Question

question 9: standard g.gmd.1
what is cavalieris principle?
a. cavalieris principle states that two solids lying between parallel planes have equal heights and all cross sections at equal distances from their bases have equal areas, then the solids have equal volumes.
b. cavaliens principle states that the volume of a sphere is equal to the volume of a cylinder with the same height and radius.
c. cavalieris principle claims that if two solids have the same surface area, they also have the same volume, regardless of their shapes.
d. cavaliens principle indicates that any polygon can be divided into a finite number of triangles, and the sum of the areas of these triangles equals the area of the polygon.

question 10: standard g.gmd.1
if 2 shapes have the same base area and same height, we know they have the same _____.
a. height
b. base
c. volume
d. area

question 11: standard g.gmd.1
these 2 shapes have the same volume because?
two 3d shapes, one with a curved side and one rectangular prism
a. cavalieri’s principle
b. archimedes principle
c. pythagorean theorem
d. postulates of volume

Explanation:

Question 9

Cavalieri's Principle is about volumes of solids between parallel planes with equal cross - sectional areas at equal distances from bases. Option a matches this definition. Option b is incorrect (sphere and cylinder volume relation is different). Option c is wrong as surface area doesn't determine volume. Option d is about polygon area via triangles, not Cavalieri's Principle.

When two shapes have the same base area and same height, by Cavalieri's - like reasoning (or the formula for volume of many solids like prisms, cylinders which is \(V = Bh\) where \(B\) is base area and \(h\) is height), they have the same volume. Option a (height) and b (base) are given as same, so not the answer. Option d (area) is about 2 - D, but we are talking about 3 - D shapes' volume.

The two shapes (a non - rectangular prism and a rectangular prism) have the same volume because of Cavalieri's Principle, which states that if two solids between parallel planes have equal cross - sectional areas at equal distances from the base, they have equal volumes. Archimedes' principle is about buoyancy, Pythagorean theorem is about right - triangles, and there's no "Postulates of Volume" related to this.

Answer:

a. Cavalieri's principle states that two solids lying between parallel planes have equal heights and all cross sections at equal distances from their bases have equal areas, then the solids have equal volumes.

Question 10