Problem
A cylinder seen from the front is its vertical cross-section
A cylinder is shown from above and from the front. We need its surface area.
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — The two views hand over the only two numbers a cylinder needs: the top view gives the base and the front view gives the height. After that it is the surface-area rule -- two bases and a rolled-up rectangle.
1STEP 1
Read the radius and the height
The top view is the base circle, so its diameter is the base's; the front view's height is the cylinder's height.
10 ÷ 2 = 5
Radius 5 cm, height 10 cm.
7.G.A.3Visualize Spatial RelationshipsThe front view's width is the base's diameter and its height is the cylinder's height.
Why?
Looking straight at the cylinder from the front, the widest thing in the way is the base circle at its full width across the centre.
🧱A circle's radius is constantEvery point on a circle sits exactly one radius from the center, so all radii of a circle are equal.
Why?
The cylinder rises the same height everywhere, so the outline from the front is a rectangle that tall.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
2STEP 2
Measure the two bases
Each base is a circle of that radius, and there are two of them.
5 × 5 × 3.14 × 2 = 157
The bases come to 157 cm².
7.G.B.4Draw A Diagram3STEP 3
Unroll the curved side
Cut the side open and it flattens into a rectangle as wide as the base's circumference and as tall as the cylinder.
5 × 2 × 3.14 = 31.4, 31.4 × 10 = 314
The side is 314 cm².
7.G.B.4Draw A Diagram4STEP 4
Add them
The surface is the two bases plus the side.
157 + 314 = 471
The surface area is 471 cm².
7.G.B.6Identify SubproblemsAnswer
471 cm²
Check the shape: the front view is square because the height equals the diameter, and 10 = 5 × 2, so the two views agree.
Takeaway
A cylinder needs only two numbers, and its views hand both of them over.
- Read the radius and the height
- Measure the two bases
- Unroll the curved side
- Add them