Surface Area Formulas and Volume Formulas of 3D Shapes


Surface area formulas and volume formulas appear time and again in calculations and homework problems. Pressure is a force per area and density is mass per volume. These are just two simple types of calculations that involve these formulas. This is a short list of common geometric shapes and their surface area formulas and volume formulas.

Sphere Surface Area Formula and Sphere Volume Formula

Sphere

A sphere is a solid figure where every point on the surface is equidistant from the center of the sphere. This distance is the radius, r, of the sphere.

Surface area = 4πr2

Volume = 43πr3


Prism Surface Area Formula and Prism Volume Formula

Prism

A prism is a geometric shape consisting of a stack of identical base shapes stacked on top of each other to a depth d. This prism is a prism formed by a stack of triangles.

Surface Area of a Prism = 2 × (Area of the base shape) + (Perimeter of base shape) × (d)

Volume of a Prism = (Area of base shape) × d

To find the area and perimeter of the base shape, check out Area Formulas and Perimeter Formulas.


Box Surface Area Formula and Box Volume Formula

Box

A box can be thought of a stack of rectangles L long and W wide piled on top of each other to a depth of D.

Surface Area of a Box = Sum of the areas of each face of the box, or

Surface Area of a Box = 2(L × W) + 2(L × D) + 2(W × D)

Volume of a Box = L × W × D


Cube Surface Area Formula and Cube Volume Formula

Cube with dimensions shown

A cube is a special case box where all the sides are the same length.

Surface Area of a Cube = 6a2

Volume of a Cube = a3


Cylinder Surface Area Formula and Cylinder Volume Formula

Cylinder

A cylinder is a prism where the base shape is a circle.

Surface Area of a Cylinder = 2πr2 + 2πrh

Volume of a Cylinder = πr2h


Square Pyramid Surface Area Formula and Pyramid Volume Formula

Pyramid Solid

A pyramid is a solid shape consisting of a polygon base and triangular faces meeting at a common point above the base. A square pyramid is a pyramid where the base polygon is a square.

In the picture above, side a is the same length as side b. All of the face triangles are isosceles triangles meeting at a point h above the base.

volume of a square based pyramid

For pyramids with identical face triangles (a = b = c)

surface area of a equilateral pyramid
volume of a equilateral pyramid

Surface Area Formula of a Cone and Volume Formula of a Cone

Cone

A cone is a pyramid with a circular base with radius r and height h. The side length s can be found using the Pythagorean Theorem.

s2 = r2 + h2
or
s = √( r2 + h2 )

Surface Area of a Cone = πr2 + πrs

Volume of a Cone = 13( πr2h )

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