Surface area Notes

Surface Area Measures The Outside Of A 3D Object

1. When you work with 2D (flat) shapes, you calculate area, the amount of surface inside the boundary of the shape (measured in square units, such as cm$^2$).
2. For 3D solids, you often need surface area, which tells you how much material is needed to cover the outside of the object (also measured in square units).

Although this article focuses on surface area, it helps to remember the big picture:

1. Surface area is about covering the outside (paint, wrapping paper, metal sheet).
2. Volume is about filling the inside (water, air, sand).

Polyhedra Have Flat Faces, While Some Solids Have Curved Surfaces

A 3D solid may have flat faces, curved faces, or both.

Surface Area Is Usually Found By Splitting A Solid Into Familiar 2D Shapes

A powerful strategy is to identify the faces of a solid and match each face to a 2D shape you already know how to find the area of.

Pyramids Use Base Area Plus Triangular Face Areas

A pyramid has a polygon base, and the other faces are triangles meeting at a point called the apex.

Slant Height And Vertical Height Are Different Measurements

For many pyramids you will see two heights:

The slant height is used to find the area of each triangular face because each face is a triangle whose height is the slant height.

Surface Area Of A Regular-Based Pyramid

1. If the pyramid has a regular polygon base with $n$ equal sides, then all triangular faces are congruent.
2. If $A_{\text{base}}$ is the area of the base and $A_{\triangle}$ is the area of one triangular face, then: $$S = A_{\text{base}} + nA_{\triangle}}$$
3. To find $A_{\triangle}$ for each lateral face:
   1. triangle base = one side of the base polygon
   2. triangle height = slant height
4. So for a square-based pyramid with base side length $a$ and slant height $l$: $$A_{\triangle}=\tfrac{1}{2}al$$ $$\text{and} \quad S=a^2+4\left(\tfrac{1}{2}al\right)=a^2+2al$$

Finding Slant Height Using Pythagoras

1. Sometimes you are given:
   1. the base side length, and
   2. the vertical height $h$
2. You can often find the slant height $l$ using Pythagoras' theorem on a right triangle formed by:
   1. one leg = vertical height $h$
   2. other leg = distance from the center of the base to the midpoint of a base edge
   3. hypotenuse = slant height $l$
3. For a square base with side $a$, the distance from the center to the midpoint of an edge is $\tfrac{a}{2}$, so: $$l=\sqrt{h^2+\left(\tfrac{a}{2}\right)^2}$$

Cones Have A Circular Base And A Curved Surface

A cone has:

1. one circular base (radius $r$)
2. one curved surface meeting at the apex

Slant Height Of A Cone

1. The slant height (often written $s$) is the distance along the curved surface from the edge of the base circle to the apex.
2. In a right cone, $r$, vertical height $h$, and slant height $s$ form a right triangle: $$s=\sqrt{r^2+h^2}$$

Surface Area Of A Cone

1. Cone surface area has two parts:
   1. base circle area: $\pi r^2$
   2. curved surface area: $\pi rs$
2. So the total surface area is: $$S=\pi r^2+\pi rs$$

Spheres Have One Curved Surface And No Edges Or Vertices

A sphere is a perfectly round 3D solid where every point on the surface is the same distance from the center.

Surface Area Of A Sphere

1. If the sphere has radius $r$, its surface area is: $$S=4\pi r^2$$
2. This formula is not found by adding flat faces, because a sphere has no flat faces, so use it directly.

A Reliable Method For Surface Area Problems

Most errors in surface area come from missing a face, using the wrong height, or mixing up units. The following routine helps.

Step-By-Step Approach

1. Sketch the solid and label all given measurements.
2. Identify each surface (triangles, rectangles, circles, curved surface).
3. Find any missing lengths (often slant height) using geometry such as Pythagoras.
4. Calculate each area separately, keeping exact values (like $56\pi$) when helpful.
5. Add areas to get total surface area.
6. Check units: surface area should be in square units.