given shapes A, B, C, D, E, and F. 2. Two triangles are similar if their corresponding angles are equal

1. **Problem Statement:** Identify which two shapes among A, B, C, D, E, and F are similar based on their geometric properties. 2. **Understanding Similarity:** Two shapes are simi

1. **State the problem:** We need to find which two triangles among A, B, C, D, and E are congruent based on their side lengths. 2. **Recall the congruence rule for right triangles

1. The problem asks to identify which two shapes among A, B, C, D, E, F, and G are congruent. 2. Congruent shapes have the same size and shape, meaning their corresponding sides an

1. **State the problem:** We have a square window with area 81 cm², so each side is $\sqrt{81} = 9$ cm.

1. **Problem statement:** Given a pyramid $S.ABCD$ with base $ABCD$ a trapezoid where $AB \parallel CD$. Point $E$ lies on segment $SA$. Plane $(P)$ passes through $E$ and is paral