In the Geometry section of GRE quantitative reasoning syllabus, three-dimensional figures hold an important place. And every GRE aspirant should be certain that they would encounter and solve at least one or two questions based on this topic in the exam. Three-dimensional figures even though seem intimidating needs just a bit of effort to master it. So, don’t neglect it, in place master it to score well on D- day.

So, what is a three- dimensional figure?

A three-dimensional figure always has length, breadth, and height whereas; a two-dimensional will only have length and breadth. Common three dimensional figures to appear in the syllabus of GRE are cube, cuboid, prism, cone etc.

Let us understand these shapes one at a time along with their volume and surface area formulas.

Cube: It is a square box whose all sides are equal and spread at an angle of 90°.

Cube

Volume  of  Cube=a×a×a=a3Volume \; of \; Cube = a \times a \times a = a^{3}

Surface Area of Cube =2×length×height+2×base×length+2×base×height=6a2= 2 \times length \times height + 2 \times base \times length + 2 \times base \times height = 6a^{2}

Where, a = base = length = height and a = side of the cube

Cuboid: Cuboid is a rectangular solid.

Cuboid

Volume  of  Cuboid=Length×Breadth×HeightVolume \; of \; Cuboid = Length \times Breadth \times Height

Surface Area of Cuboid =

Surface Area of Cuboid =2×length×height+2×base×length+2×base×height=2(lh+bl+bh)= 2 \times length \times height + 2 \times base \times length + 2 \times base \times height = 2(lh + bl + bh)

Sphere: Balls, football, cricket ball etc. are example of spheres.

Sphere

Volume  of  Sphere=V=43πr2Volume \; of \; Sphere = V = \frac{4}{3} \pi r^{2} Surface  Area  of  Sphere=A=4πr2Surface \; Area \; of \; Sphere = A = 4 \pi r^{2}

Pyramid: You can even calculate the volume and surface area of the great Pyramids of Egypt. Shocked? Don’t be! The Great Pyramids of Egypt are example of the pyramids that we study in three- dimensional geometry.

Pyramid

Volume  of  Pyramid=13×Area  of  base×height=13×A×hVolume \; of \; Pyramid = \frac{1}{3} \times Area \; of \; base \times height = \frac{1}{3} \times A \times h Surface  Area  of  Pyramid=l×b+l(b2)2+h2+b(l2)2+h2Surface \; Area \; of \; Pyramid = l \times b + l \sqrt{\left ( \frac{b}{2} \right )^{2} + h^{2}} + b \sqrt{\left ( \frac{l}{2} \right )^{2} + h^{2}}

Where, l = length, b = breadth, h = height

Square Pyramid: Square pyramid is a type of pyramid that has square base. It is a polyhedron in nature.

Square-Pyramid

Volume  of  a  Square  Pyramid=(side  of  square)2×height3Volume \; of \; a \; Square \; Pyramid = (side \; of \; square)^{2} \times \frac{height}{3} Surface  Area  of  a  Square  Pyramid=a2+2aa24+h2Surface \; Area \; of \; a \; Square \; Pyramid = a^{2} + 2a \sqrt \frac{a^{2}}{4} + h^{2}

Cone: Ever enjoyed the delicious and chilling ice-cream cone? Who hasn’t! Ice- creams are favorites of all but have you ever paid attention to the shape of the cone? It is circular at one side and pointed at the other.

Cone

Volume  of  a  Cone=V=πr2h3Volume \; of \; a \; Cone = V = \pi r^{2} \frac{h}{3}Surface  Area  of  a  Cone=A=πr(r+h2+r2)Surface \; Area \; of \; a \; Cone = A = \pi r (r + \sqrt{h^{2} + r^{2}})

Surface  Area  of  a  Cone=A=πr(r+h2+r2)Surface \; Area \; of \; a \; Cone = A = \pi r (r + \sqrt{h^{2} + r^{2}})

Cylinder: Cylinder is a solid geometrical figure with straight parallel sides and a circular or oval cross section.

Cylinder

Volume  of  a  Cylinder=V=πr2hVolume \; of \; a \; Cylinder = V = \pi r^{2}h Surface  Area  of  a  Cylinder=A=2πrh+2πr2Surface \; Area \; of \; a \; Cylinder = A = 2\pi rh + 2 \pi r^{2}

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